Lecture 6: Ensemble of non-interacting spins 1 particle: Ψ(x, y, z, cα) 28 particles: Ψ(x(O), x(H1), x(H2), x(e1), x(e2), . . . 1 000 000 000 000 000 000 000 000 000 particles: Ψ ??? Ψ(x, y, z, cα) Ψ = 1 h3 · e i pxx · e i pyy · e i pzz ·   1 0   Ψ = φ(x(O), x(H1), x(e1), . . .) · ψ(cα,1, cα,2) ? • electron motions: > 1016 s−1 • molecular rotations: 108 s−1 (20 kDa protein) to 1012 s−1 (water) • magnetic moment precession: ∼ 109 s−1 electrons as ”blurred cloud of a given shape”: • electron motions: > 1016 s−1 • molecular rotations: 108 s−1 (20 kDa protein) to 1012 s−1 (water) • magnetic moment precession: ∼ 109 s−1 at B0 = 24 T electrons as ”blurred cloud of a given shape”: • electron motions: 1016 s−1 • molecular rotations: 108 s−1 (20 kDa protein) to 1012 s−1 (water) • magnetic moment precession: ∼ 109 s−1 at B0 = 24 T molecular motions ← almost independent → magnetic moment precession ← almost independent → Ψ = φ(x1, x2, x3, . . .) · ψ(c1, c2, c3, . . .) ψ(c1, c2, c3, . . .) =                              c1 c2 c3 c4 c5 c6 ... ... ... c99999999999999999999999998 c99999999999999999999999999 c100000000000000000000000000                              ψ(c1, c2, c3, . . .) = ψ1(c1) · ψ2(c2) · ψ3(c3) . . . ? Is it possible to separate ψ of individual magnetic moments? Yes, if interactions of magnetic moments • depend only on external fields ⇒ interactions change energy eigenvalues, not eigenfunctions • the external fields are homogeneous (same in the whole sample) not true in MRI. Then,    1 0    and    0 1    form basis for all ψ’s ⇒ operators are represented by 2 × 2 matrices. Pure state: Expected value A of a quantity A for single nucleus: A = Tr        cαc∗ α cαc∗ β cβc∗ α cβc∗ β        A11 A12 A21 A22       Mixed state: Expected value A for multiple nuclei with the same basis: A = Tr      cα,1c∗ α,1 cα,1c∗ β,1 cβ,1c∗ α,1 cβ,1c∗ β,1     A11 A12 A21 A22   +   cα,2c∗ α,2 cα,2c∗ β,2 cβ,2c∗ α,2 cβ,2c∗ β,2     A11 A12 A21 A22   + · · · = Tr        cα,1c∗ α,1 cα,1c∗ β,1 cβ,1c∗ α,1 cβ,1c∗ β,1   +   cα,2c∗ α,2 cα,2c∗ β,2 cβ,2c∗ α,2 cβ,2c∗ β,2   + · · ·     A11 A12 A21 A22      = NTr      cαc∗ α cαc∗ β cβc∗ α cβc∗ β   ˆρ   A11 A12 A21 A22   ˆA    = NTr ˆρ ˆA . ˆρ is the (probability) density matrix • Two-dimensional basis for N uncoupled nuclei. • Statistical approach: macroscopic result – mixed state no insight into microscopic states. • Choice of the basis of ψ is encoded in definition of ˆρ (eigenfunctions of ˆIz) • The state is described not by a vector, but by a matrix ˆρ is a matrix like matrices representing the operators. Populations • Diagonal elements of ˆρ or matrices with diagonal elements only. • describe longitudinal polarization of µ (distribution along B0) • real numbers, cαc∗ α + cβc∗ β = 1 • cαc∗ α = 1/2: no net polarization along B0 equal populations of the α and β states It does not indicate that all spins must be either in α or β state! Any combination of superposition states, µ pointing in all possible directions as long as Mz = 0 Probability of 50 % spins in α state, 50 % spins in β state negligible M B0 Coherences • Off-diagonal elements or matrices with diagonal elements only • Pure state: cβc∗ α = |cα||cβ|e−i(φα−φβ) • Mixed state: cβc∗ α is complex number |cα||cβ| · e−i(φα−φβ) amplitude |cα||cβ|, phase given by e−i(φα−φβ) • cβc∗ α · cαc∗ β = 1 • Describe transverse polarization of µ in the plane ⊥ B0 with magnitude |cα||cβ| and in direction given by the phase. • Incoherent superposition of states α, β: e−i(φα−φβ) = 0 ⇒ cβc∗ α = 0 • Coherent superposition of states α, β: cβc∗ α = 0 • Coherent evolution: φα,j and φβ,j vary, but with identical frequency ω0 for all j: e−i(φα−φβ) = e−i(φα(0)−φβ(0)) · eiω0t M B0 Phases and coherences |ϑj, ϕj =       cos ϑj 2 e−i ϕj 2 sin ϑj 2 e+i ϕj 2       =    cα,j cβ,j    = cα,j|α + cβ,j|β cβc∗ α = cos ϑ 2 sin ϑ 2 e+iϕ = 1 2 sin ϑe+iϕ 2 sin a cos a = sin(2a) Phases and coherences Independent distribution of ϑ and ϕ: cβc∗ α = cos ϑ 2 sin ϑ 2 e+iϕ = 1 2 sin ϑe+iϕ = 1 2 sin ϑ · e+iϕ Evolution in B0: Hamiltonian ˆH = −γB0ˆIz = ω0ˆIz cα(t) = cα(t = 0)e+i γB0 2 t = cos ϑ 2 e−iϕ(t=0) 2 e−i ω0 2 t cβ(t) = cβ(t = 0)e−i γB0 2 t = sin ϑ 2 e+iϕ(t=0) 2 e+i ω0 2 t cβc∗ α(t) = 1 2 sin ϑ eiϕ(t=0) eiω0t Coherent evolution if ω0 = γB0 is the same for all j Phases and coherences Random distribution of ϕ(t = 0) ⇒ ϑ distributed on the whole sphere eiϕ(t=0) = 0 ⇒ cos ϑ = 0 sin ϑ = 0 Incoherent superposition of |α and |β Phases and coherences Random distribution of ϕ(t = 0) ⇒ ϑ distributed on a meridian eiϕ(t=0) = 1 ⇒ cos ϑ = 0 sin ϑ = 1 2 Coherent superposition of |α and |β Basis: • Any 2 × 2 matrix can be written as a linear combination of four 2 × 2 matrices. Such four matrices can be used as a basis Example:    a b c d    = a    1 0 0 0    + b    0 1 0 0    + c    0 0 1 0    + d    0 0 0 1    Basis: • A good basis is a set of orthonormal matrices: Tr{ ˆA † j ˆAk} = δjk j, k ∈ {1, 2, 3, 4}, δjk = 1 for j = k, δjk = 0 for j = k, ˆA † j is an adjoint matrix of ˆAj (adjoint matrix: matrix obtained from ˆAj by exchanging rows and columns and replacing all numbers with their complex conjugates.) E.g., for    a b c d    and    e f g h   : calculate    a∗ c∗ b∗ d∗       e f g h    =    a∗e + c∗g a∗f + c∗h b∗e + d∗g b∗f + d∗h    Trace: a∗e + c∗g + b∗f + d∗h Example:    a b c d    = a    1 0 0 0    + b    0 1 0 0    + c    0 0 1 0    + d    0 0 0 1    The basis is orthonormal, e.g.:    1 0 0 0       0 1 0 0    =    0 1 0 0   , trace: 0 + 0 = 0    1 0 0 0       1 0 0 0    =    1 0 0 0   , trace: 0 + 0 = 1 Basis sets: Cartesian √ 2It = 1 √ 2    1 0 0 1    √ 2Iz = 1 √ 2    1 0 0 −1    √ 2Ix = 1 √ 2    0 1 1 0    √ 2Iy = 1 √ 2    0 −i i 0    (1) It = 1 2 ˆ1 Ix = 1 ˆIx Iy = 1 ˆIy Iz = 1 ˆIz (2) Basis sets: Single-element Iα = It + Iz =    1 0 0 0    Iβ = It − Iz =    0 0 0 1    I+ = Ix + iIy =    0 1 0 0    I− = Ix − iIy =    0 0 1 0    (3) Basis sets: Mixed √ 2It = 1 √ 2    1 0 0 1    √ 2Iz = 1 √ 2    1 0 0 −1    I+ =    0 1 0 0    I− =    0 0 1 0    (4) Liouville - von Neumann equation evolution in time dˆρ dt = i (ˆρ ˆH − ˆHˆρ) = i [ˆρ, ˆH] = − i [ ˆH, ˆρ] (5) or in the units of (angular) frequency dˆρ dt = i(ˆρH − H ˆρ) = i[ˆρ, H ] = −i[H , ˆρ]. (6) H = 1 ˆH (7) If ˆρ = cIj, H = ωIl, and [Ij, Ik] = iIl, then the density matrix evolves as ˆρ = cIj −→ cIj cos(ωt) + cIl sin(ωt) rotation about Ik in abstract 3D space defined by the basis Ij, Ik, Il. I1z −I1x I1y −I1z ˆρ −I1y I1x ωt H = ωI1z General strategy 1. Define ˆρ at t = 0 2. Describe evolution of ˆρ using the relevant Hamiltonians usually several steps 3. Calculate the expectation value of the measured quantity (magnetization components in the x, y plane) according to Eq. 1 M+ = Mx + iMy = NTr ˆρ ˆM+ The procedure requires knowledge of 1. relation(s) describing the initial state of the system (ˆρ(0)) 2. all Hamiltonians (H ) 3. the operator representing the measurable quantity ( ˆM+) HOMEWORK: ˆρ(0) = Iy H = ωIz ω = π × 105 rad/s t = 2.5 × 10−5 s I1z −I1x I1y −I1z ˆρ −I1y I1x ωt H = ωI1z SOLUTION: ˆρ(0) = Iy H = ωIz ω = π × 105 rad/s t = 2.5 × 10−5 s ˆρ(0) = Iy → Iy cos(ωt) − Ix sin(ωt) ωt = 2.5π = 2π + π 2 cos(ωt) = cos 2π + π 2 = cos π 2 = 0 sin(ωt) = sin 2π + π 2 = sin π 2 = 1 ˆρ(0) = Iy → −Ix