Physics 580
Handout 15
9 November 2010
Quantum Mechanics I
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Homework 11
Prof. P. M. Goldbart
University of Illinois
1) Coherent states
: In quantum optics, the classical limit of quantum mechanics, quantum
ﬁeld theory and quantum statistical mechanics, there is a very important set of states, known
as
coherent states
, which may be constructed from the harmonic oscillator energy eigenstates
{
n
⟩}
. Each coherent state is labelled by a single complex number
z
; it is denoted

z
⟩
, and
deﬁned by

z
⟩ ≡
exp(
za
†
)

0
⟩
,
where the ket

0
⟩
is the harmonic oscillator ground state.
a) Show that the coherent state

z
⟩
is an eigenstate of the annihilation operator
a
. What
is its eigenvalue?
b) Evaluate
⟨
z

a
†
in terms of
⟨
z

and
z
∗
.
c) Show that (
d/dz
)

z
⟩
=
a
†

z
⟩
.
d) Show that the inner product of two coherent states is given by
⟨
z

z
′
⟩
= exp(
z
′
z
∗
)
.
Consider the function of two complex variables
f
(
z, z
′
). The operator obtained by inserting
a
†
for
z
and
a
for
z
′
is generally ambiguous because the order of the operators is not speciﬁed.
For any
f
we can unambiguously deﬁne an operator
f
(
a
†
, a
), in which the operator ordering
ambiguity is resolved by the prescription that, in any product, all annihilation operators
lie to the right of all creation operators. Operators of this form are called
normal ordered
operators.
e) Show that for normal ordered operators we have
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 Fall '10
 Staff
 mechanics, Work, coherent, coherent states

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