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Exercise [21.07]
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Joined: 12 Jul 2010, 07:44
Posts: 154
Exercise [21.07]
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.

Last edited by deant on 01 Sep 2012, 12:10, edited 4 times in total.

01 Sep 2012, 11:42

Joined: 12 Jul 2010, 07:44
Posts: 154
Re: Exercise [21.07]
Note: I'm not entirely happy with this solution, since it relies on knowing the fact that commuting operators have simultaneous eigenstates (i.e. functions that are eigenstates of both operators) - something that hasn't really been discussed in the book.

If anyone has a better way of demonstrating that commutation with the operator implies conservation, please post it!

01 Sep 2012, 11:58
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