Quantum Mechanics for Mathematicians

plot gives the wavefunctions, which in this case are real and can be negative.
The square of this function is what has an interpretation as the probability
density for measuring a given position.

| 2 〉

E=^52 ~ω

| 1 〉

E=^32 ~ω

| 0 〉

E=^12 ~ω

Figure 22.1: Harmonic oscillator energy eigenfunctions.

While we have preserved constants in our calculations in this section, in what
follows we will often for simplicity set~=m=ω= 1, which can be done by
an appropriate choice of units. Equations with the constants can be recovered
by rescaling. In particular, our definition of annihilation and creation operators
will be given by

a=

1

√

2

(Q+iP), a†=

1

√

2

(Q−iP)

22.3 The Bargmann-Fock representation

Working with the operatorsaanda†and their commutation relation

[a,a†] = 1

makes it clear that there is a simpler way to represent these operators than
the Schr ̈odinger representation as operators on position space functions that we
have been using, while the Stone-von Neumann theorem assures us that this will
be unitarily equivalent to the Schr ̈odinger representation. This representation
appears in the literature under a large number of different names, depending on
the context, all of which refer to the same representation:

Definition(Bargmann-Fock or oscillator or holomorphic or Segal-Shale-Weil
representation).The Bargmann-Fock (etc.) representation is given by taking as