Chapter 11

Fourier Analysis and the

Free Particle

The quantum theory of a free particle requires not just a state spaceH, but also
an inner product onH, which should be translation invariant so that translations
act as unitary transformations. Such an inner product will be given by the
integral

〈ψ 1 ,ψ 2 〉=C

∫

R^3

ψ 1 (q)ψ 2 (q)d^3 q

for some choice of normalization constantC, usually taken to beC= 1.Hwill
be the spaceL^2 (R^3 ) of square-integrable complex-valued functions onR^3.
A problem arises though if we try and compute the norm-squared of one of
our momentum eigenstates|k〉. We find

〈k|k〉=C

∫

R^3

(e−ik·q)(eik·q)d^3 q=C

∫

R^3

1 d^3 q=∞

As a result there is no value ofCwhich will give these states a finite norm, and
they are not in the expected state space. The finite dimensional spectral theorem
4.1 assuring us that, given a self-adjoint operator, we can find an orthonormal
basis of its eigenvectors, will no longer hold. Other problems arise because our
momentum operatorsPjmay take states inHto states that are not inH(i.e.,
not square-integrable).
We’ll consider two different ways of dealing with these problems, for sim-
plicity treating the case of just one spatial dimension. In the first, we impose
periodic boundary conditions, effectively turning space into a circle of finite ex-
tent, leaving for later the issue of taking the size of the circle to infinity. The
translation group action then becomes theU(1) group action of rotation about
the circle. This acts on the state spaceH=L^2 (S^1 ), a situation which can be
analyzed using the theory of Fourier series. Momentum eigenstates are now in
H, and labeled by an integer.