L8.5 Time dependence of expectation values

published on July 20, 2020

PROFESSOR: It's a statement about the time dependence of the expectation values It's a pretty fundamental theorem So here it goes You have d dt of the expectation value

Of Q This is what we want to evaluate We Now this would be d dt of integral psi star of x and t, Q psi of x and t And the d dt acts on the two of them So it gives you integral partial psi star dt Q psi of x and t plus psi star Q partial psi dt

And this is the integral over the x You've seen that kind of stuff And what is it? Well, integral dx, this is this Schrodinger equation,

D psi star dt is i over h bar, H psi star From the Schrodinger equation Then you have the Q psi of x and t On this term, you will have a very similar thing Minus i over H bar this time, psi star QH psi of x and t

So we use the Schrodinger equation in the form, i d psi dt– i H bar d psi dt– equal H psi I used it twice So then, it's actually convenient to multiply here

By i H bar d dt of Q So I multiplied by i H bar, and I will cancel the i and the H bar in this term, minus them this term So we'll have d cube x psi star Q H hat psi minus H hat psi star cube psi

OK Things have simplified very nicely And there's just one more thing we can do Look, this is the product of Q and H But by hermiticity, H in here can be brought to the other side to act on this wave function

So this is actually equal to the integral d cube dx psi star QH hat psi minus– the H can go to the other side– psi star H cube psi But then, what do we see there?

We recognize a commutator This commutator is just like we did for x and p, and we started practicing how to compute them They show up here And this is maybe one of the reasons commutators are so important in quantum mechanics

So what do we have here? i H bar d dt of the expectation value of Q is equal to the integral dx of psi star, QH minus HQ psi This is all of x and t Well, this is nothing else but the commutator of Q and H

So our final result is that iH bar d dt of the expectation value of Q is equal to– look It's the expectation value of the commutator Remember, expectation value for an operator– the operator is the thing here– so this

Is nothing else than the expectation value of Q with H This is actually a pretty important result It has all the dynamics of the physics in the observables Look, the wave functions used to change in time Due to their change in time, the expectation values of the operators change in time

Because this integral can't depend on time But here what you have succeeded is to represent the change in time of the expectation value– the change in time of the position that you expect you find your particle– in terms of the expectation value of a commutator

With a Hamiltonian So if some quantity commutes with a Hamiltonian, its expectation value will not change in time If you have a Hamiltonian, say with a free particle, well, the momentum commutes with this Therefore the expected value of the momentum,

You already know, since the momentum commutes with H This is 0 The expected value of this is 0 And the expected value of the momentum will not change, will be conserved So conservation laws in quantum mechanics

Have to do with things that commute with the Hamiltonian And it's an idea we're going to develop on and on

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