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Advanced Quantum Mechanics Schwabl
· Problem 12.7
Free Fields
Physics — Quantum Mechanics · coherent-states-linear-chain
Problem
The coherent states for the linear chain are defined as eigenstates of the annihilation operators
a
k
a_k
a
k
. Calculate the expectation value of the operator
q
n
(
t
)
=
∑
k
1
2
N
m
ω
k
[
e
i
(
k
n
a
−
ω
k
t
)
a
k
(
0
)
+
e
−
i
(
k
n
a
−
ω
k
t
)
a
k
†
(
0
)
]
q_n(t) = \sum_k \frac{1}{\sqrt{2Nm\omega_k}} \left[ e^{i(kna - \omega_k t)} a_k(0) + e^{-i(kna - \omega_k t)} a_k^\dagger(0) \right]
q
n
(
t
)
=
k
∑
2
N
m
ω
k
1
[
e
i
(
k
na
−
ω
k
t
)
a
k
(
0
)
+
e
−
i
(
k
na
−
ω
k
t
)
a
k
†
(
0
)
]
for coherent states.
Related theory
Coherent States
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