What Is the Expectation Value of Energy After an Infinite Square Well Expansion?

[Raze2dust]

Homework Statement

A particle of mass m is in the ground state of the infinite square well. Suddenly the well expands to twice its original size, the right wall moving from a to 2a- leaving the wave function (momentarily) undisturbed. The energy of the particle is now measured. What will be the expectation value of energy? Hint: If you find yourself confronted with an infinite series, try ANOTHER method

Homework Equations

The Attempt at a Solution

The Hint is what is troubling me. I did encounter an infinite series, but I was able to find its sum using some mathematical manipulations.

But I'm curious to know what is the OTHER method the author is talking about. I thought a lot but I can't find any other method, which will not lead to an infinite series.. Any ideas?

Thanks =)

Edit: The answer I got by explicitly summing the series(which is not what the author wants me to do) is $$\hbar\omega$$, where $$\omega=\frac{\pi^{2}\hbar}{2ma^{2}}$$. I would like to verify this as well.

 

[Raze2dust]

please I just need the concept part...what other method to find <E> is he author expecting?
Do I need to show my working too? I thought it won't be relevant here, since my working is based on summation of the infinite series...

 

[arunma]

You know the particle is initially in the ground state. Since the process of moving the wall is (more or less) instantaneous, the wavefunction remains momentarily undisturbed. Write out the ground state wavefunction, noting that it goes to zero at the boundaries of the well (that is, the well before the size is increased), and then note that this will be the value of the new time-dependent wavefunction at time t=0. When the well expands, you'll have a whole new set of orthogonal time-independent eigenfunctions and eigenenergies. You can write the t=0 wavefunction as a linear combination of these eigenstates.

Well, I hope that helps!

 

[Dick]

I think the point of the problem is that you don't HAVE to split the wavefunction into energy eigenfunctions of the new well, though you could, of course. You can evaluate the expectation value of the energy by just sandwiching the energy operator between the wavefunction and it's complex conjugate. Thinking about it that way, ask yourself why should the energy change?? The sum of your infinite series should be exactly the original energy. Just as the sum of the infinite series of energy eigenfunctions is exactly the original wavefunction.

 

[arunma]

Now that Dick mentions it, there's a piece of information that I forgot to provide. The energy is just the expectation value of the Hamiltonian. Of course to evaluate this you need to know the wavefunction. You know its form at t = 0, since it's just the ground state of the well before the increase in size. As long as the cat's out of the bag, the energy should be the same, since you're using the same eigenstate. However, it wouldn't be a bad idea to see if you can find the time evolution of the wavefunction anyway, since this is a pretty classic problem in quantum mechanics.

 

[Dick]

Now that Dick mentions it, there's a piece of information that I forgot to provide. The energy is just the expectation value of the Hamiltonian. Of course to evaluate this you need to know the wavefunction. You know its form at t = 0, since it's just the ground state of the well before the increase in size. As long as the cat's out of the bag, the energy should be the same, since you're using the same eigenstate. However, it wouldn't be a bad idea to see if you can find the time evolution of the wavefunction anyway, since this is a pretty classic problem in quantum mechanics.

The OP managed to sum the series for the expectation value of the energy using the eigenstate expansion. And indeed, got the energy of the initial state, so I think he/she knows what they are doing. As you say, to get the time evolution you would have to do that. But luckily they don't ask for that. This is completely analogous to the classical problem. If you have a particle bouncing around in a compartment and suddenly double the size, the energy of the particle doesn't change.

 

[Raze2dust]

I think the point of the problem is that you don't HAVE to split the wavefunction into energy eigenfunctions of the new well, though you could, of course. You can evaluate the expectation value of the energy by just sandwiching the energy operator between the wavefunction and it's complex conjugate. Thinking about it that way, ask yourself why should the energy change?? The sum of your infinite series should be exactly the original energy. Just as the sum of the infinite series of energy eigenfunctions is exactly the original wavefunction.

ok..So you mean to say energy does not change. It sits perfect since my answer after series summation also tells me the same.

I was trying to use <E>=(<p^2>)/2m

Thanks a lot for the effort Dick and arunma :)