Quantum Mechanics for Mathematicians

This is what one expects physically, sincep

′
mis the velocity corresponding to
momentump′for a classical particle.
Note that the choice of square root ofiin 12.9 is determined by the condition
that one get an analytic continuation from the imaginary time version forτ >0,
so one should take in 12.9
√
m
i 2 πt

=e−i

π

4

√

m

2 πt

We have seen that an initial momentum eigenstate

ψ(q 0 ,0) =

1

√

2 π

eik

′q 0

evolves in time by multiplication by a phase factor. An initial position eigenstate

ψ(q 0 ,0) =δ(q 0 −q′)

evolves to

ψ(qt,t) =

∫+∞

−∞

U(t,qt−q 0 )δ(q 0 −q′)dq 0 =

√

m

i 2 πt

e−

im 2 t(qt−q′)^2

Neart= 0 this function has a rather peculiar behavior. It starts out local-
ized atq 0 att= 0, but at any later timet >0, no matter how small, the
wavefunction will have constant amplitude extending out to infinity in position
space. Here one sees clearly the necessity of interpreting such a wavefunction
as a distribution.
For a physical interpretation of this calculation, note that while a momentum
eigenstate is a good approximation to a stable state one can create and then
study, an approximate position eigenstate is quite different. Its creation re-
quires an interaction with an apparatus that exchanges a very large momentum
(involving a very short wavelength to resolve the position). By the Heisenberg
uncertainty principle, a precisely known position corresponds to a completely
unknown momentum, which may be arbitrarily large. Such arbitrarily large
momenta imply arbitrarily large velocities, reaching arbitrarily far away in arbi-
trarily short time periods. In later chapters we will see how relativistic quantum
theories provide a more physically realistic description of what happens when
one attempts to localize a quantum particle, with quite different phenomena
(including possible particle production) coming into play.

12.6 Propagators in frequency-momentum space

The propagator defined by equation 12.4 will take a wavefunction at time 0 and
give the wavefunction at any other timet, positive or negative. We will find it
useful to define a version of the propagator that takes into account causality,
only giving a non-zero result fort >0: