Mathematical Methods for Physics and Engineering - Matematica.NET

19.2 PHYSICAL EXAMPLES OF OPERATORSother set of four operators with the same commutation structure would resultin the same eigenvalue spectrum. In fact, quantum mechanically, orbital angularmomentum is restricted to cases in which n is even and so l is an integer; thisis in accord with the requirement placed on l if solutions to ∇ 2 ψ = f(r) thatarefinite on the polar axis are to be obtained. The non-classical notion of internalangular momentum (spin) for a particle provides a set of operators that are ableto take both integral and half-integral multiples of as their eigenvalues.We have already seen that, for a state |l, m〉 that has a z-component ofangular momentum m, the state U|l, m〉 is one with its z-component of angularmomentum equal to (m +1). But the new state ket vector so produced is notnecessarily normalised so as to make 〈l, m +1| l, m +1〉 = 1. We will concludethis discussion of angular momentum by calculating the coefficients µ m and ν m inthe equationsU|l, m〉 = µ m |l, m +1〉 and D|l, m〉 = ν m |l, m − 1〉on the basis that 〈l, r | l, r〉 = 1 for all l and r.To do so, we consider the inner product I = 〈l, m |DU| l, m〉, evaluated in twodifferent ways. We have already noted that U and D are Hermitian conjugatesand so I canbewrittenasI = 〈l, m |U † U| l, m〉 = µ ∗ m〈l, m | l, m〉µ m = |µ m | 2 .But, using equation (19.31), it can also be expressed asThus we are required to haveI = 〈l, m |L 2 − L 2 z − L z | l, m〉= 〈l, m |l(l +1) 2 − m 2 2 − m 2 | l, m〉=[l(l +1) 2 − m 2 2 − m 2 ] 〈l, m | l, m〉=[l(l +1)− m(m +1)] 2 .|µ m | 2 =[l(l +1)− m(m +1)] 2 ,but can choose that all µ m are real and non-negative (recall that |m| ≤l). Asimilar calculation can be used to calculate ν m . The results are summarised in theequationsU| l, m〉 = √ l(l +1)− m(m +1)| l, m +1〉, (19.34)D| l, m〉 = √ l(l +1)− m(m − 1) | l, m − 1〉. (19.35)It can easily be checked that U|l, l〉 = |∅〉= D|l, −l〉.663