Physics 580
Handout 11
12 October 2010
Quantum Mechanics I
webusers.physics.illinois.edu/
∼
goldbart/
Prof. P. M. Goldbart
3135 (& 2115) ESB
University of Illinois
1) Measurements
: A hermitean operator Λ has orthonormal eigenkets

λ
1
⟩
,

λ
2
⟩
,

λ
3
⟩
and

λ
4
⟩
.
The corresponding eigenvalues are
λ
1
= 3, and
λ
2
=
λ
3
=
λ
4
= 1.
In terms of the
{
λ
i
⟩}
basis, a certain state,

ψ
⟩
, is given by

ψ
⟩
=
i

λ
1
⟩
+

λ
2
⟩ −
i

λ
3
⟩ − 
λ
4
⟩
.
a) Calculate the probability of obtaining the result
λ
= 1 upon measurement of the
physical quantity which the operator Λ represents.
b) Calculate the probability of obtaining
λ
= 2. Why do you get this result?
c) Calculate the probability of obtaining
λ
= 3.
d) Suppose that the observation is made a large number of times on an identically prepared
state

ψ
⟩
. Calculate the mean of the values obtained for
λ
.
e) Calculate the meansquare value.
f) Is the meansquare value equal to the square of the mean value?
g) In the light of your answer to part (f), would you say that this quantum mechanical
system has ﬂuctuations?
h) Do classical systems ﬂuctuate?
i) Suppose
λ
is measured and the result
λ
= 1 is obtained. Calculate the state vector
immediately after the measurement is made.
j)
{
λ
i
⟩}
are simultaneously eigenkets of the operator Ω corresponding to the observable
ω
. Their
ω
eigenvalues are
ω
1
=
ω
2
=
ω
3
= 7, and
ω
4
= 5. Suppose Ω is measured
immediately after the result
λ
= 1 is obtained. Calculate the possible outcomes, and
their probabilities.
k) Suppose the result
ω
= 7 is obtained. Calculate the state vector immediately there
after? What are the possible results if an immediate measurement of
λ
is now made?
1
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2) Angular momentum (after Shankar, 4.2.1)
: Consider the following matrices repre
senting operators on the Hilbert space
V
3
(
C
):
L
x
↔
1
√
2
0
1
0
1
0
1
0
1
0
,
L
y
↔
1
√
2
0
−
i
0
i
0
−
i
0
i
0
,
L
z
↔
1
0
0
0
0
0
0
0
−
1
.
a) If
L
z
is measured, determine the possible values one can obtain.
b) Take the state in which
L
z
= 1. In this state, determine the values of
⟨
L
x
⟩
,
⟨
L
2
x
⟩
and
∆
L
x
.
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 Fall '10
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