# Statistical mechanics: Particle on a spring

## Homework Statement

A classical particle with mass m is in thermal equilibrium with a fluid at temperature T. The particle is stuck to a harmonic ('Hookean') spring and can only move on a horizontal line ($-\infty < x < \infty$). The position of the particle is x = 0 if the spring is in its equilibrium position, but thermal movement can cause it to stretch or compress. The probability distribution of finding the particle in position x is proportional to $e^{-\frac{x^2}{2\sigma^2}}$.

a) Calculate the spring constant C as a function of m, $\sigma$, $k_b$ and/or T.

b) Calculate the average quadratic displacement of the particle.

## The Attempt at a Solution

The probability of finding the particle in position x is proportional to $e^{-\frac{x^2}{2\sigma^2}}$. The energy of the spring is $\frac{1}{2}C x^2$, so the chance of finding the particle in position x is also proportional to $e^{-\frac{1}{2}C x^2 \beta}$. This gives $\frac{1}{2}C x^2 \beta = \frac{x^2}{2\sigma^2}$ or $C = \frac{k_B T}{\sigma^2}$

Is this correct?

b) Do they mean finding $\langle x^2 \rangle$?

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## Answers and Replies

This is a weird question, but I think based on what the question says, you're giving the right answer. I actually think the question must be screwed up.

It's weird because:
I've never heard the term "average quadratic displacement" but <x2> sounds right.

The question is thermodynamically nonsense because:
Here σ is supposedly a constant, and the spring constant depends on temperature. (Problems I've seen have a constant spring constant, and the probability distribution has thermal dependence.) Since σ is nonzero, even at zero temperature, there are fluctuations in the particle's position. If the fluctuations are thermal, this is a contradiction, since at zero temperature there is no thermal motion. This also implies that at zero temperature, the spring constant is zero, which is very odd.

Thanks. The problem might lie in my translation of the question, but I made sure to do it as faithfully as possible and looking at the original question it is pretty much a word for word translation. I will have to look at it some more and come back to it.

(If anyone speaks Dutch it's questions 4a and 4b http://www.a-eskwadraat.nl/tentamens/NS-201b/NS-201b.2009-03-16.tent.pdf [Broken].)

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