Quantum Mechanics for Mathematicians

A.2 Fourier transforms

The Fourier transform is defined by

f ̃(k) =√^1

2 π

∫+∞

−∞

f(q)e−ikqdk

except for the case of functions of a time variablet, for which we use the opposite
sign in the exponent (and â·instead of ̃·), i.e.

f̂(ω) =√^1

2 π

∫+∞

−∞

f(t)eiωtdω

A.3 Symplectic geometry and quantization

The Lie bracket on the space of functions on phase spaceMis given by the
Poisson bracket, determined by

{q,p}= 1

Quantization takes 1,q,pto self-adjoint operators 1 ,Q,P. To make this a uni-
tary representation of the Heisenberg Lie algebrah 3 , multiply the self-adjoint
operators by−i, so they satisfy

[−iQ,−iP] =−i 1 , or [Q,P] =i 1

In other words, our quantization map is the unitary representation ofh 3 that
satisfies
Γ′(q) =−iQ, Γ′(p) =−iP, Γ′(1) =−i 1
Dynamics is determined classically by the Hamiltonian functionhas follows
d
dt
f={f,h}

After quantization this becomes the equation

d

dt

O(t) = [O,−iH]

for the dynamics of Heisenberg picture operators, which implies

O(t) =eitHOe−itH

whereOis the Schr ̈odinger picture operator. In the Schr ̈odinger picture, states
evolve according to the Schr ̈odinger equation

−iH|ψ〉=

d

dt

|ψ〉

If a groupGacts on a spaceM, the representation one gets on functions on
Mis given by
π(g)(f(x)) =f(g−^1 ·x)

Examples include