PH403: Quantum Mechanics I

Tutorial Sheet 6
Problems in this tutorial sheet deal with the Stern-Gerlach experiment and the related
problems as discussed in chapter 4 of the book by Cohen-Tannoudji et al.
1. Consider a spin-1/2 particle of magnetic moment M = γS. The spin state space is
spanned by the basis of the |+i and |−i, eigenvectors of Sz with eigenvalues ~/2 and
−~/2. At the time t = 0, the state of the system is |ψ(0)i = |+i.

(a) If the observable Sx is measured at time t = 0, what results can be found and
with what probabilities?
(b) Instead of performing the preceding measurement, we let the system evolve un-
der the inuence of a magnetic eld of magnitude B0 , parallel to the y−axis.
Calculate in the {|+i, |−i} basis the state of the system at time t.
(c) At this time t, we measure the observables Sx , Sy , and Sz . What values can we
nd, and with what probabilities? What relation must exist between B0 and t
for the nal result of one of the measurements to be certain? Give a physical
interpretation of this condition.

2. Consider a spin-1/2 particle, as in the previous exercise.

(a) At time t = 0, we measure Sy and nd +~/2. What is the state vector |ψ(0)i
immediately after the measurement?
(b) Immediately after this measurement, we apply a uniform time-dependent eld
along the z -axis. The spin Hamiltonian H(t) is then written as
H(t) = ω0 (t)Sz

Assume that ω0 (t) is zero for t < 0 and t > T , and linearly increases from 0 to
ω0 when 0 ≤ t ≤ T (T is a given parameter whose dimensions are those of time).
Show that at time t the state vector can be written
1
|ψ(t)i = √ {eiθ(t) |+i + ie−iθ(t) |−i},
2
where θ(t) is a real function of t which you have to compute.
(c) At a time t = τ > T , we measure Sy . What results can we nd, and with what
probabilities? Determine the relation which must exist between ω0 and T in order
for us to be sure of the result. Give a physical interpretation.

3. Consider a spin-1/2 particle placed in a magnetic eld B0 = . Notation

√1 (B0 , 0, B0 )
2
is the same as in exercise 1.

(a) Calculate the matrix representing, in the {|+i, |−i} basis, the Hamiltonian op-
erator H , of the system.

1
 (b) Calculate the eigenvalues and eigenvectors of H .
(c) The system at time t = 0 is in the state |−i. What values can be found if the
energy is measured, and with what probabilities?
(d) Calculate the state vector |ψ(t)i at time t. At this instant, Sx is measured:
what is the mean value of the results that can be obtained? Give a geometrical
interpretation.
4. Consider a spin-1/2, of magnetic moment M = γS, placed in a magnetic eld B0 of
components Bx = −ωx /γ , By = −ωy /γ , Bz = −ωz /γ . We set ω0 = −γ|B0 |.
(a) Show that the time evolution operator of this spin is
U (t, 0) = e−iM t
where M is given by
1
M= {ωx Sx + ωy Sy + ωz Sz }
~
Calculate the matrix representation of M in the {|+i, |−i} basis and show that
1 ω0
M 2 = (ωx2 + ωy2 + ωz2 ) = ( )2
4 2
(b) Put the time evolution operator into the form
   
ω0 t 2i ω0 t
U (t, 0) = cos − M sin
2 ω0 2
(c) Consider a spin which at time t = 0 is in the state |ψ(0)i = |+i. Show that the
probability P++ (t) of nding it in state |+i at time t is
P++ (t) = |h+|U (t, 0)|+i|2
and derive the relation
ωx2 + ωy2
 
ω0 t
P++ (t) = 1 − sin2
ω02 2
5. Consider a two-level system with orthonormal basis vectors {|φ1 i, |φ2 i} and the Hamil-
tonian H = H0 + W dened by
H0 |φi i = Ei |φi i (i = 1, 2)
and
W |φ1 i = W21 |φ2 i; W |φ2 i = W12 |φ1 i
where Wji 's are the matrix elements of the operator W , and W12 = W21
∗
.
(a) Construct the matrix representing the Hamiltonian H in the given basis.
(b) Obtain the eigenvalues and eigenvectors of H .
(c) Consider a ket which at time t = 0 is given by |ψ(0)i = |φ1 i. Compute the
expression for the probability P12 (t) that at a later time t, the system will be in
state |φ2 i. This probability is dened by P12 (t) = |hφ2 |ψ(t)i|2 . Plot P12 (t) as a
function of time t.