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