Angular momentum III Eigenfunctions Using Operator Algebra

Abstract

In Chapter 13 we had shown that we can have a set of simultaneous eigenkets of the operators J² and J_z:

$$ J^2 \left| {\left. {j,m} \right\rangle = j\left( {j + 1} \right)\hbar ^2 } \right|\left. {j,m} \right\rangle $$

((1))

$$ J_z \left| {\left. {j,m} \right\rangle = m\hbar ^2 } \right|\left. {j,m} \right\rangle $$

((2))

where

$$ j = 0,\quad \frac{1} {2},\quad 1,\quad \frac{3} {2},\quad 2,\quad \frac{5} {2}, \ldots $$

((3))

and

$$ m = - j, - j + 1, \ldots ,j $$

((4))

It seems that if one is working from the point of view of getting beauty in one’s equations, and if one has a really sound insight, one is on a sure line of progress.

P.A.M. Dirac in The Evolution of the Physicist’s Picture of Nature, Scientific American, May 1963.