Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Insights Visualizing the 2-D Particle in a Box - comments

  1. May 4, 2015 #1

    kreil

    User Avatar
    Gold Member

  2. jcsd
  3. May 4, 2015 #2
    kewl. Thanks for the lucid tutorial!
     
  4. May 4, 2015 #3
    Great first entry kreil!
     
  5. May 4, 2015 #4
    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.
     
  6. May 4, 2015 #5
    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 ;)
     
  7. May 5, 2015 #6

    kreil

    User Avatar
    Gold Member

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

    If you have any suggestions for future topics please let me know.
     
  8. May 6, 2015 #7
    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 accomodated. 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?
     
  9. May 6, 2015 #8

    jtbell

    User Avatar

    Staff: Mentor

    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:

    Snapshot1.jpg

    to something like this:

    Snapshot2.jpg

    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. :

    Snapshot3.jpg

    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 Ak 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 $$P(x,t) = |\Psi(x,t)|^2$$
     
  10. May 6, 2015 #9
    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. o_O 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.
     
  11. May 6, 2015 #10
    No,no, sorry, it's just plain periodic.
     
  12. May 6, 2015 #11

    Atlas3

    User Avatar
    Gold Member

    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?
     
  13. May 7, 2015 #12

    jtbell

    User Avatar

    Staff: Mentor

    Sure, for a rectangular 3D "particle in a box."
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Visualizing the 2-D Particle in a Box - comments
  1. Particle in a 1-D box (Replies: 1)

Loading...