Quantum Mechanics for Mathematicians

We thus see that the Heisenberg group acts on annihilation and creation
operators by shifting the operators by a constant. The Heisenberg group acts
by automorphisms on its Lie algebra by the adjoint representation (see section
15.5), and one can check that the Γ(α,γ) are intertwining operators for this
action (see chapter 20). The constructions of this chapter can easily be general-
ized fromd= 1 to general values of the dimensiond. For finite values ofdthe
Γ(α,γ) act on states as an irreducible representation, as required by the Stone-
von Neumann theorem. We will see in chapter 39 that in infinite dimensions
this is no longer necessarily the case.

23.4 The harmonic oscillator propagator

In section 12.5 we saw that for the free particle quantum system, energy eigen-
states were momentum eigenstates, and in the momentum space representation
time evolution by a time intervalTwas given by a kernel (see equation 12.6)

U ̃(T,k) =√^1

2 π

e−i

21 mk^2 T

The position space propagator was found by computing the Fourier transform
of this. For the harmonic oscillator, energy eigenstates are no longer momentum
eigenstates and different methods are needed to compute the action of the time
evolution operatore−iHT.

23.4.1 The propagator in the Bargmann-Fock representa-

tion

In the Bargmann-Fock representation the Hamiltonian is the operator

H=ω

(

a†a+

1

2

)

=ω

(

z

d

dz

+

1

2

)

(here we choose~= 1 andm= 1, but no longer fixω= 1) and energy eigenstates
are the states
zn
√
n!

=|n〉

with energy eigenvalues

ω

(

n+

1

2

)

e−iHTwill be diagonal in this basis, with

e−iHT|n〉=e−iω(n+

(^12) )T
|n〉
Instead of the Schr ̈odinger picture in which states evolve and operators are
constant, one can instead go to the Heisenberg picture (see section 7.3) where
states are constant and operatorsOevolve in time according to
d
dt
O(t) =i[H,O(t)]