independent of (say)

implies that:

but from the SchrÃ¶dinger equation,

so we have:

i.e.

and

commute.

Now on p.521 it is noted that if quantum operators commute, you can always find wavefunctions that are eigenstates of both operators simultaneously. When

is a momentum eigenstate,

, where

is the classical momentum along the

axis - a function of time, not a differential operator. Similarly for an energy eigenstate,

, where

E is the energy (a constant).

So let

be an eigenfunction both of energy and of momentum along

. Then:

But

only when

is not actually a function of

t, i.e. when

is a constant that does not change with time.

So when

commutes with

,

is conserved.