Quantum Mechanics for Mathematicians

as the eigenvector equation. This has an orthonormal basis of solutions|n〉,
with

E=

~^2 n^2
2 m
The Schr ̈odinger equation is first-order in time, and the space of possible
solutions can be identified with the space of possible initial values at a fixed
time. Elements of this space of solutions can be characterized by

• The complex-valued square-integrable functionψ(φ,0)∈L^2 (S^1 ), a func-
  tion on the circleS^1.


• The square-summable sequencecnof complex numbers, a function on the
  integersZ.

Thecncan be determined from theψ(φ,0) using the Fourier coefficient formula

cn=

1

2 π

∫ 2 π

0

e−inφψ(φ,0)dφ

Given thecn, the corresponding solution to the Schr ̈odinger equation will be

ψ(φ,t) =

+∑∞

n=−∞

cneinφe−i

~ 2 nm^2 t

To get something more realistic, we need to take our circle to have an arbi-
trary circumferenceL, and we can study our original problem with spaceRby
considering the limitL→∞. To do this, we just need to change variables from
φtoφL, where

φL=

L

2 π

φ

The momentum operator will now be

P=−i~

d

dφL

and its eigenvalues will be quantized in units of^2 πL~. The energy eigenvalues
will be

E=

2 π^2 ~^2 n^2

mL^2

Note that these values are discrete (as long as the sizeLof the circle is finite)
and non-negative.

11.2 The groupRand the Fourier transform

In the previous section, we imposed periodic boundary conditions, replacing
the groupRof translations by the circle groupS^1 , and then used the fact
that unitary representations of this group are labeled by integers. This made