Visualizing the 2-D Particle in a Box - comments

[kreil]

kreil submitted a new PF Insights post

Visualizing the 2-D Particle in a Box

Continue reading the Original PF Insights Post.

 

[Jimster41]

kewl. Thanks for the lucid tutorial!

 

[Greg Bernhardt]

Great first entry kreil!

 

[Fooality]

Absolutely excellent. I am not a physicist, but have an interest in understanding QM and this article was perfect: The visualizations and instructions for computer simulation relate the ideas better than anything I've read.

 

[PWiz]

Nice! The blog's mathematical presentation is pretty simple and easy to understand as well. Gotta have one of these in the starting chapters of every introductory QM book ;)

 

[kreil]

Thanks guys, glad it was clear enough for non-physicists to follow!

If you have any suggestions for future topics please let me know.

 

[Jimster41]

Re-reading after thinking about it. I'd like to ask questions in context, but I don't want to distract from a great, clear tutorial.

Trying to picture what happens if you suddenly increase the size of the box. My guess was that the response is sort of quantized (or harmonic) due to the periodicity of the wave equation, that there would be no noticeable change until a new peak, or a new chorent wave frequency or pattern across the whole box was accommodated. Is that threshold Plank size? Also, was picturing the overall energy density probability going down, which seems naively consistent with thermodynamic expectation? At the moment when a new wave function suddenly "fits" the "expectation" wave spontaneously re-forms instantaneously everywhere to take this new shape? What if the box is huge?

 

[jtbell]

Trying to picture what happens if you suddenly increase the size of the box.

This is actually a fairly common exercise for students studying the one-dimensional particle in a box. Basically you start by assuming that the wave function ##\Psi(x,t)## does not change at the instant the box increases in size. If it was originally in the ground state, it goes from this:

to something like this:

Which doesn't look very interesting, does it? But, just let some time elapse!

In the original box, ##\Psi## is an energy eigenstate ##\psi_k(x) e^{-iE_k t / \hbar}## with a fixed energy. The probability distribution ##|\Psi|^2## maintains the same shape, so we call it a "stationary state".

In the new box, ##\Psi## is a superposition (linear combination) of the new energy eigenstates, e.g. :

for the new ground state and first excited state. Each of these eigenstates oscillates at a different frequency, so the probability distribution of the superposition does not maintain the same shape, that is, it is not a "stationary state." The probability distribution "sloshes" around inside the new box as time passes, starting from the probability distribution of the original wave function.

To see what this actually looks like for a specific case, you have to work out the coefficients A_k of the linear combination that expresses the spatial part of the original ##\psi(x)## in terms of the spatial part of the new energy eigenstates: $$\psi(x) = \Sigma {A_k \psi_k^\prime(x)}$$ Then you can find the new time-dependent wave function $$\Psi(x,t) = \Sigma A_k \psi_k^\prime(x) e^{-iE_k t / \hbar}$$ and the new probability distribution $$P(x,t) = |\Psi(x,t)|^2$$

 

[Jimster41]

I think I made (or am working on) a connection that I hadn't made, and I hope it is "not wrong", but please tell me if it is.

The periodic shape of wavefunction (the wavenumber?) for the given box "size" and the given energy, is dictated by way(s) that energy is divided by h (which, like the size dimension x of the box, is a "size"). The result is discrete scale invariant, is that correct?

Sorry for treating the insights as a regular old thread, but they are little bit like a class where you would love to be able to raise your hand. and get yourself squared up. Moreso IMHO than regular question threads which can be pretty chaotic and often you are just trying to figure out what the conversation is about. Or you are having to frame a question without such lucid context, and are not even sure if your terms are right.

 

[Jimster41]

No,no, sorry, it's just plain periodic.

 

[Atlas3]

Can the plot above labeled by "Here is the time-simulation for (nx,ny)=(4,4). In this case, the spots where the wave function is always zero are more numerous, and form a grid." fill a volume if stacked in another dimension?

 

[jtbell]

Sure, for a rectangular 3D "particle in a box."