Chapter 10

Momentum and the Free

Particle

We’ll now turn to the problem that conventional quantum mechanics courses
generally begin with: that of the quantum system describing a free particle
moving in physical spaceR^3. This is something quite different from the classical
mechanical description of a free particle, which will be reviewed in chapter 14.
A common way of motivating this is to begin with the 1924 suggestion by de
Broglie that, just as photons may behave like either particles or waves, the
same should be true for matter particles. Photons were known to carry an
energy given byE=~ω, whereωis the angular frequency of the wave. De
Broglie’s proposal was that matter particles with momentump=~kcan also
behave like a wave, with dependence on the spatial positionqgiven by

eik·q

This proposal was realized in Schr ̈odinger’s early 1926 discovery of a version
of quantum mechanics in which the state space is

H=L^2 (R^3 )

which is the space of square-integrable complex-valued functions onR^3 , called
“wavefunctions”. The operator

P=−i~∇

will have eigenvalues~k, the de Broglie momentum, so it can be identified as
the momentum operator.
In this chapter our discussion will emphasize the central role of the momen-
tum operator. This operator will have the same relationship to spatial trans-
lations as the Hamiltonian operator does to time translations. In both cases,
the operators are the Lie algebra representation operators corresponding to a
unitary representation on the quantum state spaceHof groups of translations
(translations in the three space and one time directions respectively).