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Classical particle distribution in a harmonic potential question

  1. Feb 29, 2008 #1

    I have a question regarding how to find the probability of finding a classical particle at position x, in a harmonic potential.

    I have a general probability function P(x) = C*exp(-1/T * V(x) ), where T is the temperature, but this just gives a Gauss-function naturally, but what I want is to make a graph like this:


    I know this one is quantum mechanical, but for high quantum numbers I should get a similar looking graph (smooth, without all the peaks).

    Can anybody tell me what I am doing wrong? I can't really see what I should do to get the correct result using this probability function.
  2. jcsd
  3. Feb 29, 2008 #2
    I can add some more information about my question:

    What I have is a list of position, velocity and acceleration values at each timestep in a verlet simulation.

    If I make a histogram of the positions I get what I should get (high bars at the turning points).

    This histogram should give a clear indication of the probability function describing the probability of finding the particle at some position x.

    But, according to the Gibbs distribution we got:

    P(x,v) = 1/Z * exp( -1/T * (½mv^2 + ½m*w^2*x^2 )

    Where Z is the partitionfunction, normalizing P(x,v).

    So integrating v out of the above equation gives

    P(x) = K * exp( -1/T * ½m*w^2*x^2)

    Where K is just some new normalizationconstant.

    But how am I supposed to use this P(x) to compare with my histogram? As I wrote before P(x) is just a Gauss-function, and there must be something I'm missing or misunderstood.

    I hope someone can help me :)
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