Chapter 23

Coherent States and the

Propagator for the

Harmonic Oscillator

In chapter 22 we found the energy eigenstates for the harmonic oscillator us-
ing annihilation and creation operator methods, and showed that these give a
new construction of the representation of the Heisenberg group on the quantum
mechanical state space, called the Bargmann-Fock representation. This repre-
sentation comes with a distinguished state, the state| 0 〉, and the Heisenberg
group action takes this state to a set of states known as “coherent states”. These
states are labeled by points of the phase space and provide the closest analog
possible in the quantum system of classical states (i.e., those with a well-defined
value of position and momentum variables).
Coherent states also evolve in time very simply, with their time evolution
given just by the classical time evolution of the corresponding point in phase
space. This fact can be used to calculate relatively straightforwardly the har-
monic oscillator position space propagator, which gives the kernel for the action
of time evolution on position space wavefunctions.

23.1 Coherent states and the Heisenberg group action

Since the Hamiltonian for thed = 1 harmonic oscillator does not commute
with the operatorsaora†which give the representation of the Lie algebrah 3
on the state spaceF, the Heisenberg Lie group and its Lie algebra are not
symmetries of the system. Energy eigenstates do not break up into irreducible
representations of the group but rather the entire state space makes up such
an irreducible representation. The state space for the harmonic oscillator does
however have a distinguished state, the lowest energy state| 0 〉, and one can ask