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Given a potential, qualitatively deducing the energy levels?

  1. Dec 30, 2011 #1
    Given the basic result that the harmonic potential well has the energy levels it has, are there ways to convincingly deduce the qualitative properties of the energy levels given a certain potential well?

    Take for example a Morse potential, is there a way to deduce (vague if need be, exact if possible) that there will be an infinite amount of energy levels inside the well and that they monotonically get closer to each other? (without solving the Schrödinger equation)

    As a side question, are there convincing arguments to qualitatively describe the shape of a free wave packet as it crosses a local potential? For example, if a free quantum wave approaches from the left across a potential initally constant but beginning to drop, will the amplitude decrease, increase or stay the same?
     
  2. jcsd
  3. Dec 31, 2011 #2
    I think the WKB ("semiclassical") approximation provides some of the intuition you are looking for. For example in the WKB approximation the modulus-squared of the wave function (for bound or free states) is inversely proportional to the classical momentum

    [itex]\sqrt{2m(E-V(x))}[/itex]

    which is intuitive because it is what you would expect for the probability density of a stochastic classical system.

    The WKB method also gives approximate energy levels of bound states which should let you determine whether there are a finite or infinite number of them more easily than exactly solving the Schrodinger equation.
     
    Last edited: Dec 31, 2011
  4. Dec 31, 2011 #3
    Thanks for the reply. (PS: the tag for latex is
    Code (Text):
    [itex] ... [/itex]
    )

    That's indeed something that I was looking for :)

    The energy levels sounds useful, but maybe not "quick" enough if you understand what I mean.
     
  5. Dec 31, 2011 #4
    I drew myself a random potential to test if I could qualitatively deduce (the real part of) an unbounded wave function. Could someone give their opinion on it?

    The random potential is as follows: the dark line indicates the potential, the grey line is the energy level of the psi function that will be drawn. The green lines, marked A and B, denote the section where E < V (and hence where psi will be exponential)

    attachment.php?attachmentid=42353&stc=1&d=1325385939.gif

    To draw Re(psi), we use that in the region where E > V:
    [itex]\left\{ \begin{array}{ll} \textrm{freq} &\propto \sqrt{E-V} \\
    \textrm{ampl} &\propto \sqrt{|\psi|^2} \propto \frac{1}{\sqrt{ p_\textrm{classical} }} \propto \frac{1}{\sqrt[4]{E-V}} \quad \textrm{(cf. WKB approximation)}
    \end{array} \right.[/itex]
    Using this we draw the points indicating the (local) wavelength (blue dots) and the amplitude (blue lines), both only for where there is oscillatory motion (i.e. E > V). The way the exponential is chosen indicates that it is somehow "a particle/wave coming for the left", if that is not too vague. Then [itex]Re(\psi)[/itex] (in red) is simply drawn using the blue construction:

    attachment.php?attachmentid=42354&stc=1&d=1325386472.gif
     

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