Disclaimer for casual readers: I write posts which vary wildly in technical difficulty, this one is a little more mathematical than most. Don’t let it scare you off! Even if you’re a little lost, it’s good to have seen it.
The various worked problems I’ve been doing recently have mostly been on the intro undergraduate level. This is mainly because we’re between semesters, and I don’t really have the time to write up long and difficult problems from scratch. Once the fall semester starts up I’ll start writing up solutions to my homework problems.
To assuage my guilt over mainly doing easy problems, how about at least something that’s undergraduate physics major difficulty?
Consider a particle in the ground state of a 1-d infinite square well with sides at positions 0 and L. Suddenly the right-hand wall is moved outward so the walls are now at new positions 0 and 2L. What is the probability of observing the particle in the ground state of this new potential?
The main thing to understand with this problem is the idea that for any potential, there’s only certain quantum states that are allowed. If a particle is in an allowed state of one potential and suddenly the potential changes to something else, the particle’s new state must be expressible in terms of the new allowed states. But a wavefunction can’t just snap instantly from being one thing to being another, so there’s a problem.
Or there would be a problem if there weren’t the superposition principle. Just like you can expand a function in terms of the polynomials when you use Taylor series or in terms of trig functions when you use Fourier series, you can expand the old state in terms of the new allowed states no matter how different the new states are. It seems like magic, but it’s mathematically sound. The allowed eigenfunctions for a given potential form a complete set, so you can expand just about any square-integrable function in terms of the allowed functions over the appropriate interval.
Shall we try it here? The old ground state wavefunction was
The new allowed functions are
So the old ground state has to be expressible in terms of those new allowed functions. In other words,
With the constants Cn to be determined. How can we find those constants? Skipping the derivation (which is in any intro quantum textbook), we use the fact that
Why L for the upper limit of integration? Remember that the original ground state wavefunction is 0 outside of the box. If we forget that and instead calculate as though the sine function keeps going, we’ll get a very wrong answer. The star represents complex conjugation which doesn’t matter here, but it’s good to be accurate.
So plug in the functions with n = 1 (since we’re going from one ground state to another) and integrate. I get
Almost done. Remember it’s not the wavefunction which represents probability, it’s the square of the absolute value of the wavefunction. So the final answer is
It’s actually somewhat more probable that the particle will end up in the n = 2 state. The probability for that happens to be 0.5 exactly. The n = 3 state has a probability of about .1296, the n = 4 state has probability of about 0.00735, and for higher states the probability continues to fall like a stone. Of course as a check you can add up these values to find that it’s very nearly equal to 1, as expected.
Clear as crystal? Probably not. It took me a while to understand this stuff. But don’t worry, there’s tons and tons of good practice problems out there and we’ll end up doing a lot of them.