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## 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 ([itex]-\infty < x < \infty[/itex]). 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 [itex]e^{-\frac{x^2}{2\sigma^2}}[/itex].

a) Calculate the spring constant C as a function of m, [itex]\sigma[/itex], [itex]k_b[/itex] 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 [itex]e^{-\frac{x^2}{2\sigma^2}}[/itex]. The energy of the spring is [itex]\frac{1}{2}C x^2[/itex], so the chance of finding the particle in position x is also proportional to [itex]e^{-\frac{1}{2}C x^2 \beta}[/itex]. This gives [itex]\frac{1}{2}C x^2 \beta = \frac{x^2}{2\sigma^2}[/itex] or [itex]C = \frac{k_B T}{\sigma^2}[/itex]

Is this correct?

b) Do they mean finding [itex]\langle x^2 \rangle[/itex]?

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