Chapter 12

Position and the Free

Particle

Our discussion of the free particle has so far been largely in terms of one observ-
able, the momentum operator. The free particle Hamiltonian is given in terms
of this operator (H=P^2 / 2 m) and we have seen in section 11.5 that solutions
of the Schr ̈odinger equation behave very simply in momentum space. Since
[P,H] = 0, momentum is a conserved quantity, and momentum eigenstates will
remain momentum eigenstates under time evolution.
The Fourier transform interchanges momentum and position space, and a
position operatorQcan be defined that will play the role of the Fourier trans-
form of the momentum operator. Position eigenstates will be position space
δ-functions, but [Q,H] 6 = 0 and the position will not be a conserved quantity.
The time evolution of a state initially in a position eigenstate can be calculated
in terms of a quantity called the propagator, which we will compute and study.

12.1 The position operator

On a state spaceHof functions (or distributions) of a position variableq, one
can define:

Definition(Position operator).The position operatorQis given by

Qψ(q) =qψ(q)

Note that this operator has similar problems of definition to those of the
momentum operatorP: it can take a function inL^2 (R) to one that is no longer
square-integrable. LikeP, it is well-defined on the Schwartz spaceS(R), as well
as on the distributionsS′(R). Also likeP, it has no eigenfunctions inS(R) or
L^2 (R), but it does have eigenfunctions inS′(R). Since
∫+∞

−∞

qδ(q−q′)f(q)dq=q′f(q′) =

∫+∞

−∞

q′δ(q−q′)f(q)dq