Quantum Mechanics for Mathematicians

As we saw in section 11.1, one way to deal with this issue is to do what
physicists sometimes refer to as “putting the system in a box”, by imposing
periodic boundary conditions

ψ(x+L) =ψ(x)

for some numberL, effectively restricting the relevant values ofxto be consid-
ered to those on an interval of lengthL. For our eigenfunctions, this condition
is
eip(x+L)=eipx

so we must have
eipL= 1

which implies that

p=

2 π

L

j≡pj

forjan integer. Then the momentum will take on a countable number of
discrete values corresponding to thej∈Z, and

|j〉=ψj(x) =

1

√

L

eipjx=

1

√

L

ei

2 πj

Lx

will be orthonormal eigenfunctions satisfying

〈j′|j〉=δjj′

This use of periodic boundary conditions is one form of what physicists call
an “infrared cutoff”, a way of removing degrees of freedom that correspond to
arbitrarily large sizes, in order to make the quantum system well-defined. One
starts with a fixed value ofL, and only later studies the limitL→∞.
The number of degrees of freedom is now countable, but still infinite, and
something more must be done in order to make the single-particle state space
finite dimensional. This can be accomplished with an additional cutoff, an
“ultraviolet cutoff”, which means restricting attention to|p|≤Λ for some finite
Λ, or equivalently|j|<Λ 2 πL. This makes the space of solutions finite dimensional,
allowing quantization by use of the Bargmann-Fock method used for the finite
dimensional harmonic oscillator. The Λ→ ∞andL→ ∞limits can then be
taken at the end of a calculation.
The Schr ̈odinger equation is a first-order differential equation in time, t and
solutions can be completely characterized by their initial value att= 0

ψ(x,0) =

+∑Λ 2 πL

j=−Λ 2 πL

α(pj)ei

(^2) Lπjx
determined by a choice of complex coefficientsα={α(pj)}. At later times the
solution will be given by
ψ(x,t) =
+∑Λ 2 Lπ
j=−Λ 2 Lπ
α(pj)eipjxe−i
p^2 j
2 mt (36.5)