Quantum Mechanics for Mathematicians

known as the “heat equation”. This equation models the way temperature
diffuses in a medium, it also models the way probability of a given position
diffuses in a random walk. Note that here it isψ(q) that gives the probability
density, something quite different from the way probability occurs in measure-
ment theory for the free particle quantum system. There it is|ψ|^2 that gives
the probability density for the particle to have position observable eigenvalueq.
Taking as initial condition

ψ(q 0 ,0) =δ(q 0 −q′)

the heat equation will have as solution at later times

ψ(qτ,τ) =

√

m

2 πτ

e−

m 2 τ(q′−qτ) 2

(12.8)

This is physically reasonable: at timesτ >0, an initial source of heat localized
at a pointq′diffuses as a Gaussian aboutq′with increasing width. Forτ <0,
one gets something that grows exponentially at±∞, and so is not inL^2 (R) or
evenS′(R).
In real timetas opposed to imaginary timeτ(i.e.,z=it, interpreted as the
limit lim→ 0 +(+it)), equation 12.7 becomes

U(t,qt−q 0 ) =

√

m

i 2 πt

e−

im 2 t(qt−q^0 )^2

(12.9)

Unlike the case of imaginary time, this expression needs to be interpreted as a
distribution, and as such equation 12.4 makes sense forψ(q 0 ,0)∈ S(R). One
can show that, forψ(q 0 ,0) with amplitude peaked around a positionq′and with
amplitude of its Fourier transform peaked around a momentumk′, at later times
ψ(q,t) will become less localized, but with a maximum amplitude atq′+k

′

mt.

q=q′

ψ(q,0)

q=q′+

k′

m

t

ψ(q,t)

Figure 12.1: Time evolution of an initially localized wavefunction.