
independent of (say)

implies that:

but from the Schrödinger 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:
![\frac{\partial}{\partial t}\left[P^3(t)\psi\right]<br />= i\hbar\frac{\partial}{\partial t}\frac{\partial}{\partial x^3}\psi<br />= i\hbar\frac{\partial}{\partial x^3}\frac{\partial}{\partial t}\psi<br />= i\hbar\frac{\partial}{\partial x^3}\frac{E}{i\hbar}\psi<br />= P^3(t)\frac{E}{i\hbar}\psi\\ \\<br />= P^3(t)\frac{\partial}{\partial t}\psi](latexrender/pictures/39a59737d0c7b7116248d73801758560.png)
But
![\frac{\partial}{\partial t}\left[P^3(t)\psi\right] = P^3(t)\frac{\partial}{\partial t}\psi \frac{\partial}{\partial t}\left[P^3(t)\psi\right] = P^3(t)\frac{\partial}{\partial t}\psi](latexrender/pictures/5745f7ed692b0279f760e3a61015cd06.png)
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.
