Blundell Thermal Physics · Problem 19.4

This problem asks you to find the mean square displacement of a particle in a harmonic potential using the equipartition theorem. The key insight is recognizing that a quadratic term in the energy—like $\alpha_i x_i^2$—contributes equally to kinetic and potential energy on average, with each quadratic degree of freedom storing $\frac{1}{2}k_\mathrm{B}T$ of energy. By writing the potential energy and applying equipartition directly, you can solve for $\langle x_i^2 \rangle$ without doing any integrals.

This problem asks you to find the mean square displacement of a particle in a harmonic potential using the equipartition theorem. The key insight is recognizing that a quadratic term in the energy—like $\alpha_i x_i^2$—contributes equally to kinetic and potential energy on average, with each quadratic degree of freedom storing $\frac{1}{2}k_\mathrm{B}T$ of energy. By writing the potential energy and applying equipartition directly, you can solve for $\langle x_i^2 \rangle$ without doing any integrals.