Quantum Mechanics for Mathematicians

This is an easily solved simple constant coefficient second-order partial differ-
ential equation. One method of solution is to separate out the time-dependence,
by first finding solutionsψEto the time-independent equation

HψE(q) =

−~^2

2 m

∇^2 ψE(q) =EψE(q) (10.3)

with eigenvalueEfor the Hamiltonian operator. Then

ψ(q,t) =ψE(q)e−

i

~tE

will give solutions to the full time-dependent equation

i~

∂

∂t

ψ(q,t) =Hψ(q,t)

The solutionsψE(q) to the time-independent equation 10.3 are complex expo-
nentials proportional to

ei(k^1 q^1 +k^2 q^2 +k^3 q^3 )=eik·q

satisfying
−~^2
2 m
i^2 |k|^2 =

~^2 |k|^2

2 m

=E

We have thus found that solutions to the Schr ̈odinger equation are given by
linear combinations of states|k〉labeled by a vectork, which are eigenstates of
the momentum and Hamiltonian operators with

Pj|k〉=~kj|k〉, H|k〉=

~^2

2 m

|k|^2 |k〉

These are states with well-defined momentum and energy

pj=~kj,E=

|p|^2

2 m

so they satisfy exactly the same energy-momentum relations as those for a clas-
sical non-relativistic particle.
While the quantum mechanical state spaceHcontains states with the clas-
sical energy-momentum relation, it also contains much, much more since it
includes linear combinations of such states. Att= 0 the state can be a sum

|ψ〉=

∑

k

ckeik·q

whereckare complex numbers. This state will in general not have a well-
defined momentum, but measurement theory says that an apparatus measuring
the momentum will observe value~kwith probability

|ck|^2

∑

k′|ck′|

2