Mathematical Methods for Physics and Engineering : A Comprehensive Guide

19.2 PHYSICAL EXAMPLES OF OPERATORS

other set of four operators with the same commutation structure would result

in the same eigenvalue spectrum. In fact, quantum mechanically, orbital angular

momentum is restricted to cases in whichnis even and sois an integer; this

is in accord with the requirement placed onif solutions to∇^2 ψ=f(r)thatare

finite on the polar axis are to be obtained. The non-classical notion of internal

angular momentum (spin) for a particle provides a set of operators that are able

to take both integral and half-integral multiples of
as their eigenvalues.

We have already seen that, for a state|, m〉that has az-component of

angular momentumm
, the stateU|, m〉is one with itsz-component of angular

momentum equal to (m+1)
. But the new state ket vector so produced is not

necessarily normalised so as to make〈, m+1|, m+1〉= 1. We will conclude

this discussion of angular momentum by calculating the coefficientsμmandνmin

the equations

U|, m〉=μm|, m+1〉 and D|, m〉=νm|, m− 1 〉

on the basis that〈, r|, r〉= 1 for allandr.

To do so, we consider the inner productI=〈, m|DU|, m〉, evaluated in two

different ways. We have already noted thatUandDare Hermitian conjugates

and soIcanbewrittenas

I=〈, m|U†U|, m〉=μ∗m〈, m|, m〉μm=|μm|^2.

But, using equation (19.31), it can also be expressed as

I=〈, m|L^2 −L^2 z−
Lz|, m〉

=〈, m|(+1)
^2 −m^2 
^2 −m
^2 |, m〉

=[(+1)
^2 −m^2 
^2 −m
^2 ]〈, m|, m〉

=[(+1)−m(m+1)]
^2.

Thus we are required to have

|μm|^2 =[(+1)−m(m+1)]
^2 ,

but can choose that allμmare real and non-negative (recall that|m|≤). A

similar calculation can be used to calculateνm. The results are summarised in the

equations

U|, m〉=

√

(+1)−m(m+1)
|, m+1〉, (19.34)

D|, m〉=

√

(+1)−m(m−1)
|, m− 1 〉. (19.35)

It can easily be checked thatU|, 〉=|∅〉=D|,−〉.