Solutions:

Homework Set 6
Due October 8, 2020

1. Find the eigenvalues and eigenvectors of

!
0 −i
σy = . (1)
i 0

!
α
Suppose an electron is in the spin state . If Sy is measured, what is the probability of
β
the result +~/2, in terms of α and β?

The two eigenvalues are λ± = ±1. The corresponding eigenvectors are

!
1 −i
ψ+ = √ (2)
2 1
!
1 i
ψ− = √ (3)
2 1

The spin operator is Sy = ~/2σy , and so has eigenvalues ±~/2.

The probability of measuring +~/2 is
 ! 2  2
1  α  1  1 
 √ (iα + β)
hψ+ | i = 2 (4)

2 2
α +β  α + β2  2
β  

α2 + β 2 + 2Im(αβ ∗ )
= (5)
2(α2 + β 2 )
1 Im(αβ ∗ )
= − 2 (6)
2 α + β2

1
2. In the 2-dimensional space of a spin-1/2 particle, consider a sequence of Euler rotations
represented by
     
−iσ3 α −iσ2 β −iσ3 γ
U (α, β, γ) = exp exp exp (7)
2 2 2
!
−i(α+γ)/2 β −i(α−γ)/2 β
e cos 2 −e sin 2
= β
. (8)
e i(α−γ)/2
sin 2 e i(α+γ)/2
cos β2

Because of the group properties of rotations, we expect that this sequence of operations is
equivalent to a single rotation about some axis by an angle θ. Find θ.

In terms of an axis n̂ and an angle θ, the rotation operator in spin-1/2 space is (see, e.g.,
Sakurai 3.2.45)
!
− iθ n̂·~
σ cos 2θ − inz sin 2θ (−inx − ny ) sin 2θ
U (n̂, θ) = e 2 = (9)
(−inx + ny ) sin 2θ cos 2θ + inz sin 2θ

Setting these equal, one can solve for (n̂, θ) in terms of (α, β, γ), or vice versa. The easiest
way to solve for θ is to equate the trace of each matrix

θ
TrU (n̂, θ) = 2 cos (10)
2
β α+γ
= TrU (α, β, γ) = 2 cos cos (11)
2 2 
−1 β α+γ
=⇒ θ = 2 cos 2 cos cos (12)
2 2

3. (a) Consider a pure ensemble of identically prepared spin-1/2 systems. Suppose the expec-
tation values hSx i and hSz i are known, as well as the sign of hSy i.
Show how we may determine the state vector. Why is it unnecessary to know the
magnitude of hSy i?

A general state in a spin 1/2 system can be written as a superposition of eigenstates of

Sz

2
 |ψi = C+ |+i + C− |−i (13)

with

~
Sz |+i = |+i (14)
2
~
Sz |−i = − |−i (15)
2

Normalization gives the condition |C+ |2 + |C− |2 = 1, and so the state can be written as

β β
|ψi = cos eiφ+ |+i + sin eiφ− |−i (16)
2 2

for some (real) parameter β. Since the overall phase of a ket is irrelevant, we can
multiply by e−i(φ+ +φ− )/2

β β
|ψi = cos ei(φ+ −φ− )/2 |+i + sin e−i(φ+ −φ− )/2 |−i (17)
2 2
β iα/2 β −iα/2
= cos e |+i + sin e |−i (18)
2 2

with α = (φ+ − φ− )/2.

So the state is completely determined by two (real) parameters α and β.
hSx i and hSz i are

~  
hSz i = hψ| |+ih+| − |−ih−| |ψi (19)
2 
~ 2 β 2 β
= cos − sin (20)
2 2 2
~
= cos β (21)
2
~  
hSx i = hψ| |+ih−| + |−ih+| |ψi (22)
2 
~ β −iα/2 β −iα/2 β iα/2 β iα/2
= cos e sin e + sin e cos e (23)
2 2 2 2 2
β β
= ~ sin cos cos α (24)
2 2
~
= sin β cos α (25)
2

3
 so knowing these, we can solve for β and cos α.
For the average hSy i, we have

~  
hSy i = −i hψ| |+ih−| − |−ih+| |ψi (26)
2 
~ β −iα/2 β −iα/2 β iα/2 β iα/2
= −i cos e sin e − sin e cos e (27)
2 2 2 2 2
~
= − sin β sin α. (28)
2

If we already know β and cos α, then we only need go know the sign of sin α, and so we
only need the sign of hSy i to completely determine the system.

(b) Consider a mixed ensemble of spin-1/2 systems. Suppose the ensemble averages [Sx ],
[Sy ], and [Sz ] are all known. Show how we may construct the 2×2 density matrix that
characterizes the ensemble.

Again using the Sz basis, we can parameterize the density matrix as

!
a b
ρ= . (29)
c d

The ensemble averages are

" ! !#
~ a b 0 1 ~
[Sx ] = Tr(ρSx ) = Tr = (b + c) (30)
2 c d 1 0 2
" ! !#
~ a b 0 −i ~
[Sy ] = Tr = i (b − c) (31)
2 c d i 0 2
" ! !#
~ a b 1 0 ~
[Sz ] = Tr = (a − c) (32)
2 c d 0 −1 2

We also need the normalization condition for any density matrix

Trρ = (a + d) = 1 (33)

4
Solving the 4 equations for (a,b,c,d):
 
1 ~
a = h+|ρ|+i = [Sz ] + (34)
~ 2
1
b = h+|ρ|−i = ([Sx ] − i[Sy ]) (35)
~
1
c = h−|ρ|+i = ([Sx ] + i[Sy ]) (36)
~  
1 ~
d = h−|ρ|−i = − [Sz ] − (37)
~ 2