This note explains where the formulae of chapter 4.5.4 come from.
The general assertions are readily checked by simply writing out both sides of the equation and comparing. And some are just rewrites of earlier ones.
Position and potential energy operators commute since they are just ordinary numerical multiplications, and these commute.
The linear momentum operators commute because the order in which differentiation is done is irrelevant. Similarly, commutators between angular momentum in one direction and position in another direction commute since the other directions are not affected by the differentiation.
The commutator between the position and linear momentum in the -direction was worked out in the previous subsection to figure out Heisenberg's uncertainty principle. Of course, three-dimensional space has no preferred direction, so the result applies the same in any direction, including the and directions.
The angular momentum commutators are simplest obtained by just
” “cyclic permutationscheme, as in:
For the commutators with square angular momentum, work out
A commutator like is zero because everything commutes in it. However, in a commutator like , does not commute with , so multiplying out and taking the out of at its own side, you get , and the commutator left is the canonical one, which has value . Plug these results and similar into and you get zero.
For a commutator like , the term produces zero because commutes with , and in the remaining term, taking the various factors out at their own sides of the commutator produces
Instead you can work out the parenthetical expression further by
substituting in the definitions for and :
The commutators between linear and angular momentum go almost identically, except for additional swaps in the order between position and momentum operators using the canonical commutator.
To derive the first commutator in (4.73), consider the
-component as the example: