As discussed in Pathria and many other texts, the number of states for one particle in a two-dimensional box is given by the values of nx and ny that satisfy the condition
where the quantum numbers nx and ny are nonzero positive integers. R is related to the energy of the system by
where L is the linear dimension of the box, m is the mass of the particle, and h is Planck's constant.
In the semiclassical limit where E is large, the number of states with energy less than or equal to E is given by the area of the positive quadrant of a circle of radius r:
However, the number of states for finite values of R (and E) is different than this asymptotic expression (see the figure).
The program shows the asymptotic expression and the actual number of states. The latter is given by
where d is the spatial dimension of the box and
Updated 4 March 2009.