Counting States

For a quantum particle in $d$ dimensions (or equivalently $d$ particles in one dimension) each microstate is represented by a point labeled by $n_1,n_2,\ldots n_d$, where the $n_i$ can each take on any positive integer.

This program allows us to calculate finite size effects for the number of microstates below a certain energy $E$ by comparing the actual count with the approximate count given in 1d by $\Gamma_1 = R$, in 2d by $\Gamma_2 = \pi R^2/4$ and in 3d by $\Gamma_3 = \pi R^3/6$, where $E = h^2R^2/8mL^2$, $L$ is the length of the infinite well containing the particles, $m$ is the mass of the particle, and $h$ is the Planck constant.

Problem: Finite-size effects in two dimensions

\begin{equation} \Gamma(E) = {1 \over 4}\pi\! R^2 = \pi {L^2 \over h^2} (2mE). \label{eq:4/phipart} \end{equation}

Equation (\ref{eq:4/phipart}) for $\Gamma(E)$ is valid only for large $E$ because the area of a quadrant of a circle overestimates the number of lattice points $n_x,n_y$ inside a circle of radius $R$. Use Program CountingStates to explore how well the relation $\Gamma = \pi\!R^2/4$ approximates the actual number of microstates. The program computes the number of nonzero, positive integers that satisfy the condition $n_x^2 + n_y^2 \leq R^2$. What is the minimum value of $R$ for which the difference between the asymptotic relation and the exact number is less than 1%?

Resources

Problems 4.10 and 4.11 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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