Quantum Mechanics for Mathematicians

Theorem 11.1(Fourier series).Ifψ∈L^2 (S^1 ), then

|ψ〉=ψ(φ) =

+∑∞

n=−∞

cneinφ=

∑+∞

n=−∞

cn|n〉

where

cn=〈n|ψ〉=

1

2 π

∫ 2 π

0

e−inφψ(φ)dφ

This is an equality in the sense of the norm onL^2 (S^1 ), i.e.,

lim

N→∞

||ψ−

+∑N

n=−N

cneinφ||= 0

The condition thatψ∈L^2 (S^1 )corresponds to the condition

+∑∞

n=−∞

|cn|^2 <∞

on the coefficientscn.

One can easily derive the formula forcnusing orthogonality of the|n〉. For
a detailed proof of the theorem see for instance [27] and [83]. The theorem
gives an equivalence (as complex vector spaces with a Hermitian inner product)
between square-integrable functions onS^1 and square-summable functions on
Z. As unitarySO(2) representations this is the equivalence of equation 11.2.
The Lie algebra of the groupS^1 is the same as that of the additive group
R, and theπ′(X) we have found for theS^1 action on functions is related to the
momentum operator in the same way as in theRcase. So, we can use the same
momentum operator

P=−i~

d

dφ

which satisfies
P|n〉=~n|n〉

By changing space from the non-compactRto the compactS^1 we now have
momenta that instead of taking on any real value, can only be integral numbers
times~.
Solving the Schr ̈odinger equation

i~

∂

∂t

ψ(φ,t) =

P^2

2 m

ψ(φ,t) =

−~^2

2 m

∂^2

∂φ^2

ψ(φ,t)

as before, we find

EψE(φ) =

−~^2

2 m

d^2

dφ^2

ψE(φ)