1 The formalism
1. In a two dimensional Hilbert space with basis {|1⟩ , |2⟩} what is the
matrix representation of the operator ˆA = |1⟩⟨2|?
2. Show that a product of unitary operators is unitary.
3. Show that Unitary operators preserve the inner product between the
states they act on.
4. What is the Hermitean conjugate of an operator ˆA = |α⟩⟨β|?
5. Define the trace of an operator by using an orthonormal basis |n⟩ as
Tr( ˆA) =
n
⟨n| ˆA |n⟩ .
Show that the definition is independent of the choice of basis by introducing
a different orthonormal basis |n′
⟩ and using that both sets of
basis vectors are complete.
6. If {|n⟩} and {|n′
⟩} are two different sets of orthonormal basis vectors.
We may define the operator
ˆU =
n′=n
|n′
⟩⟨n|
which maps a state |n⟩ in the first basis to a state |n′
⟩ in the second
basis. Show that ˆU is a unitary operator.
7. Show that the eigenvalues of a unitary operator are complex numbers
of unit modulus.
8. Show that the eigenvectors of a unitary operator are mutually orthogonal
(if no degeneracy).
9. Show that cyclicity of the trace holds Tr( ˆA ˆB) = Tr( ˆB ˆA).
10. Show that Tr(|ψ⟩⟨χ|) = ⟨χ|ψ⟩.
11. Show that (|ψ⟩⟨χ|)†
= |χ⟩⟨ψ|.
12. By using the sesquilinearity of (·, ·), show that (|ψ⟩ , |χ⟩) = (|χ⟩ , |ψ⟩)⋆
.
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13. In a space with three basis vectors {|1⟩ , |2⟩ , |3⟩} we define an operator
ˆR according to its action on the basis states as
ˆR |1⟩ = |2⟩ ˆR |2⟩ = − |1⟩ ˆR |3⟩ = |3⟩
What is the matrix representative of this operator? If we have a state
|ψ⟩ = a |1⟩ + b |2⟩ + c |3⟩, what is its matrix representative? How does
the matrix representative of ˆR act on the matrix representative of |ψ⟩?
14. Consider the operator ˆD = −i d
dx
, defined on the space of differentiable
functions of x on the interval a ≤ x ≤ b with the inner product defined
as (f(x), g(x)) = b
a dxf⋆
(x)g(x). We may define various subspaces of
the space of differentiable functions by imposing boundary conditions.
• What are the boundary conditions that one have to impose to
make ˆD hermitian?
• What are the eigenfunctions and eigenvalues for the operator ˆD?
• Are the eigenfunctions part of the space on which we define ˆD?
For what boundary conditions are the eigenfunctions part of the
space on which ˆD is defined?
• What if a = −∞ and b = ∞?
15. What boundary conditions must be imposed on the functions {f(¯x)}
defined in some finite or infinite volume of space in order for the Laplace
operator ∆ = ∇2
to be Hermitian?
2 Path Integrals
1. Show that if |n⟩ are eigenstates of the Hamiltonian with energy En, the
propagator can be written as K(x, t; x′
, t′
) = n e− i
¯h
En(t−t′)
⟨x|n⟩⟨n|x′
⟩.
2. In a two dimensional Hilbert space with a basis of normalized eigenstates
of the hamiltonian |1⟩ and |2⟩ with energy eigenvalue E1 and
E2, write the time evolution operator in terms of the states |±⟩ =
1√
2
(|1⟩ ± |2⟩).
3. Assume that space consists of two points, x and y. We will try to
find the time evolution of the system by assuming that the probability
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amplitude at each time step ∆t to stay at the same point is given by
1 + iω∆t and the probability amplitude to change points is given by
iβ∆t where ω and β are arbitrary real numbers. Define the probability
amplitude (i.e the propagator)
Kxx(T) = To go from x at t=0 to x at t=T
Kxy(T) = To go from y at t=0 to x at t=T
Kyx(T) = To go from x at t=0 to y at t=T
Kyy(T) = To go from y at t=0 to y at t=T
If we divide the time interval into N pieces so that ∆t = T
N
, show that
Kxx(T) = Kxx(T − ∆t)(1 + iω∆t) + Kyx(T − ∆t)iβ∆t
Kyx(T) = Kyx(T − ∆t)(1 + iω∆t) + Kxx(T − ∆t)iβ∆t
Show that this gives a recursion relation that can be solved as
Kxx(T) =
1
2
(1 + i(ω + β)∆t)N
+ (1 + i(ω − β)∆t)N
Kyx(T) =
1
2
(1 + i(ω + β)∆t)N
− (1 + i(ω − β)∆t)N
which when we let N → ∞ becomes
Kxx = eiωT
cos(βT)
Kyx = ieiωT
sin(βT)
Is the probability conserved? What is the wavefunction at T for a
particle which is localized at x when t = 0? What is the wavefunction
at T for a particle with an initial wavefunction ψ(x) = 1√
2
, ψ(y) = 1√
2
?
4. A model of a moving wave-packet in 1 dimension is given by the wave-
function
N dpe−a
2
(p−p0)2
|p⟩
where a is a constant and N is the normalization factor. Determine N
and use the propagator of a free particle to find how the packet moves
in time. Interpret your result!
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5. Let |n⟩ be a complete set of eigenstates of the time independent Hamiltonian
ˆH where ˆH |n⟩ = En |n⟩ and with configuration space representation
ψn(x) = ⟨x|n⟩. Using these elements, write expressions for the
time evolution operator in the |k⟩ and |x⟩ basis i.e. find Akl and B(x, x′
)
in the expressions
ˆU(t, t′
) =
k,l
Akl |k⟩ ⟨l| = dx dx′
B(x, x′
) |x⟩ ⟨x′
|
6. Calculate the propagator for a particle in a linear potential
S[x(t)] = dt(
1
2
m ˙x2
− Fx)
using path integral methods. Here are some useful observations that
you might want to use
a) In the path integral, we sum over all paths with the prescribed
boundary conditions.
b) The sum will be the same if we shift all paths by some particular
fixed path.
c) Define the new path y(t) as the old path shifted by a solution of
the equations of motion xcl(t) so that y = x − xcl.
d) However, shifting a path satisfying a particluar boundary condition
by a fixed path gives a new path that usually does not satisfy
the same boundary condition. What boundary conditions should
y(t) fulfil if the classical solution xcl satisfies the same boundary
conditions as x?
e) Find a particular xcl with the same boundary conditions as x, i.e.
that begins at x′
at time t′
and ends at x at time t.
f) Show that the action S[y(t)] consists of only of a kinetic term and
a term dependent only on the boundary conditions. In particular
there is no potential for y(t).
g) The path integral over y(t) can now be done using the result for
the path integral of a free particle. Remeber that is is given by
Dxe
i
¯h
Sfree[x(t)]
=
m
2πi¯h(t − t′)
e
im(x−x′)2
2¯h(t−t′)
for a path that starts at x′
at time t′
and ends at x at time t.
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Check that your result agrees with the result of the previous problem.
3 Scattering theory
1. Analysis of the scattering of particles of mass m and energy E = ¯h2
k2
2m
from a fixed scattering center with characteristic length a finds the pase
shifts
sin δl =
(iak)l
(2l + 1)!
a) Derive a closed expression for the total cross section as a function
of the incident energy E.
b) At what values of E does the S-wave (l = 0) scattering give a
good estimate of σ?
2. Using the Born approximation, obtain an expression for the total cross
section for scattering of particles of mass m from the attractive Gaussian
potential
V (r) = −V0e− r2
a2
3. Consider a scattering situation in which only the l = 0 and l = 1 partial
waves have appreciable phase shifts. Discuss how the contribution of
the l = 1 wave affects the total cross section. How does it affect the
angular distribution of scattered particles? What sort of measurements
should be made to obtain an accurate value of δ0 and δ1 respectively?
4. Determine in the first Born approximation the differential cross-section
for the potential
V =
0 for r > R
−V0 for r < R
with V0 > 0. Sketch the dependence (using a computer if you wish) of
the cross-section on 1) the angle θ and 2) the energy.
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5. Consider the scattering of a particle by a repulsive delta function shell
potential
V (r) =
¯h2
γ2
2m
δ(r − R),
a) Set up an equation that determines the s-wave phase shift δ0 as a
function of k (remember that E = ¯h2
k2
2m
).
b) Assume now that γ is very large,
γ ≫
1
R
, k.
Show that if tan kR is not close to zero, the s-wave phase shift resembles
the hard-sphere result discussed in the lectures. Show also
that for tan kR close to (but not exactly equal to) zero, resonance
behavior is possible; that is, cot δ0 goes through zero from the positive
side as k increases. Determine approximately the positions
of the resonances keeping terms of order 1
γ
.
4 Relativistic QM
1. For the Dirac equation written in the ϕA, ϕB basis used in the lecture
notes, find the explicit form of the gamma-matrices and show that
they satisfy the Clifford algebra {γµ
, γν
} = 2gµν
. Find the plane wave
solutions.
2. In non-relativistic physics, the transformation between two inertial systems,
moving with a relative speed v, is through the Galileo transfor-
mation
x′
= x + vt
t′
= t
Assume that the wave function ψ(x′
, t′
) is a solution to the Schr¨odinger
equation
i¯h
∂
∂t′
ψ(x′
, t′
) = −
¯h2
2m
∇′2
ψ(x′
, t′
) + V (x′
, t′
)ψ(x′
, t′
)
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Show that the wave function ψ(x + vt, t) is not a solution of the
Schr¨odinger equation in the unprimed system
i¯h
∂
∂t
ψ(x + vt, t) ̸= −
¯h2
2m
∇2
ψ(x + vt, t) + V (x + vt, t)ψ(x + vt, t)
However, if we allow for a phase factor in the transformation of the
wave function ψ(x′
, t′
) → e
i
¯h
f(x,t)
ψ(x + vt, t) find the form of f that
makes it a solution. Interpret the result in the case where ψ is a plane
wave.
In relativistic physics on the other hand, the transformation between
two inertial systems, moving with a relative speed v (in the x-direction
for simplicity), is through the Lorentz transformation
t′
= γt + γ
v
c
x
x′
= γ
v
c
t + γx
y′
= y
z′
= z
where γ−2
= 1 − v2
c2 . If ϕ(t′
, x′
, y′
, z′
) is a solution to the Klein-Gordon
equation in the primed system, show that ϕ(γt+γ v
c
x, γ v
c
t+γx, y, z) is a
solution to the Klein-Gordon equation in the unprimed system without
any phase factor.
3. A plane wave solution to the Dirac equation can be written as
ψ(x) = u(p)e− i
¯h
p·x
where u(p) is a spinor. Find the matrix equation that u(p) has to satisfy
and analogously find an equation for ¯u = u†
γ0
. Use this to show
¯u(p)q/u(p) =
p · q
mc
if we chose u(p) to be normalized to 1. (q/ = γµ
qµ).
4. How would the Dirac equation look like in 2,3,4 and 5 space-time dimensions?
Find explicit representations of the gamma matrices in all
these cases and show that they satisfy the appropriate Clifford algebra.
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5. In three space-time dimensions, verify that one can choose the gamma
matrices as following
γ0
= σ3
γ1
= iσ1
γ2
= −iσ2
i.e., verify that they satisfy the appropriate Clifford algebra. Construct
the matrices
M01
=
1
4i
γ0
, γ1
M20
=
1
4i
γ2
, γ0
M12
=
1
4i
γ1
, γ2
and show that they satisfy the SO(1, 2) algebra
M01
, M20
= −iM12
M12
, M01
= iM20
M20
, M12
= iM02
which exept for the minus sign in the first row is the same as the algebra
of the rotation group SO(3). Show that under a rotation with angle θ
in the 12-plane, the spinors transform as
eiθM12
ψ =
ei θ
2 0
0 e−i θ
2
ψ1
ψ2
whereas under a boost in the 2 direction, the spinor transforms as
eiαM20
ψ =
cosh(θ
2
) sinh(θ
2
)
sinh(θ
2
) cosh(θ
2
)
ψ1
ψ2
How do ψ†
ψ and ψ†
γ0
ψ transform under these transformations?
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