While this deals with the problem of eigenvectors not being inH, it ruins
an important geometrical structure of the free particle quantum system, by
treating positions (taking values in the circle) and momenta (taking values in
the integers) quite differently. In later chapters we will see that physical systems
like the free particle are best studied by treating positions and momenta as real-
valued coordinates on a single vector space, called phase space. To do this, a
formalism is needed that treats momenta as real-valued variables on a par with
position variables.
The theory of Fourier analysis provides the required formalism, with the
Fourier transform interchanging a state spaceL^2 (R) of wavefunctions depending
on position with a unitarily equivalent one using wavefunctions that depend on
momenta. The problems of the domain of the momentum operatorP and its
eigenfunctions not being inL^2 (R) still need to be addressed. This can be done
by introducing
- a spaceS(R)⊂L^2 (R) of sufficiently well-behaved functions on whichP
is well-defined, and - a spaceS′(R)⊃L^2 (R) of “generalized functions”, also known as distri-
butions, which will include the eigenvectors ofP.
Solutions to the Schr ̈odinger equation can be studied in any of the three
S(R)⊂L^2 (R)⊂S′(R)contexts, each of which will be preserved by the Fourier transform and allow
one to treat position and momentum variables on the same footing.
11.1 Periodic boundary conditions and the groupU(1)
In this section we’ll describe one way to deal with the problems caused by non-
normalizable eigenstates, considering first the simplified case of a single spatial
dimension. In this one dimensional case, the spaceRis replaced by the circle
S^1. This is equivalent to the physicist’s method of imposing “periodic boundary
conditions”, meaning to define the theory on an interval, and then identify the
ends of the interval. The position variableqcan then be thought of as an angle
φand one can define the inner product as
〈ψ 1 ,ψ 2 〉=1
2 π∫ 2 π0ψ 1 (φ)ψ 2 (φ)dφThe state space is then
H=L^2 (S^1 )
the space of complex-valued square-integrable functions on the circle.
Instead of the groupRacting on itself by translations, we have the standard
rotation action of the groupSO(2) on the circle. Elementsg(θ) of the group