Quantum Mechanics for Mathematicians

we find that

δ(E−

k^2

2 m

) =

√

m

2 E

(δ(k−

√

2 mE) +δ(k+

√

2 mE))

and
ψ ̃E(k) =c+δ(k−

√

2 mE) +c−δ(k+

√

2 mE) (11.10)

The two complex numbersc+,c−give the amplitudes for a free particle solution
of energyEto have momentum±

√

2 mE.

In the physical case of three spatial dimensions, one gets solutions

ψ ̃(k,t) =e−i^21 m|k|^2 tψ ̃(k,0)

and the space of solutions is a space of functions (or distributions)ψ ̃(k,0) on
R^3. Energy eigenstates with energyEwill be given by distributions that are
non-zero only on the sphere|k|^2 / 2 m=Eof radius

√

2 mEin momentum space
(these will be studied in detail in chapter 19).

11.6 For further reading

The use of periodic boundary conditions, or “putting the system in a box”, thus
reducing the problem to that of Fourier series, is a conventional topic in quan-
tum mechanics textbooks. Two good sources for the mathematics of Fourier
series are [83] and [27]. The use of the Fourier transform to solve the free parti-
cle Schr ̈odinger equation is a standard topic in physics textbooks, although the
function space used is often not specified and distributions are not explicitly
defined (although some discussion of theδ-function is always present). Stan-
dard mathematics textbooks discussing the Fourier transform and the theory of
distributions are [88] and [27]. Lecture 6 in the notes on physics by Dolgachev
[18] contains a more mathematically careful discussion of the sort of calcula-
tions with theδ-function described in this chapter and common in the physics
literature.
For some insight into, and examples of, the problems that can appear when
one ignores (as we generally do) the question of domains of operators such as the
momentum operator, see [34]. Section 2.1 of [90] or Chapter 10 of [41] provide a
formalism that includes a spectral theorem for unbounded self-adjoint operators,
generalizing appropriately the spectral theorem of the finite dimensional state
space case.