PHYS 455
Homework IV
Assigned: November 29, 2011, Tuesday.
Due: December 7, 2011, Wednesday.
1.
Consider an ensemble for a spin1/2 particle where the particle is in
“spinup along
z
” state with probability
p
1
= 1
/
6,
“spindown along
z
” state with probability
p
2
= 1
/
3,
and “spinup along
x
” state with probability
p
3
= 1
/
2.
(a)
Compute the density matrix
ρ
of this ensemble.
(b)
As an example, compute the expectation value of
σ
z
, by using (i) the density matrix
and (ii) the ensemble states and probabilities.
2.
Consider a mixed state with the following matrix
ρ
=
1
5
•
2
1 +
i
1

i
3
‚
.
(a)
Is this really a valid density matrix? (In other words, does
ρ
satisfy all of the condi
tions that a density matrix should satisfy?)
(b)
Compute
h
σ
x
i
,
h
σ
y
i
and
h
σ
z
i
. Express your results compactly by expressing
h
~σ
i
.
(b
0
)
The vector
h
~σ
i
expresses the position of the mixed state
ρ
inside the Bloch sphere.
Verify that the length of this vector is less than or equal to 1.
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 Spring '11
 starg
 Probability, Hilbert space, density matrix

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