1
Introduction to Quantum and Statistical Mechanics
Homework 2
Prob. 2.1.
We will observe the variance of an operator through the wave functions expanded by
the eigenfunctions of the operator.
The mathematical form is very helpful when we choose only
a few eigenfunctions for expansion in the later finitebase approximation.
Give an operator A
with eigenfuntions of
φ
i
and corresponding eigenvalues of
a
i
.
For a particle in an arbitrary state
described by
∑
=
=
n
i
i
i
c
t
x
1
)
,
(
φ
ψ
, where
c
i
’s are properly normalized, and
φ
i
forms an orthogonal
set, i.e.,
ij
j
i
δ
φ
φ
=

(a)
Express the variance by summation of terms with
c
i
and
a
i
.
(5 pts)
(b)
Prove that if
A
is Hermitian, the variance is always positive.
You will need to use the
CauchySchwarz inequality:
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
≤
⎟
⎠
⎞
⎜
⎝
⎛
∑
∑
∑
=
=
=
n
i
i
n
i
i
n
i
i
i
y
x
y
x
1
2
1
2
2
1
when
x
i
and
y
i
are real
numbers.
(10 pts)
(c)
Prove that if
A
is Hermitian, the variance is zero only if
ψ
(x,t)
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 '09
 KAN
 pts, wave function, Quantum and Statistical Mechanics

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