C H A P T E R
3
The Ups and Downs of Classical
and Quantum Formulations of
Magnetic Resonance
Lars G. Hanson
Danish Research Centre for Magnetic Resonance, Copenhagen University Hospital, Hvidovre,
Denmark and DTU Elektro, Technical University of Denmark, Copenhagen, Denmark
O U T L I N E
3.1. Introduction 142
3.2. Quantum Mechanics in General 142
3.3. Misconceptions in NMR
Introductions 144
3.4. Where Did It Go Wrong? 148
3.5. A Limited Introduction to Classical
and Quantum Mechanics 149
3.6. Indeterminism vs. Uncertainty and
the Role of Measurement 151
3.7. The Role of Eigenstates in
Single-Particle Measurements 153
3.8. Entanglement 155
3.9. Superpositions 156
3.10. The Missing Role of Eigenstates
in Ensemble Measurements 157
3.11. The Role of Eigenstates in
Mathematical Descriptions 158
3.12. Visualization of Spin
Distributions 160
3.13. Thermal Equilibrium 162
3.14. Classical Eigenstates, Resonance,
and Couplings 164
3.15. The Eigenmode Structure for
Nuclear Excitation 167
3.16. J-Coupling 169
3.17. The Aftermath 170
Acknowledgments 171
References 171
141Anthropic Awareness # 2015 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/B978-0-12-419963-7.00003-1
3.1 INTRODUCTION
This chapter is triggered by misunderstandings that often appear when quantum mechanical
(QM) descriptions of nuclear magnetic resonance (NMR) are interpreted. The errors are
abundant in NMR introductions, including those in the literature on magnetic resonance imaging
(MRI), and they provide examples of the Mental Traps described in Chapter 1. It will be
argued that several of those Traps are responsible, and that even excellent and QM-savvy scientists
are affected, which is apparent from much of the literature on NMR. The mathematical
descriptions are typically accurate, but the accompanying explanations are often not. Misunderstandings
make explanations nonintuitive and give rise to false expectations. It is here
argued that many differences between classical and QM are a matter of formalism and that
the two approaches are more similar than they initially appear. The connection between
descriptions based on quantum and classical mechanics, and also important concepts such
as eigenstates, superpositions, interference and entanglement, are discussed. The different
roles of measurement for individual nuclei and ensembles are also covered. The dynamics
involved in basic NMR are shown to be similar to those of coupled classical oscillators
(e.g., pendulums), which gives insight into the resonance phenomenon itself as well as spectral
features resulting from intramolecular, scalar J-coupling of atomic nuclei. It is discussed
how classical and quantum mechanics give rise to similar expectations for the basic NMR
phenomenon and why a classical intuitive understanding is central: classically described
NMR provides an excellent basis not only for understanding phenomena like excitation,
echoes, and relaxation but also for digging deeper and for understanding QM formulations.
These aspects are of equal relevance for NMR in the context of chemical analysis and MRI.
They are also relevant for electron spin resonance, which is not discussed explicitly in this
chapter, although much of the discussion pertains to this also.
Neither the classical nor the quantum descriptions of NMR are covered in detail here, as
that is done elsewhere, for example, in Chapter 2 and in Levitt’s Spin Dynamics: Basics of
Nuclear Magnetic Resonance1
for the classical and quantum cases, respectively. Focus is instead
shifted to the connection between the two, and to the Mental Traps that lie behind common
misunderstandings in NMR introductions. This chapter, although self-contained, is largely a
continuation of a paper2
from 2008 entitled “Is quantum mechanics necessary for understanding
magnetic resonance?” that addressed the equivalence of classical and quantum mechanics
for describing the basic NMR phenomenon (and only that). Both that paper and the current
chapter have sections aimed at a broad audience and sections with mathematical details
needed to substantiate claims that may otherwise be considered controversial. The texts
are largely complementary.
3.2 QUANTUM MECHANICS IN GENERAL
Some simplified characteristics of QM are widely known and are more or less implicitly
brought forward in many NMR introductions:
1. Microscopic systems such as atoms can only exist in discrete states with specific energies.
142 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
2. Transitions between these discrete states happen in sudden “quantum jumps” and involve
exchange of energy (emission or absorption). An atom in an excited state may, for example,
spontaneously jump back to the ground state, in which case a photon with the difference
energy is emitted.
3. The timing of such jumps is truly unpredictable.
These statements represent “old” QM as first proposed by Bohr3
and they contradict both
classical mechanics and the mature, general versions of QM that followed soon after the initial
formulations. The claims above are often used to explain phenomena like optical emission
spectra such as the discrete frequencies present in light from a sodium vapor lamp. Quantitative
descriptions of optical spectra were indeed among the first victories of QM, as classical
mechanics failed miserably when explanations for these experimental observations were
sought. QM has since been developed into a general and quantitative theory (modern
QM), applicable in theory to all observed phenomena. The general claims above need to
be interpreted with care, since they are somewhat misleading simplifications of the general
quantum theory.4
When the complexity of a physical problem has allowed for modern QM’s full invocation,
and when it has been applied properly, we were never let down. The predictions match experiments
to the precision of the latter, and QM has proved its worth in countless situations. It
is crucial for understanding atomic structure and has resulted in developments such as computers
and lasers. The relative agreement between QM predictions and experiment has in
certain situations been experimentally verified to be better than one part in a billion (10À9
),
which shows that QM is amazingly accurate, when experimental conditions allow for sensitive
testing.5
Though QM is typically called a theory, it has gained status of a set of physical laws, in the
same sense as, for example, “Newton’s laws” forming the basis for classical mechanics. This is
appropriate since there have been no experimental data revealing fundamental limits of QM.
In contrast, the limitations of classical mechanics are well established. Both have distinct
properties that make their use advantageous in different contexts, typically QM for atomic
scale systems and classical mechanics for macroscopic systems. Engaging in QM when building
skyscrapers or when doing normal MRI makes little sense, for example, since classical and
quantum descriptions are consistent for both.
Among the highlights of the theoretical QM framework are the Schr€odinger and Liouvillevon
Neumann equations that describe how systems evolve over time, that is, they provide
“equations of motion.” When QM is combined with the theory of relativity that also has scientific
status as a confirmed theory, spin appears as a natural but highly surprising characteristic
of elementary particles. It is mind-boggling that all hydrogen nuclei (protons), for
example, are equally magnetic and thus appear to rotate at exactly the same frequency. There
are subtle properties of spin that makes it even more exotic, but that is of limited relevance for
basic NMR. For now, we will just acknowledge that hydrogen nuclei largely behave as
spinning, magnetic particles.
The rest of this chapter will focus on spin-½ nuclei such as hydrogen. Those are the simplest
and the most important nuclei in the context of NMR. They constitute model systems suited for
establishing the basic understanding, but there is more beyond. When not explicitly written, the
scope is also limited to “normal” NMR with macroscopic samples. When NMR is done with
1433.2 QUANTUM MECHANICS IN GENERAL
II. EXAMPLES FROM NMR THEORY
isolated nuclei in atomic beams, quantum aspects become more important. Such experiments
are not done in the context of chemical analysis or structure determination, but typically only in
fundamental science. In this context, “macroscopic” does not need to be more than a thousand
molecules, e.g., which is much less than a cell. Keeping the very exotic NMR situations out of
the discussion is warranted since the focus here is on introductions to the basic NMR phenomenon,
which is an important beginning, even for those who want to dig deeper. However, in
order to discuss the connection between classical and quantum mechanics, measurements
on isolated nuclei are also covered in this chapter.
3.3 MISCONCEPTIONS IN NMR INTRODUCTIONS
The paragraphs below provide a typical but problematic description of basic NMR in terms
of QM. It should be printed in blinking red for warning, but we will have to go with reduced
size text. Normal typesetting is used for the paragraphs that are largely free of problematic
statements. As a practical exercise in critical thinking, you should consider carefully which
sentences below should raise concerns because they are illogical or give limited predictive
power. Since QM has nonintuitive elements, a certain amount of oddity in the explanation
may be warranted.
A partially flawed introduction to NMR inspired by “old” QM:
Due to proton spin, each hydrogen nucleus is weakly magnetic. In the absence of magnetic field, the nuclei
are oriented randomly and the net magnetization of a macroscopic sample is zero. When the nuclei are exposed
to a magnetic field, QM tells us that each spin will align either nearly parallel or antiparallel with the field as
shown in Fig. 3.1. These orientations correspond to the up/down “eigenstates,” which are the only possible
states of the nuclei. The parallel orientation has the smallest energy, and is therefore the preferred state. A net
magnetization of the sample is formed by the small surplus of nuclei in the parallel state (the ground state).
Mx
My
Mz
|Sz = ½
|Sz = −½
Energy
¯hγB0
FIGURE 3.1 These graphs illustrate the energy eigenstates. The left side is often erroneously used to illustrate
thermal equilibrium also: the low-energy “up” state Sz ¼ 1
⁄2j i (upper cone) with spin oriented mostly parallel to
the applied B0 field is slightly more populated than the “down” state Sz ¼ À1
⁄2j i (lower cone). This is a correct statement
but the graphical representation is inappropriate. The corresponding energy levels are shown to the right. Radio
waves with a photon energy matching the energy difference between the up and the down states can move population
into the excited state from where it will relax back, sometimes while emitting radio waves (the NMR signal).
There is truth in this, but mentioning of “radio waves” and “photons” is arguably bad practice in the context of NMR.6
The graphs are not bad as illustrations of some of the up/down eigenstate properties, but the eigenstates are not central
for understanding the basic NMR phenomenon.
144 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
QM indeterminism/uncertainty is reflected in Fig. 3.1 that shows the “up” and “down”
eigenstates as spins precessing on one of two corresponding cones oriented parallel and antiparallel
to the B0 field directed along the z axis. These states are often labeled a and b as done
in Chapter 2. Several different choices of quantization axis will be made in this chapter, and
the eigenstates will therefore be denoted Sz ¼ 1
⁄2j i and Sz ¼ À1
⁄2j i in agreement with another
popular convention discussed in detail later in the chapter. The figure illustrates reasonably
well that the eigenstates cannot be perceived as purely longitudinal magnetization. It is a verified
peculiarity of QM that all three components of the spin cannot have specific values simultaneously.
Spin is classically represented as an angular momentum vector S ¼ Sx, Sy, Sz
À Á
along the spin axis. Sz is well-defined in the up/down states with values Æ1
⁄2, respectively, and
the magnitude of S somewhat larger,
ffiffiffi
3
p
=2 (angular momentum is given in units of Planck’s
reduced constant ℏ throughout this chapter). The uncertainty of the transversal magnetization
resulting from Sz being well defined is apparent in Fig. 3.1.
The problematic description of NMR continues:
In equilibrium, the orientation of nuclei is as illustrated in Fig. 3.1 with most nuclei in the ground state (upper
cone). In order to measure the magnetization, the nuclei need to be excited, which can be accomplished with a
burst of radio waves moving population to the higher-energy state (lower cone). QM tells us that the photon
energy associated with these radio waves need to be matched to the energy difference between the states. This
is given by the Zeeman splitting (relative difference of energy levels proportional to the magnetic field). The radio
wave frequency is therefore given by the gyromagnetic ratio times the magnetic field (the Larmor frequency).
When the nuclei fall back to the ground state, they re-emit photons that are detected by the receiver coil. These
quantum jumps are responsible for the NMR signal that is detected and analyzed.
You may consider this frequently heard explanation of NMR to be understandable, or even
true, but it has serious problems. There are many elements of truth in it, but it contains misunderstood
QM and leaves the reader with little predictive power. Considering that this is a
common explanation of NMR, you should be skeptical of the claim of it being wrong (but also
of it being right; cf. Trap #6). Here are a few questions for you to think about when evaluating
the validity and usefulness of the provided explanation:
• Does QM really tell us that nuclear spin can only be parallel or antiparallel to an applied
field? Does it apply to thermal equilibrium or more generally? How strong does the field
need to be for this to apply? Is the earth magnetic field strong enough to give this very
drastic effect, for example? Do the nuclei immediately jump into the up/down states
when the field is applied, and if so, do they emit radio waves while doing so? What would
the frequency of these be? If not, which states do they occupy in the brief period right
before transition to the up/down states?
• If the preferred orientation is parallel to the field, why are nearly half the nuclei pointing
opposite the field in thermal equilibrium, that is, the classically least favorable of all
orientations? QM may tell us that they do, but it is surprising, considering that we
never see compasses pointing opposite the magnetic field.*
*In fact, you may think that you have seen exactly this. If a compass is brought rapidly into a strong magnetic
field or enters in a way that prevents free needle movement all along, the needle may end up pointing opposite
the expected direction since it is made of material that can be remagnetized. The magnetic moment of the
needle may get inverted by a strong external field. This does not apply to nuclei since they are permanently
magnetic. Do not fall into Mental Trap #25 or #39!
1453.3 MISCONCEPTIONS IN NMR INTRODUCTIONS
II. EXAMPLES FROM NMR THEORY
• What happens if the radio waves are not exactly on resonance? Nothing apparently,
according to the above explanation, but that is actually incorrect. How close does the radio
wave frequency need to be to the Larmor frequency? If anything, the explanation seems to
indicate that the lifetime determines this, which is only true for very weak radio-frequency
(RF) fields.
• If the NMR signal is the result of relaxation from the excited state back to the parallel
ground state, then why does excitation by a 180° inversion pulse not result in subsequent
emission of signal, which seems to be a consequence of the level diagram in Fig. 3.1
contradicting experiments? Such a pulse moves all ground-state population into the
excited state and could therefore be expected to give the maximum signal considering
the explanation above. You may correctly argue that such an inversion pulse would not
create any transversal magnetization and therefore no signal. In the process, however,
you have wisely extended the description above to include transversal magnetization
as the source of signal. Sometimes, this is done as illustrated in Fig. 3.2 showing how
transversal magnetization can arise, even when only the up/down states are available,
as claimed in the explanation above.
Though not offering any proof, Fig. 3.2 at least indicates that the magnetization may be
transversal while each nucleus is in one of the two eigenstates, which is often said to be of
central importance in QM. The role of transversal magnetization was missing in the initial
explanation, but the added claim of its importance seems to resolve the problem of the missing
NMR signal after a 180° pulse that the level diagram in Fig. 3.1 introduces. It is time to ask
yourself if you are now happy with the explanation above.
In fact, Fig. 3.2 raises new questions more serious than the one it was aimed at answering:
Why are the nuclei affected differently by the radio waves? How do they “know” which way
to point? Is the size of the magnetization changed by excitation, like it seems? If so, can this
phenomenon be used to create magnetization? Don’t bother to find the answers, as the figure
is misleading, considering that homogeneous magnetic fields cannot change relative spin
directions of nuclei exposed to such fields. This is true for both the static field B0 and the
RF field B1, so all they can do is to rotate the spin distribution in Fig. 3.1 around a vertical
axis (precession around the field vector B0) or any other axis, depending on the characteristics
of the applied fields (e.g., a transversal axis in the rotating frame of reference for a combination
of B0 and a resonant B1 field).
Mx
My
Mz
M
FIGURE 3.2 Illustration of how transversal magnetization could in principle be
formed by nuclei in the conical eigenstates (Fig. 3.1) if the RF field somehow could
reduce the angular spread. In reality, it is relatively easy to show using classical or
quantum mechanics that homogeneous magnetic fields like B0 and B1 cannot change
the relative orientation of nuclear spins, so the figure has no basis in reality. Instead, all
spins are rotated equally during excitation, and the net magnetization consequently
follows this rotation.
146 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
Even worse, the rest of the explanation above is also pretty bad for a number of reasons.
One concern is the mentioning of “radio waves” in the explanation, which is problematic
when understood as traveling waves as used for radio communication, for example. David
Hoult, in particular, correctly insists that traveling waves play little role in NMR6
and the term
“radio waves” should therefore arguably not be used in this context. Detailed accounts of
other problems are given in my introductory paper on myths in NMR,2
including calculations
based on QM and they will not be repeated. Here is a partial summary:
1. There is nothing in QM telling us that the nuclear spin can only be in the up or the down
state. It can point in any direction, just like a classical dipole.
2. In thermal equilibrium, the most probable direction of the nuclear spin is indeed along
the external magnetic field. But all the other directions are almost equally probable at
relevant fields and temperatures. In equilibrium, the antiparallel orientation is the least
probable of all states.
3. Neither photon energies nor the very concept of photons is needed in the explanations of
normal NMR.6–8
Photons are widely understood as particles or quanta of light, which is
reasonably consistent with a more precise definition involving excitation of
electromagnetic field modes.7
Do not worry about what that really means, since field
quantization is of little relevance to the basic NMR understanding. Even if you do QM
calculations for NMR, you can typically treat the field classically with no loss of insight or
accuracy.6,8
An NMR description with simplified field quantization will nevertheless be
used later in this chapter, since it gives insight into QM concepts from a classical
perspective. For now, the RF field is treated classically, and a sinusoidal field is therefore
characterized by the amplitude, phase, and frequency. The latter indeed has to be near
the Larmor frequency to cause transitions. How near depends on the amplitude of the
field, which is a natural consequence of the alternative NMR explanation discussed in
Chapter 2, and not a consequence of the limited QM-inspired pseudoexplanation
above, since amplitude translates into photon count.
4. As often done, it was stated in the flawed NMR explanation above that QM is
responsible for the resonance condition, that is, the need of the RF to match the Larmor
frequency (the Zeeman splitting) for transitions to occur. This is true but overly
complicated since the text might equally well have stated that classical mechanics is
responsible. Better yet, the explanation could have provided a simple argument that this is
so. Since the principles of QM are believed to be generally valid, we can trivially claim that
QM is responsible for everything we ever observe. With this in mind, a claim of QM being
responsible for something has close to zero information content. Typically, when
something is said to follow from QM, it is therefore implicitly meant that it is a nonclassical
aspect, but this is not the case for the NMR resonance condition, which is exactly as you
would expect it to be classically (including near-resonance effects as hinted above).
5. Abrupt transitions between states (quantum jumps) play no role in normal NMR. In
particular, sudden changes in the quantum state caused by measurement are irrelevant
(the so-called state reduction or collapse).
While the above QM-inspired description of NMR may sound simple and has elements
of truth, it is misleading and even wrong in several important respects, as explained in detail
elsewhere.2,6
Of course, the validity of an explanation does not depend on whether it is simple
1473.3 MISCONCEPTIONS IN NMR INTRODUCTIONS
II. EXAMPLES FROM NMR THEORY
or not, whereas the practical value of it may. What matters mostly is whether it gives the user
insight and predictive powers. If more explanations are equally good in these respects, the
simpler should be preferred, and unnecessary postulates avoided (see the principle of parsimony,
Occam’s razor, mentioned also in Chapter 1 in connection with Trap #24). To be of value,
any explanation has to be reasonably correct, and the typical QM interpretation above does not
qualify in this respect. It is largely misunderstood and reminiscent of early QM formulations.
In some contexts, it is important and necessary to describe NMR and field interactions
quantum mechanically, but far from all, and important insights are missed if basic QM
descriptions are not supplemented with analogous classical descriptions. Many will benefit
from a quantum description, especially if it is preceded by a classical description providing
intuitive insight. Even more are probably best left with a classical description alone, for
example, the vast majority of people working with MRI where this is typically fully sufficient
and leads to less confusion and misunderstanding than QM counterparts. A classical description
can be relatively easy, intuitive, and even accurate, as it can be shown to be a direct
consequence of QM. In that sense, it also qualifies as a QM explanation, despite the lack of
typical QM formalism. Aimed with a classical understanding, many aspects of NMR will give
more meaning, including a quantum description.
3.4 WHERE DID IT GO WRONG?
The correspondence between QM and classical NMR descriptions deserves explanation
and is a focus of the remaining part of this chapter. Before engaging in that, it is worth considering
why severe misunderstandings are present in much of the NMR literature many
years after the phenomenon was first described. Has much of the scientific community really
misunderstood the very basics of the quantum formulation that many rely on? Apparently,
yes, but it is not unique to NMR4
and the practical consequences are limited. The problems
can largely be attributed to Mental Traps as discussed in Chapter 1. Even though not all the
classic literature is flawed, a significant part is, and errors track back to the early days of QM
and NMR (Traps #1, #7, #8, and #28). The wrong statements may sound simple and probable
at first (Traps #4, #10, #22, and #42) but make NMR basics incomprehensible, which many fail
to notice since the errors have little practical consequence. These are repeated by many
authors and lecturers who do not have experience interpreting QM, including brilliant and
widely recognized quantum-savvy scientists (Traps #6, #31, #42, and #43), who have failed
to notice the mistakes (Traps #3 and #4) or have not bothered to explore alternatives.
Although the errors are now recognized and avoided by many, they are still repeated without
much reflection by unsuspecting authors. QM is correctly known to have very nonintuitive
aspects, and though many have felt uncomfortable with the wrong explanations, most have
been told—or have assumed—that this was a natural consequence of QM. Some find joy in the
apparent necessity of exotic quantum physics for “truly understanding” NMR (Traps #23 and
#24), although “truly misunderstanding” sometimes seems more accurate. Granted, QM is
needed for NMR understanding beyond the very basics, and everybody is likely to make
mistakes when interpreting QM, including the author of this chapter. QM indeed has oddities
as described below, but none of relevance for the basic magnetic resonance phenomenon.
148 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
Other clear reasons for the abundant errors are misinterpretation of experimental results and
mathematics. The QM formalism applied to NMR may seem to suggest that the eigenstates are
more important than they really are (Traps #11, #12, and #13). The Stern-Gerlach experiment9
discussed below may similarly be wrongly interpreted to show that nuclei can only be in one of
two states (Traps #25 and #39), but the effect of measurement on the nuclei is ignored in that
interpretation (Trap #26). The spiky appearance of NMR spectra may be wrongly interpreted as
supporting this false conclusion (Trap #29). Both the mathematical and the physical aspects and
sources of confusion are discussed in the following sections that supplement earlier work2
by
bringing the classical and the quantum descriptions closer together.
The reader may think of reasons for sticking to a wrong QM interpretation, for example,
that it is based on benign approximations necessary to avoid lengthy or complicated descriptions.
Such arguments reflect self-deceit in the case of introducing basic NMR. It takes little
effort to avoid the mentioned errors and correct alternatives need to be neither lengthy nor
complicated. For explaining spectral features, however, it can arguably be simpler to think
in terms of sudden transitions between states rather than of slow transitions or coherent
superpositions of such, but it should be with awareness that this is a substitute for the
QM-predicted dynamics of normal NMR, which do not involve quantum jumps, as discussed
later in this chapter (this relates closely to the issue of a model’s legitimacy as discussed under
Pillar 13 and Trap #18 in Chapter 1 and also to the ability to treat descriptions of NMR as
models rather than “truths of nature” as mentioned in Chapter 2).
3.5 A LIMITED INTRODUCTION TO CLASSICAL
AND QUANTUM MECHANICS
The fairly typical explanation of NMR opening this chapter was inspired by QM but got
important aspects wrong. While the presented QM formalism is typically accurate, the problems
concern interpretation. It can be argued that the mathematics speaks for itself, but only
those sufficiently trained to make their own interpretations will likely benefit from
unexplained formalism. Even the most hard-core NMR scientists make use of semiclassical
mental representations when thinking about the basics since that has proved highly rewarding
(see the role of metaphoric models in scientific thinking discussed in Chapter 1 under
Pillars 10 and 11).
An attempt of explaining the QM foundation for NMR, while reconciling the classical and
quantum descriptions, is now made. Only the needed parts are presented and only superficially
so as to favor general readability. The mathematical description can be found in many
books and will not be repeated here. The concepts of QM and classical mechanics are instead
discussed in general and are related to NMR.
Initially, we focus on classical mechanics. A starting point is Newton’s second law
describing how forces influence particles. Generally, we need to supplement this with field
descriptions, how they interact with particles, and how they evolve, for example, as
expressed in the classical Maxwell equations. Taken together, all the classical equations
necessary to describe system evolution will here be called “the classical equations of motion.”
A physical system can classically be described completely by all its characteristics
1493.5 A LIMITED INTRODUCTION TO CLASSICAL AND QUANTUM MECHANICS
II. EXAMPLES FROM NMR THEORY
including fields and the constituent particle’s positions and velocities (the microstate). This
will here be called the “classical configuration.” Given such a complete description of an
isolated system, the classical equations of motion can in principle tell us how the system
will evolve forever after, as classical mechanics are deterministic. In practice, no physical
system is completely isolated from the surroundings except the system of everything. This
special case does not leave room for an external observer who can perform measurements
on the system, so we largely keep that situation out of the discussion. Another practical
issue is the obvious impossibility of measuring all system properties at a particular point
in time with zero inaccuracy. This is needed since even minor uncertainty will generally
make long-term system behavior unpredictable. These practical issues do not challenge
the deterministic nature of classical mechanics: our experimental inabilities and limited
knowledge of the configuration are irrelevant to that question. We can imagine a perfectly
isolated system (e.g., the universe, even if infinite), and the classical equations of motion
will deterministically describe how the system evolves forever, no matter whether
observers know the classical configuration or not. If the isolation is imperfect, or our knowledge
about the system incomplete, classical uncertainty about the system evolution
increases over time, for example, as reflected in next week’s weather forecast that is typically
more uncertain than tomorrow’s.
Classical mechanics is not generally valid, and we now focus on differences introduced
with QM. The starting point for most QM derivations of NMR is the Schr€odinger equation.
It describes how any physical system, for example, nuclei in magnetic fields, evolves in
periods where the system is left to itself. When written for a specific system, it constitutes
“the QM equations of motion”:
iℏ
@
@t
cj i ¼ ^H cj i: (3.1)
The skills needed to use this equation are not assumed known in the following, although
occasional mathematics will appear. As an equation for the evolution of everything, it appears
deceptively simple, but similarly to classical mechanics, the practical solving can be immensely
complicated, especially when many particles and fields are involved. ^H is here
the so-called Hamiltonian that describes interactions within the system, and it is taken to
be piecewise constant throughout this text, that is, only changing at particular moments in
time such as when the RF transmitter is turned on or off (this does not correspond to classical
nuclear interactions within the system being piecewise constant). The state of the system is in
QM represented by a time-dependent “state vector” jci that is a complete characterization of
the system state, that is, there is nothing more to know about it: from the state vector the result
of any measurement of system properties can be calculated with maximum certainty (although
not with full certainty as discussed below). The state vector is not a vector in real
space, but in an abstract multidimensional state space with a dimensionality reflecting the
degrees of freedom of the system. The notation jXi for a state with properties X is common
and needed below. Sz ¼ 1
⁄2j i is, for example, the state vector for the spin-up state a since it has
the z-component of the spin, Sz, equal to ½. Only properties of particular interest appear explicitly
when this notation is used.
When the state vector and the character of all interactions of an isolated system are known,
the Schr€odinger equation deterministically describes how it evolves forever on. The state
150 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
vector is a QM analog of the classical configuration in the sense that they both describe a system
to the maximum extent. As mentioned for classical mechanics, it is practically impossible
to isolate a system, but that does not change the fact that QM is deterministic in this sense: if
the state vector is known at one point in time, it can be known forever on. In another sense,
QM is truly indeterministic in contrast to classical mechanics. We will soon return to
that point.
Readers having worked with state vectors or similar representations will know that they
appear very different from classical configurations. First of all, they may be complex-valued,
and “i” appearing in the Schr€odinger equation is the complex unit, i2
¼ À1. The complex unit
does not appear in the classical equations of motion, but this apparent difference does not
truly distinguish classical and quantum mechanics. Complex functions can generally conveniently
be used to describe oscillatory systems, for example, precessing spins, electronics, or
waves, whether they are classical or not.
Another deceptive difference comes from state vectors being close relatives of probability
density functions. For a system of particles in normal 3-D space, for example, the state vector
only provides probabilities that particles can be found at specific positions, having specific
velocities, spin states, etc. In contrast, a classical configuration may be expressed in terms
of exact particle positions and velocities, which are concepts that are revised in QM. This does
not in itself imply any significant difference between classical and quantum mechanics. Indeed,
classical mechanics can be expressed in terms of probability functions too, and it is often
fruitful to do this. If the state of a system is known with some uncertainty, classical equations
of motion can be formulated that describe deterministically how the system and the uncertainties
evolve. An example is illustrated in Fig. 3.3.
3.6 INDETERMINISM VS. UNCERTAINTY AND THE ROLE
OF MEASUREMENT
The main message so far is that even though the QM and classical equations of motion
appear very different at first sight, they can both be expressed rigorously in terms of quantitative
deterministic, probabilistic equations, and they will then appear much more similar.
There are important fundamental differences regarding the role of measurement, however.
These differences between quantum and classical mechanics are not important for the understanding
of basic NMR, but they are an important source of QM misunderstanding and will
x0 x1
t = t0
t = t1
x
v ± Δv
P(x) FIGURE 3.3 Consider a normal particle described
by classical mechanics. Take it to move freely from approximately
position x0 at time t0 with speed v along
the x axis. The lack of precise knowledge of these quantities
may, for example, be characterized by probability
distributions with width Dx and Dv. In that case, the
probability P(x) of finding the particle at some position
x at a later time t1 will be distributed with a peak probability
at x0 + v t1 Àt0ð Þ and with a position uncertainty
increasing over time.
1513.6 INDETERMINISM VS. UNCERTAINTY AND THE ROLE OF MEASUREMENT
II. EXAMPLES FROM NMR THEORY
be discussed here in the context of the previously introduced idealized isolated systems. To
do a measurement on such a system, we will need to interact with it, independent of whether
the system is described by QM or classical mechanics. In either case, there will be an effect on
the system that we may want to minimize to limit the influence on future system evolution,
but this minimal effect differs fundamentally. Important differences between QM and classical
mechanics are as follows:
1. QM is probabilistic at the most fundamental level. In contrast to classical mechanics, it is
not optional whether to express QM in terms of probabilities or specific configurations.
Even if you have the most complete knowledge of the system (the state vector jci), you
will still not be able to predict the outcome of every given experiment. But you will be able
to accurately calculate the probabilities of different possible outcomes. Nature can thus
be truly undecided about any outcome until it is measured, and this is called “quantum
indeterminism.” For example, a particle generally has no position until it is measured or
confined—it is not just unknown. In contrast, classical mechanics can be expressed as a
probabilistic theory, but with adequate information about a system, the uncertainties can in
principle be arbitrarily small, leaving the result of experiment fundamentally predictable.
As a side note, there are actually formulations of QM that are deterministic, but such
nonlocal, hidden-variable formulations are more complicated than standard QM and have
elements that are even more bizarre than indeterminism.10
They rely on nonverifiable
elements beyond normal QM and therefore fall for Occam’s razor, but one of them could in
principle be a correct explanation of mechanisms underlying QM.
2. Gaining knowledge about a classical system via measurement does not in itself change
its evolution. The physical interaction between measurement apparatus and the system
may, but classically there is no intrinsic change of system behavior associated with
acquiring knowledge about it. In contrast, for a system described by QM any measurement
by an external observer will force a reality onto the system that does not generally exist
in advance (indeterminism is changed). During system evolution, the Schr€odinger
equation keeps all possible paths of system evolution open as jci expresses the system
properties in a probabilistic way (a probabilistic description accounts for all possible
paths to all possible outcomes). However, according to QM, a particular path of system
evolution is only selected when some interaction happens that limits the possible paths to
just one, for example, a measurement. This reduction of indeterminism is superficially
similar to measurement on a classical system characterized by uncertainty: A measurement
reduces our uncertainty, but it does not necessarily change anything else, classically.
The truly weird quantum aspect is that the Schr€odinger equation allows for interference
between different open paths. This may even result in cancellation of possibilities for
events that could otherwise have happened. A particular event that can happen in two
ways classically can be prevented by QM from happening at all due to cancellation of the
two possibilities. Only if one of the two ways is eliminated, the event can happen, although
classically it should have become less likely. The double-slit experiment is an often used
example of this, illustrated in Fig. 3.4. As a direct result of changing interference via
measurement, any part of the system may immediately after the measurement behave
entirely differently. This is very nonintuitive but has been verified in many experiments.
Even though the changes of system behavior can happen instantly over large distances, this
152 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
does not allow for superluminal communication, for example. These mysterious “spooky
actions at a distance” (Albert Einstein’s words as discussed, e.g., by Bell11
) are not violating
the laws of relativity and will only show in the statistics when correlations between
experimental results are calculated. They are classically most unexpected, but do not leave
theoretical loose ends such as inconsistencies of the formalism.
3.7 THE ROLE OF EIGENSTATES IN SINGLE-PARTICLE
MEASUREMENTS
Having established some fundamental differences between QM and classical mechanics,
we now return to another difference that is fundamental and important, but nevertheless
Detector screen
Emitter
x
Wall with slits
P(x)
FIGURE 3.4 The famous double-slit experiment showing nonclassical aspects of QM. The emitter is adjusted to
shoot electrons at a wall one by one. Some may pass through slits in the wall and hit the fluorescent screen behind it to
produce small flashes of light upon impact. The marks on the x axis show example positions of these. If only one of the
slits is open, the flashes will be distributed almost as expected classically, that is, symmetrically behind the open slit,
although more spread out due to diffraction (the smaller the hole, the more do particles passing through the hole
“forget” their direction). If both slits are open, however, an interference pattern may appear, that is, the distribution
P(x) of flashes on the screen will show a series of minima and maxima as indicated. In particular, there may be positions
on the screen where no flash will ever be detected with both slits open, but where flashes may appear if one slit
is blocked or equipped with a nonabsorbing electron detector. Flashes at such positions start appearing only when the
canceling open paths through both slits are replaced with a single certain path, which eliminates interference. A similar
pattern is seen for light, sounds, and other waves, and the distributions of flashes are consistent with electrons
propagating as waves that interfere. However, the flashes on the screen each appear at just one distinct location,
which shows that electrons are detected as particles, not waves. In fact, it is the probability amplitudes that propagate
and interfere like waves as described by QM. The wave description was reasonably straightforward here, but it
should be remembered that the waves of QM generally propagate in “configuration space,” which is not a physical
space but a mathematical construct. They should therefore not be thought of as “real.”
1533.7 THE ROLE OF EIGENSTATES IN SINGLE-PARTICLE MEASUREMENTS
II. EXAMPLES FROM NMR THEORY
sometimes overestimated—the role of eigenstates. For the sake of simplifying arguments, we
assume consistently in this chapter that there is no traditional measurement noise, and that
our measurement apparatus is perfect and disturbs the probed system as little as theoretically
possible.
It is often said that QM dictates that systems can only be in particular discrete states, for
example, spin eigenstates. The situation is more complicated than that, however. For every
measurement on a QM system, there are indeed only certain possible outcomes, which
may or may not be discrete, that is, being separated numerically. These numerical outcomes
are called “eigenvalues,” and they correspond to specific states called “eigenstates.” If a system
is in an eigenstate for a particular kind of measurement, the corresponding eigenvalue
will result, if the measurement is conducted. The mathematical representation of a particular
measurement involves the corresponding “measurement operator.” For a particle, for example,
there is a mathematical position operator that represents a position measurement along a
given direction, for example, the x axis. The eigenvalues of this are the possible results of a
position measurement. Each eigenvalue x0 is associated with at least one eigenstate x ¼ x0j i
meaning that if the system is in this state at time t, that is, cj i ¼ x ¼ x0j i, then a measurement
of position at that point in time is guaranteed to give the value x0. Once confined in this way,
the particle will lose its memory of direction in accordance with the Heisenberg Uncertainty
Principle. This gives rise to diffraction as described in the caption of Fig. 3.4.
A more pertinent example is now considered. Each proton has a magnetic moment, meaning
that they magnetically appear like tiny spinning bar magnets with a north and a south
pole. The direction of the magnetic moment coincides with the spin axis, since spin is responsible
for the magnetic property: spin and magnetism are proportional. With a so-called
Stern-Gerlach apparatus,9
we can measure the magnetism of isolated nuclei along any
particular axis, for example, the z axis. These nuclei are sent individually through the
apparatus, and a component of nuclear magnetism is measured by their deflection in an
inhomogeneous magnetic field oriented along the measurement axis. The measure of magnetism
is trivially turned into a measure of the proportional spin component given in units
of ℏ. It is found that the maximum measurement value of a proton spin component is ½ and
that the minimum value is À½, for example, corresponding to +15° and À15° angular deflection
of the particle trajectory for a particular Stern-Gerlach apparatus. These details of measurement
and scaling are not important in the current context, but the results are.
If the nuclear spin happens to be oriented along the z axis chosen to coincide with the B0
field (proton is in the up state), we will measure the value ½ for the spin Sz along this axis.
Likewise, we will measure the value À½ when the spin is oriented opposite the measurement
axis. If we send the same particle through two Stern-Gerlach devices to measure the direction
of spin along the same direction twice, we will get the same result twice. From a classical perspective,
there are no surprises so far, but they will soon appear.
If a nucleus happens to spin orthogonally to the measurement axis z, we will classically
expect to measure 0. This value, however, is not an eigenvalue of the spin component measurement
operator Sz, so this outcome will never appear. In fact, the only eigenvalues of spin
along any axis are ƽ, which is highly surprising from a classical perspective. Instead of
measuring 0 when the spin and measurement directions are perpendicular, we will measure
either +½ or À½ with equal probabilities, and in the process, the nucleus gets aligned with the
measurement axis: the measurement forces it into the corresponding up/down eigenstates in
154 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
a process often called “state reduction” or “collapse.” This is technically a projection onto the
subspace of eigenstates consistent with the measured eigenvalue. For a single proton, any
prior indeterminism inconsistent with the measured value disappears when the measurement
imposes a new reality onto the nucleus. However, new indeterminism of other variables
than the one measured may well be introduced simultaneously. This is particularly true
for the orthogonal spin components that become indeterminate, which can be verified by
passing the particle through more Stern-Gerlach apparatus serially. Such results are highly
surprising and nonclassical but have been verified in many different contexts.
It will be described below that nuclear spin can be oriented in any direction whether it is in
a magnetic field or not. When its component in a particular direction is forming an angle y
with the measurement direction, the probabilities of measuring the only two possible outcomes
are cos2
(y/2) and sin2
(y/2), so there is certainty of the outcome when the measurement
direction is aligned with the spin direction (parallel or antiparallel).
3.8 ENTANGLEMENT
As mentioned earlier, it can be shown with Stern-Gerlach devices and similar experimental
setups that measurements fundamentally change the state of the system. This can even
have immediate consequences in far away regions, which is most obvious when “entanglement”
is involved. This is another unique quantum aspect, and it is beyond the scope of this
text to describe it in depth, but an example is given: consider a situation where two hydrogen
nuclei with known and opposite spin orientations are brought together and left to interact
magnetically, so that each precesses in the dipole field generated by the other. For the
sake of argument, we will assume this to be the only interaction in this thought experiment.
Afterward, we do not know either nucleus’ spin orientation, but due to the conservation of
angular momentum (including spin), we know for sure that they are still opposite. We have
ended up with two nuclei whose spin states are perfectly correlated (opposite), but otherwise
unknown. We say that the two nuclei are in an entangled state. We can bring the two
nuclei far apart, and if we measure the component of spin of each along the same direction,
we are guaranteed to get opposite results. There is yet no conflict with classical expectations,
but there is an important difference that will soon show. The spin interaction may give rise
to classical uncertainty, but in addition it can result in quantum indeterminism which may
present itself similarly but is very different: the state of the nuclei after the interaction reflects
that different possibilities of outcome are still open (not just unknown) even after
the interaction has finished. If we repeat the spin preparation and measurement many times
with the measurement done in varying directions for both particles, it will be found that the
results violate classical expectations.11
It will not show for individual measurements, but in
the statistics when the results of repeated measurements performed for each nucleus are
correlated. The reason is that the state collapse for one nucleus simultaneously also forces
the other into an eigenstate even if it is far away. The statistics rule out that the outcome of
the nuclear interaction is fixed before a measurement takes place, possibly much later and
far away.11
This makes little sense since our minds are shaped by the classical experience we
have from everyday life, but nature offers such surprises when we move outside our classical
comfort zone.
1553.8 ENTANGLEMENT
II. EXAMPLES FROM NMR THEORY
3.9 SUPERPOSITIONS
From the more exotic and fundamental differences between classical and quantum mechanics,
we now return to the more practical that you will rapidly encounter when reading
NMR introductions. A superficial difference between typical QM and classical descriptions
becomes apparent when the state vector is expressed in terms of basis functions or eigenstates
as typically done in the context of NMR. Such decompositions are often perceived as something
specific to QM, which is not the case. The eigenstates of measurement operators do play
a special role, but this role is often not visible from experiments on ensembles. Before
discussing the classical analog of eigenstates, the role of these in QM descriptions will be
discussed.
As described, noninteracting protons in a magnetic field each have two eigenstates for spin
measured along any particular axis. Despite what some NMR introductions say, these are by
no means the only possible states of the nuclei. In most texts that take the QM description of
NMR seriously, you will find the state vector jci written in terms of eigenstates, typically
those associated with measurement of the spin component Sz:
cj i ¼ a Sz ¼ 1
⁄2j i + b Sz ¼ À1
⁄2j i, (3.2)
where a and b are complex coefficients in this weighted sum of states called a superposition of
Sz eigenstates. The probabilities of finding a particle characterized by state jci in the up or
down states are jaj2
and jbj2
, respectively, for spin measurements along the z axis. Since
+ 1
⁄2 and À1
⁄2 are the only two outcomes, the corresponding probabilities therefore must
sum to 1: jaj2
+ jbj2
¼ 1.
Does Eq. (3.2) imply that the particle in state jci is in either the Sz ¼ 1
⁄2j i or the Sz ¼ À1
⁄2j i
state? No. Before a measurement, all weighting factors a and b satisfying jaj2
+ jbj2
¼ 1 are
valid. Any such pair describes a spin state jci according to the expression above. It can be
shown that it is possible to write the coefficients a and b differently, so it becomes clear that
any valid combination of these describe exactly one spin orientation in ordinary 3D space.
Let y and ’ be the polar and azimuthal angles in normal spherical coordinates. A spin
state expressing the closest QM analog to classical spin around the (y,’) direction can be
written as
y, ’j i ¼ cos y=2ð Þ Sz ¼ 1
⁄2j i + sin y=2ð Þexp i’ð Þ Sz ¼ À1
⁄2j i: (3.3)
jy,fi is here used as a shorthand for Sy,f ¼ 1
⁄2



, that is, spin along the (y,’) direction. The
probability normalization condition jaj2
+ jbj2
¼ 1 is clearly fulfilled for this superposition.
The polar angle is seen to be reflected in the amplitude of the up and down states, whereas
the azimuthal angle is their phase difference. It is a general feature of QM that the overall
phase factor of any state vector is without physical significance (jci and exp(iw)jci describe
the same reality, whereas the relative phase of states in superpositions matters). The coefficient
of Sz ¼ 1
⁄2j i was therefore chosen real in Eq. (3.3) without loss of generality, and the state
jci is seen to be fully characterized by the polar and the azimuthal angle only, just like a classical
dipole with fixed amplitude. Although we can assign such specific angles to the spin
state, there is still intrinsic indeterminism associated with orthogonal components of the spin.
A spin cone similar to the upper part of Fig. 3.1 but oriented in the direction of (y,’) is a
156 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
reasonable graphical representation of jy,fi, but a unit vector with these spherical coordinates
is in most ways a better visualization. This is the so-called Bloch vector that can be defined
for any QM two-level system in an abstract parameter space.12
In the case of basic spin ½
NMR, it is simply proportional to the magnetic moment in normal 3-D space. Generalizations
to mixed states and to more levels can be made, but the discussion is limited to the Bloch vector
of pure states for single spin ½ particles in this chapter. This vector offers a good graphical
representation of jci for several reasons:
(1) Unlike the vectors in the cone in Fig. 3.1, the Bloch vector has a definite physical meaning
stated in Eq. (3.3), and it actually points up for the “up” state Sz ¼ 1
⁄2j i and down for the
“down” state Sz ¼ À1
⁄2j i (y¼0° and 180°, respectively).
(2) It is the closest QM analog of a classical magnetic dipole, since it evolves like one when
subject to magnetic fields.
(3) Apart from relaxation effects, it keeps these properties when generalized to statistical
ensembles of weakly interacting nuclei. The description of magnetic resonance in
Chapter 2 is therefore equally valid for QM and classical mechanics. Note that eigenstates
played no role in the NMR introduction of Chapter 2, so they are not crucial for the
understanding of the resonance phenomenon. As a shorthand, it is warranted to call the
Bloch vector of a nucleus its magnetic moment in analogy with classical mechanics, but
the differences should be remembered. Entanglement, for example, is not easily shown
with Bloch vectors or cone figures, that is, that the Bloch vector of two nuclei may be
correlated without having specific directions.
The problematic QM-inspired NMR explanation opening this chapter ignored superpositions,
and it therefore invites rate equations for the populations of the eigenstates. It is tempting
to introduce transition probabilities and write simple differential equations with reference
to the energy level diagram in Fig. 3.1. Such rate equations are appropriate when T2 is so short
that it limits the possibility of coherence buildup, but they cannot describe coherent evolution
essential to NMR. It is an important insight that rate equations offer no shortcut to the QM
equations of motion that need to be based on the Schr€odinger equation or equivalent forms,
which describe vector dynamics similar to those of classical mechanics.12
3.10 THE MISSING ROLE OF EIGENSTATES IN ENSEMBLE
MEASUREMENTS
If a measurement of the nucleus’ individual spin along the z axis is performed, for example,
with a Stern-Gerlach experiment as described above, this forces the nucleus into state
Sz ¼ 1
⁄2j i or Sz ¼ À1
⁄2j i. So it may be thought that a measurement of a sample’s magnetization
along the z direction forces each nucleus into one of its Sz eigenstates, but that is actually not
true. Measurements of the total magnetization in other directions do not force individual
spins into eigenstates either. The reason is that a complete ensemble consisting of many nuclei
has spin eigenstates that are not eigenstates of any individual nuclear spin measurement.
Whereas a series of measurements on individual nuclei force each into an eigenstate, it
can be shown that a measurement on an entire spin ensemble changes the state of the
1573.10 THE MISSING ROLE OF EIGENSTATES IN ENSEMBLE MEASUREMENTS
II. EXAMPLES FROM NMR THEORY
individual nuclei only insignificantly.2
The orientations of individual nuclei remain
undetermined, both classically and in the special QM sense. In particular, a measurement
of the total magnetization leaves the individual nuclei almost unaffected in entangled superpositions
of the up and down states (the more nuclei, the less effect). Even if nuclei did get into
single-nucleus eigenstates, they would soon interact with other nuclei, which results in more
general superpositions.
3.11 THE ROLE OF EIGENSTATES IN MATHEMATICAL
DESCRIPTIONS
Since we do not actually perform measurements along the longitudinal direction z in NMR,
but rather along a transversal direction, for example, x, it may seem more natural to express
the state jci in terms of the eigenstates characteristic for measurement along that direction,
Sx ¼ 1
⁄2j i and Sx ¼ À1
⁄2j i. In analogy with the up/down nomenclature, these states could be
called right/left eigenstates, as they have similar properties to the Sz eigenstates, but for a
transversal spin component. Any spin state can be written as a superposition of these states,
but with other coefficients: cj i ¼ c Sx ¼ 1
⁄2j i + d Sx ¼ À1
⁄2j i where jcj2
+ jdj2
¼ 1.
No matter which spin state a single nucleus is in, it can be written in terms of the two
eigenstates associated with measurement along any particular axis. As illustrated above,
we may or may not choose this “quantization axis” to coincide with the measurement axis.
That choice is only a matter of mathematical convenience since jci itself is not influenced by
our mathematical description of it. The choice of quantization axis and corresponding
eigenstates is something as mundane as selecting a basis or coordinate system for our mathematical
descriptions. This is illustrated by a classical example. Suppose you want to describe
the throw of a stone mathematically. When leaving the hand, the stone’s velocity can be characterized
by a vector v pointing in the direction of the throw. Ignoring air resistance, the stone
is only affected by the gravitational downward pull, and the vertical velocity will therefore
change linearly with time. In the horizontal direction there is no force, and the stone will move
with constant velocity. The natural choice of basis defining the coordinate space is therefore
unit vectors {^x,^y} oriented in the horizontal and vertical directions, which makes the equations
of motion simple to write down and solve. It is possible to choose any other rotated basis
{^x0
,^y0
}, but we then have to decompose the gravitational pull into x0
and y0
components to
solve the equations of motion. The actual flight of the stone is unaffected of our choice of basis,
but the mathematical description depends on it: v ¼ vx^x + vy^y ¼ v0
x^x0
+ v0
y^y0
.
Analogously, any spin component measurement (whether we perform it or not) has
correspondingeigenstatesthatconstituteapossiblebasisforcalculationsinNMRasexemplified
bythe twoalternativedecompositions cj i ¼ a Sz ¼ 1
⁄2j i + b Sz ¼ À1
⁄2j i ¼ c Sx ¼ 1
⁄2j i + d Sx ¼ À1
⁄2j i.The
basis Sz ¼ 1
⁄2j i, Sz ¼ À1
⁄2j if g and the basis Sx ¼ 1
⁄2j i, Sx ¼ À1
⁄2j if g are merely rotated with respect to
each other. Each set of basis state vectors is orthogonal, in contrast to the spin orientations that
the states represent: up and down are not orthogonal directions, but orthogonal eigenstates.
These aspects are given some attention here since spin decompositions are often attributed
more significance than they deserve. QM provides extreme oddities such as the “Schr€odinger
cat states,” where more or less macroscopic systems appear to be in two states simultaneously.10
158 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
Thismaybetheinspirationwhenspinsuperpositionsaresometimesclaimedtobemysteriousin
thatindividualnuclei can be inthe Sz ¼ 1
⁄2j i and the Sz ¼ À1
⁄2j i statessimultaneously.Itis nomore
mysterious thanaflyingstone movingverticallyandhorizontallyatthesametime,whichitdoes
for atypicalthrow.QMdoesindeedincludeweirdpossibilitiesofseveralthingsapparentlyhappening
at once, for example, an electron passing through both slits in a double-slit experiment,
but a simple decomposition of a nucleus’ spin state reflects no such magic.
For descriptions of NMR the Sx ¼ Æ1
⁄2 eigenstates may appear to be the natural choice since
the transversal magnetization is measured. Nevertheless, the longitudinal axis is typically
chosen as the quantization axis. For normal NMR the choice of basis is influenced more by
relaxation processes and B0 field interaction than by the measurement, since the latter is implicitly
and correctly assumed to influence the nuclear states insignificantly. This choice is
made since the up and down states are eigenstates not only of the spin measurement operator
Sz, but also of the energy which is lowest when the nuclear spin and the magnetic field are
aligned. As stated earlier, the nuclear spin can point in any direction, but only two states,
Sz ¼ 1
⁄2j i and Sz ¼ À1
⁄2j i, are associated with well-defined energies (this situation changes when
an RF field is applied). The states with a well-defined energy play a special role in QM as they
are stationary. Such a state jcEi with energy E does not evolve into other states as long as the
interaction remains unchanged. In the absence of measurement, only linear phase evolution
occurs at a rate that is proportional to the energy of the eigenstate:
cE tð Þj i ¼ exp ÀiEt=ℏð Þ cE t ¼ 0ð Þj i: (3.4)
This follows from solving the Schr€odinger Eq. (3.1) for an energy eigenstate cE t ¼ 0ð Þj i ¼ Ej i,
where the Hamiltonian ^H and the energy E satisfy the eigenvalue equation ^H Ej i ¼ E Ej i.
This insight provides a relatively simple way to solve the equations of motion. Even if no
energy measurement is made, we may still choose to express a general spin state jci in terms
of the energy eigenstates, which form a basis, cj i ¼
X
E
CE Ej i. If we, for example, want to find
the time evolution of transversal magnetization excited to be along the x axis at time zero (i.e.,
y ¼ 90° and ’ ¼ 0°), Eq. (3.3) tells us that right after excitation the magnetization is in the state
c t ¼ 0ð Þj i ¼ Sx ¼ 1
⁄2j i ¼ cos 90°=2ð Þ Sz ¼ 1
⁄2j i + sin 90°=2ð Þ Sz ¼ À1
⁄2j i ¼ 1=
ffiffiffi
2
p
Sz ¼ 1
⁄2j i + Sz ¼ À1
⁄2j ið Þ:
This is not an energy eigenstate but a weighted sum of such, and each term therefore
evolves according to Eq. (3.4). The phase difference of the energy eigenstates will consequently
evolve at a rate proportional to their energy difference. This frequency is the Larmor
frequency o0, and by comparison to Eq. (3.3) we see that the azimuthal angle of the magnetization
simply changes linearly with time:
c tð Þj i ¼ cos 90°=2ð Þ Sz ¼ 1
⁄2j i + sin 90°=2ð Þexp Àio0tð Þ Sz ¼ À1
⁄2j i ¼ y ¼ 90°,f ¼ Ào0tj i:
This is recognized as precession in the transversal plane, and it can be accurately visualized
in terms of the Bloch vector or the similar classical dipole. In direct conflict with the incorrect
NMR explanation opening this chapter, no transition between Sz eigenstates is involved, just
independent phase evolution of each coefficient in the weighted sum of states.
Please note that the frequency of the periodic phase variation in Eq. (3.4) is not unique due
to the zero-energy level being freely selectable. The frequency of each state should therefore
1593.11 THE ROLE OF EIGENSTATES IN MATHEMATICAL DESCRIPTIONS
II. EXAMPLES FROM NMR THEORY
not be confused with its precession frequency as sometimes done. Only phase differences are
meaningful in QM state vector descriptions.
In summary, since NMR measurements on ensembles are affecting the state of individual
nuclei insignificantly, the spin eigenstates are not essential for understanding the basic NMR
phenomenon. They offer options for a choice of coordinate system when we wish to make
mathematical descriptions. The choice can be practically important, and it may influence
the way we interpret the experiments, but it must be remembered that with other choices
of basis, other interpretations may be more natural. In particular, coherences and population
differences are manifestations of the same quality but expressed in different bases.2
This is
important but beyond the scope of the current text. It should also be remembered that both
classical and quantum mechanics can be formulated in different equivalent ways that each
may invite both valid and invalid interpretations.4,13
3.12 VISUALIZATION OF SPIN DISTRIBUTIONS
Figures 2.9 and 2.14 in Chapter 2 show alternatives to the cones of Fig. 3.1. Such graphs
(that are also available for download14
) illustrate near-isotropic spin distributions that are
of central importance for understanding thermal equilibrium and why it is sufficient to keep
track of the net magnetization of spin isochromates, that is, collections of nuclei experiencing
the same external fields.2
With accompanying explanation, the graphs are easily understandable
from a classical perspective, but it may be less clear which meaning they convey in a
quantum context and whether the distributions will evolve as predicted classically, that is,
precession around the immediate magnetic field that is the sum of B0 and B1. Unless you want
arguments for the validity of the figures in a QM context, you can safely ignore the following
two sections that are somewhat more technical than most.
Some familiarity with operators and particularly the density operator1,7
is assumed below.
This is defined as an ensemble average ^r ¼ cj i ch j of the outer vector product of jci with its
own conjugate vector hcj. Except for the arbitrary phase, this outer product jcihcj has the
same information as the state vector itself, but unlike state vectors, it can be averaged in a
meaningful way over ensembles of similar systems such as protons in a homogeneous sample.
It can capture both classical uncertainty and quantum indeterminism and can compactly
represent statistical properties of system behavior. When expressed in a basis, the diagonal
elements of ^r provide the probabilities of finding the system in one of the basis states, should a
measurement be performed. The off-diagonal elements represent coherences (correlations1
)
between the states. Expressed in the energy eigenstate basis, ^r is diagonal in thermal equilibrium
with diagonal elements given by the Boltzmann factors (details follow).
Figure 3.1 showing cones is a reasonable representation of the eigenstates and therefore of
the spin distribution after a series of spin measurements along the z axis have been conducted
on individual nuclei. Each measurement will give a result ƽ and the measured nucleus will in
the process be forced into the corresponding eigenstate of the Sz operator. Starting with an
ensemble of nuclei in thermal equilibrium, there will be a slight surplus of measurements that
leave the spin pointing in the direction of B0. However, Fig. 3.1 is not a good representation of
thermal equilibrium undisturbed by measurement or after a measurement of a sample’s net
160 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
magnetization. As mentioned, showing Bloch vectors of individual nuclei is a better choice for
visualization, but they are unknown since only average properties of spins in the ensemble
are available, as expressed in the density operator ^r. Furthermore, entanglement caused by
nuclear interaction challenges the concept of individual Bloch vectors. We can nevertheless
make a good graphical representation of a spin distribution by calculating the expected results
of measurements over many directions approximately uniformly distributed.
Figure 2.9 shows the resulting distribution as fairly isotropic but skewed towards north
(higher density of arrows near this direction). The creation and interpretation of this figure
are now discussed.
For each nucleus in a random subset of a nuclear ensemble, a simulated measurement is
performed, resulting in an arrow showing the Bloch vector of each nucleus after measurement.
This involves calculating the probability of measuring spin along a particular direction
(y,f), P y, fð Þ ¼ y, ’h j^r y, ’j i. In accordance with the calculated probabilities, the Bloch vector
is for each nucleus chosen semirandomly to point either along the measurement direction or
directly against it. This is done since a measurement of spin Sy,’ along direction (y,f) for a
particular nucleus will bring it into the eigenstates jy,fi or pÀy, f + pj i corresponding to
finding the spin to be parallel or antiparallel to the jy,fi direction, where the latter state is
calculated from Eq. (3.3):
pÀy,’ + pj i ¼ sin y=2ð Þ Sz ¼ 1
⁄2j iÀcos y=2ð Þexp i’ð Þ Sz ¼ À1
⁄2j i:
Hence, each simulated measurement will assign a specific Bloch vector to a nucleus of
the ensemble, and the directions will afterward be consistent with the general orientations
of nuclei in the ensemble since the ensemble density matrix is unchanged by such measurements.
The resulting distribution is sketched in Fig. 2.9 of Chapter 2 with the anisotropy of
the distribution exaggerated to enhance visibility. The operator for spin measurement in
direction (y,f) can be expressed in terms of its known eigenvalues and eigenstates,
Sy,’ ¼ 1
⁄2 y, ’j i y, ’h jÀ1
⁄2 pÀy,’ + pj i pÀy,’ + ph j.
Using Eq. (3.3), this operator can be expressed in the basis Sz ¼ 1
⁄2j i, Sz ¼ À1
⁄2j if g. After
having performed individual simulated spin measurements on a subset of the nuclei, these
have well-defined Bloch vectors pointing in many directions but with an orientation distribution
slightly skewed towards magnetic north. Since Bloch vectors are known to evolve
like classical magnetic dipoles, it is valid QM to make animations, for example, showing
how each vector is gradually rotated by magnetic fields.14
The result of rotation of the distribution
of Fig. 2.9 by a 90° excitation pulse will, for example, be as illustrated in Fig. 2.16.
Nuclear interactions will only make the assignment of Bloch vectors dubious on a timescale
T2 that is typically long relative to timescales of field interactions (precession and
excitation).
Considering that measurements change reality, it may be questioned, however, if it can be
experimentally confirmed that the distribution rotates as indicated in the figures. Indeed, it
can, since only measurements conducted in directions more or less orthogonal to the Bloch
vector will change it randomly. If we subject the spin ensemble to known magnetic fields,
we will know how each Bloch vector is rotated, and for each of the selected nuclei we can
get confirmation by measuring the spin along the direction of the expected Bloch vector.
The measurement result will be certain if the rotation was as expected. Conducting the actual
experiment is not needed, since it has been shown before in many contexts that Bloch vectors
1613.12 VISUALIZATION OF SPIN DISTRIBUTIONS
II. EXAMPLES FROM NMR THEORY
of individual nuclei evolve as expected from QM, that is, in accordance with classical mechanics
when left undisturbed by measurement or interaction. The conceptual or simulated experiments
were merely introduced to provide a specific and experimentally verifiable quantum
interpretation of Figs. 2.9 and 2.16.
Being gifted with healthy skepticism, you may have noticed there is still reason for
concern, however, which is the topic of the next section providing further necessary
arguments for Fig. 2.9 being a much better illustration of thermal equilibrium than Fig. 3.1.
3.13 THERMAL EQUILIBRIUM
Like the previous section, this one is likely of limited interest for the general reader. It
discusses a crucial question, however: which reality can we reasonably assign to individual
nuclei? The simulated measurements described in the previous section assign Bloch vectors
to a subset of a spin ensemble so they get oriented consistently with the density matrix.
Two objections can be raised, however. It suffices to discuss these for the case of thermal
equilibrium (Fig. 2.9) where the concerns are most easily expressed:
1. The outlined method provides illustrations consistent with the density matrix, but very
different and equally consistent illustrations can be made by other algorithms since the
density matrix only expresses the statistical behavior of the nuclei, not the individual
nuclei’s orientation. In particular, the infamous Fig. 3.1 shows another spin distribution,
which corresponds to the same density matrix and which therefore will give the same
experimental predictions. So how can it be claimed that Fig. 2.9 is more precise than
Fig. 3.1?
2. A related question concerns the fact that the simulated measurements may change the spin
distribution: even if each spin was in either the up or the down state before doing simulated
measurements, the prescribed method would result in a graph showing the chosen
subsample as a near-uniform distribution.
It may seem that the illustration method was crafted to give the desired result. Indeed, it
was to some extent, but it can be argued that this is also the only reasonable result. To get
there, we first have to acknowledge that the individual nuclei do not actually have a direction
in thermal equilibrium—there is only a tendency for them to have one. This is a nonclassical
consequence of QM: suppose that each had some unknown direction at a particular moment
of time, that is, that the diagonal density matrix was the result of classical lack of knowledge
rather than indeterminism. Random interactions would soon cause entanglement, and
as discussed earlier, this forces us to give up on the concept of nuclei having specific
directions—they become undecided. So the classical uncertainty is very rapidly exchanged
with quantum indeterminism, and we have to conclude that individual nuclei have no direction
in thermal equilibrium. Nevertheless, they have a preferred direction when “asked” by
measurement, as reflected in differences between the density matrix’ diagonal elements.
Having acknowledged that individual nuclei have no direction (are “undecided”) and that
Figs. 2.9 and 3.1 give rise to the same density matrix, how can it be justified that Fig. 2.9
is any better than Fig. 3.1 as claimed here? The first argument is similar to the one given above:
162 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
If all nuclei were indeed in separate Sz eigenstates at some particular moment of time, their
interactions would very rapidly bring them out of that state. There are many more configurations
that are near isotropic than configurations that have each individual spin in the up or
down state, so the entropy is much lower for the latter. Statistics tells us that nature evolves
towards high-entropy states, so the spin configuration shown in Fig. 3.1 would rapidly evolve
into a near-uniform distribution compatible with the density matrix. Such a distribution is
shown in Fig. 2.9 that is calculated as outlined in the previous section.
A similar but somewhat more technical argument in favor of Fig. 2.9 involves calculation of
the density matrix in the basis of product states for all nuclear spins, rather than for the single
spin states. This basis is normally avoided since it has 2N
basis vectors where N is the number
of nuclei in the ensemble. This is the basis of choice when we want to say something about
individual nuclei, but the dimensionality of the problem correspondingly increases enormously.
The basis vectors are each a product state
YN
i¼1
Si
z ¼ Æ1
⁄2



where Sz
i
is the Sz spin component
of nucleus i. The 2N
eigenvalue sign combinations each gives a basis vector. The
product states are not exactly energy eigenstates due to nuclear interactions, but since these
are very weak for normal NMR compared to the static field interaction, the exact energy
eigenstates deviate only insignificantly.
The corresponding density matrix has dimension 2N
squared, and it is diagonal as before.
The diagonal elements are each proportional to the Boltzmann factor exp ÀE=kTð Þ where E is
the energy of the state corresponding to the particular diagonal element (the Boltzmann distribution
expresses maximized entropy). A common normalization factor ensures that the
sum of all diagonal elements is 1. The lowest energy state is, for example, the tensor product
S1
z ¼ 1
⁄2



 S2
z ¼ 1
⁄2



ÁÁÁ SN
z ¼ 1
⁄2



with energy ÀNℏo0 relative to the highest energy state
S1
z ¼ À1
⁄2



 S2
z ¼ À1
⁄2



ÁÁÁ SN
z ¼ À1
⁄2



. In thermal equilibrium at room temperature, the probabilities
are nearly uniformly distributed over the 2N
states. This density matrix can be inappropriately
interpreted to reflect that the probabilities of the ensemble being in each of these
states are nearly equal.15
It has already been demonstrated, however, that the nuclei are not
each in any particular state (they are undecided), so instead, we have to interpret the density
operator as the 2N
different states being almost equally populated, which is consistent with
Fig. 2.9 and not at all with Fig. 3.1. We can conclude that the latter is not a valid representation
of thermal equilibrium.
The fact that measurements, and therefore the observer, play such a prominent role in QM
has been a subject of much debate. What qualifies as an observer? Does it need to be conscious,
living, or macroscopic, for example? The latter option is the most reasonable.* If a
“measurement” can be performed by interaction with anything macroscopic, like a bottle
of water, for example, this opens an interesting possibility for potentially rescuing a QM interpretation
that this text has been refuting: the random interactions between a nucleus and its
surroundings could potentially amount to something similar to a measurement of Sz, so that
the spin is brought into an eigenstate. May Fig. 3.1 thus be a reasonable representation of
*Vaguely stated, measurement involves coupling to a system with essentially infinite degrees of freedom and
states densely distributed over energies, which causes near-instant coherence loss and projection onto a
subspace. This, however, does not eliminate the “measurement problem of QM,” as it does not prevent the
“Schr€odinger cat states,” that is, entanglement involving macroscopic entities.10
1633.13 THERMAL EQUILIBRIUM
II. EXAMPLES FROM NMR THEORY
equilibrium magnetization after all? No. Even if such random interactions did have the same
effect as measurements along the direction of the magnetic field, these would necessarily have
to be weak/partial and be effective only on a timescale T2 since coherent evolution would
otherwise not be observable (we know it is). Those same nuclear interactions would make
the cone picture of Fig. 3.1 inaccurate on the same timescale.
Although there are strong arguments for Fig. 2.9 and similar, they are still just particular
illustrations of statistical distributions and of quantum realities, which arguably do not exist.
The figure shows properties of nuclei that they do not have until it is enforced by simulated
measurements (specific Bloch vector directions). However, these figures and the related and
commonly used “semiclassical” vector diagrams showing, for example, excitation,
dephasing, and echoes do capture the essential dynamics of QM for weakly interacting spin
½ ensemble dynamics. The Bloch vector diagrams leave NMR users and developers with a
practically useful and intuitive understanding of basic NMR phenomena and methods.
The remaining deviations from quantum reality are minor and of academic interest only.
In contrast, the common cone figures will cause confusion, at best. The most important argument
in favor of the Bloch vector diagrams is therefore pragmatism. Such graphs capture the
essentials of the QM descriptions, unlike cone figures.
3.14 CLASSICAL EIGENSTATES, RESONANCE, AND COUPLINGS
The limited fundamental role of quantum eigenstates in NMR may be surprising considering
that spectra are often explained as emission resulting from sudden quantum jumps between
eigenstates. There are no jumps in normal NMR, however, and quantum eigenstates
are convenient for describing NMR, but not a necessity. They share essential features with
classical eigenstates that are useful for understanding aspects of NMR and QM qualitatively.
Vaguely stated, energy eigenstates in QM represent states where no observable quantity
changes over time. They are therefore said to be stationary, but there is nevertheless linear phase
evolution of the probability amplitude as described by Eq. (3.4). This periodic variation corresponds
to a standing wave not reflected in any observable, when individual energy eigenstates
are excited. The spin-up state, for example, may reasonably be thought of as a probabilistic
representation of a single magnetic moment precessing with a specific polar angle (the “magic”
angle, 55°) but with all azimuthal angles equally probable. This interpretation is reflected in the
often seen Fig. 3.1 illustrating the eigenstates as cones that are symmetrical around the quantization
axis (the cone opening angles were drawn much smaller than 110°, however, since
otherwise the traditional up/down terminology would seem even less intuitive).
Classical systems also have resonances or modes of periodic oscillation. A pendulum will
have a characteristic frequency, for example, and it is independent of amplitude when oscillations
are small. Although energies are not quantized classically, frequencies often are. A
guitar string will support a number of periodic oscillatory eigenmodes as shown in Fig. 3.5,
each characterized by a standing wave pattern on the string. The corresponding classical
probabilistic eigenstates represent a string known to oscillate with a particular frequency,
but with unknown phase. Such modes are analogous to quantum eigenstates with all phases
of the oscillation represented simultaneously, which is consistent with the probabilistic nature
of eigenstates. Phase evolution corresponds to vibration of the string, but since all phases
164 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
are equally probable, this abstract classical state does not visibly change (it matches the
blurred visual appearance of an oscillating guitar string, which only changes slowly on the
timescale of damping).
It will be argued now that the evolution after excitation by plucking the guitar string is
similar to that described earlier for quantum superpositions of energy eigenstates. Each stationary
pattern of string movement is spatially characterized by the integer number of half
wavelengths along the string and temporally by the frequency of oscillation as shown in
Fig. 3.5. The two quantities increase together: high-pitched modes have a high number of half
wavelengths along the string. That the string supports a number of eigenmodes does not imply
that it can only oscillate in one of these standing wave patterns since any superposition of
modes is in fact possible. Any possible string motion can be described as a superposition of
the discrete eigenmode oscillations. When a guitar string is plucked sharply at one end, for
example, a traveling wave will move along the string and be reflected a few times before it is
completely dispersed.
Mathematically, such string motion can conveniently be described as excitation of a broad
spectrum of the standing wave eigenmodes, a superposition statej i ¼
X
mode
Cmode modej i.
Each mode oscillates in accordance with its characteristic spectral-spatial pattern and
independent of the other modes. The phase and amplitude of each mode are determined
by the specifics of the string plucking. The particular movement of the string, such as a traveling
wave, is expressed in the relative phases of the coefficients Cmode. Each mode’s oscillation
is damped on a timescale characteristic of the particular mode. In the absence of such
damping, the excitation of each mode is constant since there are no transitions between
the eigenmodes that are approximately uncoupled. There is just linear phase evolution happening
at a rate given by each mode’s frequency.
λ/2λ/2
FIGURE 3.5 Three standing wave
modes of oscillation are shown for an elastic
string suspended between fixed points. The
solid and dashed lines show the extreme
positions of the string for a particular amplitude
of oscillation. The vertical arrows indicate
the direction of string motion when the
string contracts from the extremes. The two
dotted arrows each indicate half a wavelength,
l/2, for the middle pattern of oscillation.
The three example modes represent
repeated oscillation with only a single temporal
frequency that increases from bottom
to top. The time evolution can therefore be
described by simple phase evolution on
timescales that are short compared to the oscillation
damping. Corresponding classical
probabilistic states with unknown phase
(even probability of all phases) are timeindependent.
Any possible motion of the
string can be expressed as a superposition
of eigenmode oscillations in accordance
with Fourier theory.
1653.14 CLASSICAL EIGENSTATES, RESONANCE, AND COUPLINGS
II. EXAMPLES FROM NMR THEORY
The advantage of this description is clear when the spectrum of the guitar sound is of interest.
It will be peaked around the frequencies of the eigenmodes, and the phase of each peak
is determined by the initial phase associated with the excitation. This classical description of
the string oscillation is analogous to the description of precession made earlier, except
that each nuclear spin supports only two eigenmodes, whereas a guitar string supports more.
Similarly, the frequency spectra are related.
A more complicated system may exhibit a spectrum of resonances more interesting than
the equally spaced harmonics of a string. A number of masses connected by springs, for
example, may exhibit a spectrum of vibrational resonances with structure similar to that of
a molecule where the nuclei are interacting magnetically, that is, a spectrum of a finite number
of discrete peaks.
Focus is now shifted to a particular classical example that will give additional insight into
eigenmodes, magnetic resonance, and spectral features: two coupled oscillators. A physical
realization can, for example, be two pendulums that are well separated but both hanging
down from a horizontal third string. If one pendulum is set in motion (excited) orthogonally
to the connecting string, it will exert a weak force on the other pendulum via their common
suspension. Energy may therefore be transferred between the pendulums, in particular if they
are similar so they have the same natural oscillation frequency. The focus is on such similar
pendulums in the following discussion.
If you do the experiment or search for “coupled pendulums” on the YouTube™ video
community, for example, you may see the following: when one pendulum is set in motion,
while the other is initially at rest, the amplitude of oscillation of the swinging pendulum gradually
decreases, while the other begins to swing. After a while, the energy is completely transferred
to the other pendulum, and simultaneously, the first one comes to a brief stop. It will
soon regain oscillation, however, since energy is transferred back from the other pendulum.
The energy will oscillate back and forth between the two pendulums at a frequency that can
be much smaller than the pendulum oscillation frequency. This frequency is proportional
to the coupling strength of the two pendulums that is characteristic of their common
suspension.
The dynamics of this system is analogous to nuclear excitation, which may not be
immediately clear, but follows from doing a quantization of the electromagnetic field. This
is implicit whenever the concept of photons is engaged, but it is seldom spelled out for
NMR, despite photons often being mentioned in that context. Field quantization will be done
here to explain the role of eigenstates, even though photons are probably best left out of QM
introductions for reasons that will soon be discussed.
To establish the correspondence between two coupled pendulums and nuclear excitation,
we first think of the nucleus and the electromagnetic field as separate storages of energy.
When a spin is aligned with the static field, it is in the ground state. During excitation, energy
is transferred from the field to the nuclear spin, mediated by dipolar interaction. It will be
transferred back if the RF field is kept on, so the tip angle exceeds 180° (yes, an RF pulse
can remove spin energy). We can express the dynamics in terms of two weakly coupled
combined states of the field and nucleus: the state Sz ¼ 1
⁄2,N + 1j i has the nuclear spin in
the ground state and N + 1 photons in the field. The state Sz ¼ À1
⁄2,Nj i has the nucleus in
the excited state after absorbing one photon from the field, which is left with N photons.
As before, any normalized superposition of the two states is possible.
166 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
Each photon carries an energy ℏo so the Sz ¼ 1
⁄2,N + 1j i and Sz ¼ À1
⁄2,Nj i states have the
same energy (are “degenerate”), if the frequency of the electromagnetic field is matched to
the Larmor frequency, that is, satisfies the resonance condition. Please note that both states
are associated with almost equally strong electromagnetic fields with only one photon difference.
These two states are only energy eigenstates in the absence of coupling between field
and nucleus. Introducing coupling corresponds to placing the nucleus in the electromagnetic
field. Rabi and coworkers realized this experimentally in 1937 when NMR was first detected
in a molecular beam sent through a region with an oscillating magnetic field matched to the
Larmor frequency.16
Once dipolar coupling between the Sz ¼ 1
⁄2,N + 1j i and the Sz ¼ À1
⁄2,Nj i states is introduced,
they are no longer energy eigenstates. Couplings between degenerate states remove the
degeneracy, and the new energy eigenstates are equal mixes of the former (superpositions
with equal amplitude of the coefficients). It is convenient for the following discussion to adopt
conventional naming: The states Sz ¼ 1
⁄2,N + 1j i and Sz ¼ À1
⁄2,Nj i specify the state of field and
nucleus separately and they are therefore called “bare” states in contrast to the “dressed”
states that are energy eigenstates after dipolar coupling is introduced, for example, when
the nucleus enters the region with field (e.g., as in the Rabi molecular beam experiment16
).
The nuclei are “dressed” in the field.7
The dressed states are equal mixes of the bare states,
having well-defined energies with an energy separation proportional to the coupling strength
between the bare states. This is itself proportional to the photon count N and therefore to the
RF field amplitude B1. Specifically, the energy difference between the dressed states is ℏgB1 on
resonance. When the system is in a superposition of dressed states, the phase difference between
the components will therefore evolve at a frequency gB1 in accordance with Eq. (3.4). As
shown below, the components will sometimes add up to the bare state Sz ¼ 1
⁄2,N + 1j i and
sometimes to the bare state Sz ¼ À1
⁄2,Nj i. This oscillation back and forth is the nutation (or
Rabi oscillation) happening during excitation.
In fact, the electromagnetic fields generated by NMR coils are far from being phase-free
monochromatic photon states jNi that are closer to representing laser light and most appropriate
for cavity electrodynamics.7
The field inside an NMR coil is instead a superposition of
modes of the electromagnetic field,6,8
so results may be misleading if the above QM description
is taken much further without representing the field realistically. This is an argument for
not mentioning photons in NMR introductions—doing it right is far from simple, and it is not
necessary as RF fields are typically better treated classically in this context.6,8
This path will
therefore not be followed far, but the relation to classical coupled oscillators (pendulums)
will be shown, as well as the role of eigenstates, while staying with monochromatic electromagnetic
fields.
3.15 THE EIGENMODE STRUCTURE FOR NUCLEAR EXCITATION
Energy eigenstates represent states of simple oscillation and their phases evolve proportionally
to their energies. As such, each of the states Sz ¼ 1
⁄2,N + 1j i and Sz ¼ À1
⁄2,Nj i is
represented as oscillators in the absence of coupling, and their phase evolves at the same rate
since their energy is equal when the photon energy is matched to the Larmor frequency. We
1673.15 THE EIGENMODE STRUCTURE FOR NUCLEAR EXCITATION
II. EXAMPLES FROM NMR THEORY
can take uncoupled pendulums as a mechanical analogy since they have the same characteristics
and eigenstate structure. Each state corresponds to a pendulum. The physical nature of
the systems and that of the oscillations are different, but similar dynamics occur and the mathematical
description can be shared.
When coupling between the bare states is introduced, the energy eigenstates are changed.
Two modes of oscillation are now stationary, that is, they represent states of simple oscillation.
For the coupled pendulums, the eigenmodes are simple to imagine: If the two pendulums
are started in synchrony, they will remain that way despite the coupling. Since they are
similar and experience the same mutual interaction, a simple symmetry argument tells us that
they will keep oscillating in synchrony. Similarly, if the two pendulums are released in an
antisymmetrical state, they will remain oscillating with opposite phase. We therefore know
the two patterns of simple oscillation for coupled pendulums (the eigenstates), symmetrical
and antisymmetrical oscillation. The two eigenstates evolve at different frequencies, however,
since the pendulums interact differently in the two situations: When the pendulums oscillate
with opposite phase, they pull each other back towards equilibrium, which increases
the oscillation frequency compared to in-phase oscillation. The energies of the eigenmodes of
coupled pendulums (corresponding to dressed states) are therefore different, so the degeneracy
of the eigenstates is removed. The corresponding normalized symmetrical and
antisymmetrical QM dressed states are
+j i ¼ 1=
ffiffiffi
2
p
Sz ¼ 1
⁄2,N + 1j i + Sz ¼ À1
⁄2,Nj ið Þ
Àj i ¼ 1=
ffiffiffi
2
p
Sz ¼ 1
⁄2,N + 1j iÀ Sz ¼ À1
⁄2,Nj ið Þ:
All of this now comes together in a discussion of NMR excitation and, equivalently, pendulum
motion. Initially, we will consider the uncoupled case starting from z-polarization, for
example, a system in the bare state Sz ¼ 1
⁄2,N + 1j i. The state does not immediately change
when the nucleus enters the electromagnetic field at time t ¼ 0, but the energy eigenstates
do. Expressed in terms of these, the state is
c t ¼ 0ð Þj i ¼ Sz ¼ 1
⁄2,N + 1j i ¼ 1=
ffiffiffi
2
p
+j i + Àj ið Þ (3.5)
A superposition of the symmetrical and antisymmetrical states is present as long as the RF
field is on, but the phases of the coefficients will evolve at a rate determined by the energy of
each eigenstate. The common phase is unimportant as it is not observable, but the energy difference
ℏgB1 mentioned above gives rise to an important phase difference gB1t:
c tð Þj i ¼ 1=
ffiffiffi
2
p
+j i + exp ÀigB1tð Þ Àj ið Þ:
A similar classical description can be made for the pendulums that each correspond to a
bare state. To determine the dynamics after the pendulum associated with Sz ¼ 1
⁄2,N + 1j i is set
in motion, and coupling is introduced, it is convenient to express the initial state in terms of
the symmetrical and antisymmetrical oscillation patterns as done above. Oscillation of just
one pendulum can be interpreted as equal amounts of symmetrical and antisymmetrical
oscillation, so the motion of the other pendulum cancels as expressed in Eq. (3.5). The components
of the superposition will evolve with different frequencies, however, since
antisymmetrical motion has a shorter period. Consequently, both pendulums will soon
168 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
swing, until the time comes where all energy is completely transferred to the other pendulum.
This follows from the analogous calculation for the NMR experiment where this characteristic
time is t ¼ p=gB1ð Þ:
c tð Þj i ¼ 1=
ffiffiffi
2
p
+j i + exp ÀigB1tð Þ Àj ið Þ
¼ 1=2 Sz ¼ 1
⁄2,N + 1j i + Sz ¼ À1
⁄2,Nj ið Þ + exp ÀigB1tð Þ Sz ¼ 1
⁄2,N + 1j iÀ Sz ¼ À1
⁄2,Nj ið Þð Þ
¼ 1=2 1 + exp ÀigB1tð Þð Þ Sz ¼ 1
⁄2,N + 1j i + 1Àexp ÀigB1tð Þð Þ Sz ¼ À1
⁄2,Nj ið Þ
c t ¼ p=gB1ð Þj i ¼ Sz ¼ À1
⁄2,Nj i:
After yet another period p/gB1, the magnetization has been rotated a full round and is back
in the longitudinal direction:
c t ¼ 2p=gB1ð Þj i ¼ Sz ¼ 1
⁄2,N + 1j i:
This is all consistent with the motion of coupled pendulums where the energy also oscillates
back and forth at a frequency determined by the coupling strength. Note that in the basis
of dressed eigenstates there are no transitions whatsoever, only independent phase evolution
of each component of the superposition of eigenstates. The bare states are only energy
eigenstates when the dipolar field coupling is ignored. Expressed in terms of these, dipolar
coupling causes smooth transitions, reflecting coherent evolution. There are no sudden jumps
between states in any basis, even though such are often said to be the source of spectral
features.
The previous example hopefully gave insight into the similarities of classical and quantum
dynamics. It was covered in considerable detail to give a feel for eigenstates, both bare vs.
dressed and quantum vs. classical. The close relation between NMR and pendulum motion
is not incidental since energy eigenstates are indeed oscillatory, as are pendulum modes of
oscillation. It was crucial to include the electromagnetic field in the quantum treatment to establish
the connection, which can be done in several ways. The analogy extends further since
the corresponding formulas remain consistent in the case where there is a frequency offset,
that is, when the RF field is not applied exactly on resonance.
3.16 J-COUPLING
The coupled oscillator analogy can also give insight into effects of scalar J-coupling.
This intramolecular interaction between nuclei is mediated by the electronic cloud. Considering
the situation from a classical perspective can give insight into the nature of spectral
features. This will be done with much less rigor compared to the description of magnetic
resonance in the last section.
It is not surprising classically that nuclei interact magnetically via electrons that are also
magnetic. J-coupling itself is a quantum effect, however, and from a classical perspective
the size of the coupling is surprising, and the effect should not be observable in the spectra
(nonclassical exchange interaction is responsible). Let us nevertheless take J-coupling for
granted and consider what the effect on the spectra may be. We focus on the simplest situation
where two nuclei after excitation have the same chemical shift but are J-coupled. This sounds
much like the pendulum case, and we may classically expect similar behavior. When two
1693.16 J-COUPLING
II. EXAMPLES FROM NMR THEORY
similar classical oscillators are coupled, the eigenmodes of oscillation are changed. Symmetrical
and antisymmetrical oscillation patterns result as described above. The degeneracy is
removed by the coupling, and the energy separation of the eigenmodes is given by the coupling
constant. Thus, two resonances exist around the uncoupled Larmor frequency with a
frequency separation J. Exciting just one nucleus—or the antisymmetrical component—will
be difficult in this case, however, since the nuclei have the same chemical shift.
If the starting point is changed to J-coupling of two nuclei with different chemical shifts,
this corresponds to the coupling of two pendulums with different oscillation frequencies, for
example, having different lengths of the strings. Each of the two eigenstates is split into a
doublet separated by a frequency J when weak coupling is introduced. This sounds like
NMR, but the sentence describes coupled pendulums, which have four eigenmodes in this
case. When one of the pendulums is set in motion, this corresponds to a superposition of
eigenmodes that each evolve at a different frequency. As in the previous case, the oscillations
will be modulated with the difference frequencies, reflecting how energy is transferred
between pendulums (or within the molecule for the case of NMR).
3.17 THE AFTERMATH
The coverage of J-coupling presented above is too simple. After all, the dynamics of just a
single nucleus in an electromagnetic field was shown above to be accurately represented by
two coupled oscillators. The much more complex system of two J-coupled nuclei is now
represented by the same classical analogy, so obviously much was swept under the carpet
using simplifying assumptions and vague statements. The main idea was not to give detailed
insight into J-coupling, but to show that spectral features and peak splittings similar to those
observed by NMR are not unexpected in classical systems. Spectral complexity is often taken
as a sign of quantum mechanics’ role in NMR, but this conclusion cannot be based on discrete
spectral features alone (Trap #25). A simple system of masses connected by springs may have
a qualitatively similar spectrum of resonances, for example, including peak splittings, amplitude
modulation, and power broadening. QM is necessary for most NMR, however, to get not
only the details right but also the larger picture, especially when coherent nuclear interactions
are involved. This is exemplified by J-coupling itself being nonclassical and by NMR quantum
computing implemented with coherently coupled nuclei, which clearly exposes
nonclassical aspects of spin systems.17
Even when a classical analogy or description can be
made, it is typically not worth using it quantitatively, as the quantum formalism is analogous,
and is known to give precise results. An exception is simple spin ½ NMR as used in almost all
MRI, since the two descriptions are equal.
For relaxation, the situation is somewhat similar. Intuitive classical arguments will tell you
much about relaxation mechanisms and the dependency on, for example, nuclear mobility,
but it certainly will not tell you the whole story, and you must always be ready to question the
validity of results obtained from classical arguments (Trap #19). QM and classical mechanics
typically give qualitatively similar results, so when they do not agree, there is reason to expect
a flaw in one of the derivations. Classical mechanics is therefore also a useful tool for sanity
checking, including avoidance of Mental Traps.
170 3. THE UPS AND DOWNS OF CLASSICAL AND QUANTUM NMR FORMULATIONS
II. EXAMPLES FROM NMR THEORY
The description of coupled oscillators was not included to suggest an alternative approach
to explaining NMR. A brief introduction to classical NMR consistent with Chapter 2, for example,
leading up to a good QM introduction is typically a better choice. Consistent classical
and quantum vector descriptions need not be long or complicated and can advantageously be
supplemented with animations and simulations.14
The focus was on deeper NMR understanding
and especially on making it clear that QM and classical descriptions of NMR can
be quite similar. Some of the mistakes of typical QM interpretations were pointed out as well
as Mental Traps responsible for their continued existence in literature describing basic NMR.
Many aspects were obviously covered superficially, and assumptions and details were left
out for the sake of focus and readability. Errors were likely also introduced or repeated. Science
involves getting in and out of Mental Traps continuously.
Acknowledgments
Dr. Steven J. van Enk is gratefully acknowledged for insightful suggestions for this chapter. I am also grateful to Dr.
Csaba Sza´ntay for the invitation to participate in this exciting book project and for the necessary enthusiastic
encouragement.
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