QUANTUM OPERATORS
operate repeatedly on it with the (down) ladder operatorD, we will generate a
state|ψd〉which, whilst still an eigenstate ofL^2 with eigenvaluea,hasthelowest
physically possible value,dsay, for the eigenvalue ofLz. If this happens aftern
operations we will have thatd=c−nand
Lz|ψd〉=(c−n)|ψd〉.Arguing in the same way as previously thatD|ψd〉must be an unphysical ket
vector, we conclude that
0 |∅〉=U|∅〉=UD|ψd〉=(L^2 −L^2 z+Lz)|ψd〉, using (19.30),=[a−(c−n)^2 +(c−n)]|ψd〉
⇒ a=(c−n)^2 −(c−n).Equating the two results foragives
c^2 +c=c^2 − 2 cn+n^2 ^2 −c+n^2 ,2 c(n+1)=n(n+1),c=^12 n.Sincenis necessarily integral,cis an integer multiple of^12 . This result is valid
irrespective of which eigenstate|ψ〉we started with, though the actual value of
the integerndepends on|ψu〉and hence upon|ψ〉.
Denoting^12 nbywe can say that the possible eigenvalues of the operatorLz, and hence the possible results of a measurement of thez-component of the
angular momentum of a system, are given by
, (−1), (−2), ... , −.The value ofafor all 2+ 1 of the corresponding states,
|ψu〉,D|ψu〉,D^2 |ψu〉, ... , D^2 |ψu〉,is(+1)^2.
The similarity of form between this eigenvalue and that appearing in Legendre’sequation is not an accident. It is intimately connected with the facts (i) thatL^2
is a measure of the rotational kinetic energy of a particle in a system centred
on the origin, and (ii) that in spherical polar coordinatesL^2 has the same form
as the angle-dependent part of∇^2 , which, as we have seen, is itself proportional
to the quantum-mechanical kinetic energy operator. Legendre’s equation and
the associated Legendre equation arise naturally when∇^2 ψ=f(r)issolvedin
spherical polar coordinates using the method of separation of variables discussed
in chapter 21.
The derivation of the eigenvalues(+1)^2 andm, with−≤m≤, dependsonly on the commutation relationships between the corresponding operators. Any