How Does Quantum Mechanics Explain the Continuity of Wave Functions Over Time?

[joker_900]

Thanks in advance for reading/replying

Homework Statement

In the interval (t+dt, t) the Hamiltonian H of some system varies in such a way that |H|psi>| remains finite. Show that under these circumstances |psi> is a continuous function of time.

A harmonic oscillator with frequency w is in its ground state when the stiffness of the spring is instantaneously reduced by a factor (f^4)<1 such that its natural frequency becomes (f^2)w. What is the probability that the oscillator is subsequently found to have energy 1.5.(h/2pi).w.f^2?

Homework Equations

En = (n + 0.5)(h/2pi)w

The Attempt at a Solution

For the first part I thought maybe I needed to use ehrenfest, as |H|psi>|^2 = <psi|H|psi>. So then I think you get

d/dt <psi|H|psi> = <psi|dH/dt|psi>

However I don't really know what to do with this and am probably way off.

For the second part I'm really confused. The energy it talks about is the energy eigenvalue of the first excited state; but I don't know how to get a probability without the state |psi> being given to me. Initially the state is the ground stationary state for the original frequency w, so initially a measurement will definitely find the energy value (h/2pi)w. So the state must change, but how do I work out what it changes to?


Argh

 

[Jhero]

, I am stuck on this problem too! It seems like we are both trying to use the Ehrenfest theorem to solve this problem, but I'm not sure if that's the right approach. Let me try to break down the problem and see if we can make some progress.

First, the problem states that in the interval (t+dt, t), the Hamiltonian H varies in such a way that |H|psi>| remains finite. This means that the Hamiltonian is changing in time, but the state |psi> does not become infinite or undefined at any point during this interval. This suggests that the Hamiltonian is changing smoothly and continuously, which would lead to the conclusion that |psi> is also a continuous function of time.

So how do we prove this? Well, we can start by using the Ehrenfest theorem, which states:

d/dt <A> = (1/i*h)<[H,A]>

where <A> is the expectation value of the observable A, [H,A] is the commutator of the Hamiltonian and the observable A, and i is the imaginary unit. In this case, we are interested in the expectation value of the Hamiltonian itself, so we have:

d/dt <H> = (1/i*h)<[H,H]>

But the commutator of a quantity with itself is always zero, so this simplifies to:

d/dt <H> = 0

This means that the expectation value of the Hamiltonian is constant in time. Now, we can use this fact to show that |psi> is also a continuous function of time. We know that the expectation value of the Hamiltonian is given by:

<H> = <psi|H|psi>

Since this is constant in time, it must also be true that:

d/dt <psi|H|psi> = 0

But we also know that:

d/dt <psi|H|psi> = <psi|dH/dt|psi> + <psi|H|dpsi/dt>

Since the first term is zero, this means that:

<psi|H|dpsi/dt> = 0

This tells us that the inner product <psi|H|dpsi/dt> is zero, which in turn means that the vector |dpsi/dt> is orthogonal to the vector <psi|H|. In other words, |dpsi/dt