Quantum Mechanics for Mathematicians

with solution
O(t) =eitHO(0)e−itH

In the harmonic oscillator problem we can express other operators in terms of
the annihilation and creation operators, which evolve according to

d

dt

a(t) =i[H,a(t)] =−iωa,

d

dt

a†(t) =i[H,a†(t)] =iωa†

with solutions
a(t) =e−iωta(0), a†(t) =eiωta†(0)

The Hamiltonian operator is time invariant.
Questions about time evolution now become questions about various prod-
ucts of annihilation and creation operators taken at various times, applied to
various Heisenberg picture states. Since an arbitrary state is given as a linear
combination of states produced by repeatedly applyinga†(0) to| 0 〉, such prob-
lems can be reduced to evaluating expressions involving just the state| 0 〉, with
various creation and annihilation operators applied at different times. Non-zero
results will come from terms involving

〈 0 |a(T)a†(0)| 0 〉=e−iωT

which forT >0 has an interpretation as an amplitude for the process of adding
one quantum to the lowest energy state at timet= 0, then removing it at time
t=T.

23.4.2 The coherent state propagator

One possible reason these states are given the name “coherent” is that they re-
main coherent states as they evolve in time (for the harmonic oscillator Hamil-
tonian), withαevolving in time along a classical phase space trajectory. If the
state att= 0 is a coherent state labeled byα 0 (|ψ(0)〉=|α 0 〉), by 23.2, at later
times one has

|ψ(t)〉=e−iHt|α 0 〉

=e−iHte−

|α 0 |^2

2

∑∞

n=0

αn 0

√

n!

|n〉

=e−i

(^12) ωt
e−
|α 0 |^2
2

∑∞

n=0

e−iωntαn 0

√

n!

|n〉

=e−i

(^12) ωt
e−
|e−iωtα 0 |^2
2

∑∞

n=0

(e−iωtα 0 )n

√

n!

|n〉

=e−i

(^12) ωt
|e−iωtα 0 〉 (23.9)
Up to the phase factore−i
(^12) ωt
, this remains a coherent state, with time de-
pendence of the labelαgiven by the classical time-dependence of the complex