Introduction to Quantum Mechanics · Problem 4.21

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · angular_momentum_ladder_operators

Problem

The raising and lowering operators change the value of

m

by one unit:

L_+ f_\ell^m = (A_\ell^m) f_\ell^{m+1}, \quad L_- f_\ell^m = (B_\ell^m) f_\ell^{m-1}

where

A_\ell^m

and

B_\ell^m

are constants. Question: What are they, if the eigenfunctions are to be normalized? Hint: First show that

L_\mp

is the hermitian conjugate of

L_\pm

(since

L_x

and

L_y

are observables, you may assume they are hermitian... but prove it if you like); then use Equation 4.112. Answer:

A_\ell^m = \hbar\sqrt{\ell(\ell+1) - m(m+1)} = \hbar\sqrt{(\ell - m)(\ell + m + 1)},

B_\ell^m = \hbar\sqrt{\ell(\ell+1) - m(m-1)} = \hbar\sqrt{(\ell + m)(\ell - m + 1)}.

Note what happens at the top and bottom of the ladder (i.e. when you apply

L_+

f_\ell^\ell

L_-

f_\ell^{-\ell}