Classical particle distribution in a harmonic potential question

In summary, the conversation discusses the process of finding the probability of finding a classical particle at a specific position in a harmonic potential. The general probability function is mentioned, but the desired result is a graph similar to a quantum mechanical one. The individual also explains their method of using a verlet simulation to gather position, velocity, and acceleration values, and how a histogram of the positions can indicate the probability function. The Gibbs distribution is mentioned, and the individual is unsure how to compare it with the histogram. They are seeking clarification on how to use the probability function to achieve their desired graph.
  • #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.
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  • #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 :)

What is a classical particle?

A classical particle refers to a particle that follows the laws of classical mechanics, which describes the behavior of macroscopic objects.

What is a harmonic potential?

A harmonic potential is a type of potential energy function that is shaped like a parabola and is used to represent the potential energy of a particle in a harmonic oscillator system.

How does a classical particle behave in a harmonic potential?

In a harmonic potential, a classical particle will oscillate back and forth around the minimum point of the potential, with its motion being described by the principles of classical mechanics.

What is the classical particle distribution in a harmonic potential?

The classical particle distribution in a harmonic potential refers to the probability distribution of finding a classical particle at a particular position in the potential. This distribution is described by the Boltzmann distribution function.

What factors affect the classical particle distribution in a harmonic potential?

The classical particle distribution in a harmonic potential is affected by the temperature of the system, the mass of the particle, and the strength of the harmonic potential. Additionally, the distribution will change over time as the particle undergoes oscillations in the potential.