A Guide to Physics Problems. Part 2. Thermodynamics, Statistical Physics, and Quantum Mechanics

QUANTUM MECHANICS 259

d) In order to demonstrate that and are also eigenstates of

compose the commutator

by (S.5.15.1). Similarly,

Now,

Substituting (S.5.15.4) into (S.5.15.5) and replacing by we have

Rearranging (S.5.15.6) yields

as required. A similar calculation gives

We see from the above results that the application of the operator on

a state has theeffectof“raising” thestateby 1, and theoperator

lowers the state by 1 (see (f) below).

e)

since, by assumption,

f) Since by (c), the number operator and the Hamiltonian
commute, they have simultaneous eigenstates. Starting with
we may generate a number state whose energy eigenvalue is 1 + 1/2
by applying the raisingoperator Applying again produces a state of
eigenvalue 2+1/2. What remains to be done is to see thatthese eigenstates
(number, energy) are properly normalized. If we assume that the state
is normalized, then we may compose the inner product