Modern Physics: Mastering Physics: Particle in a 3-D Box: Counting States

Problems that require solving the three-dimensional Schrödinger equation can often be reduced to related one-dimensional problems. An example of this would be the particle in a cubical box.
Consider a cubical box with rigid walls (i.e., $U(x,y,z)=\infty$ outside of the cube) and edges of length

. The general solution for this problem is

$\psi(x,y,z) = A \sin \left(\frac{n_x\pi x}{L} \right)\sin \left(\frac{n_y\pi y}{L} \right)\sin \left(\frac{n_z\pi z}{L} \right)$ ,

where

, and

are all positive integers. Note that this solution is just the product of three solutions to the one-dimensional particle in a box. The energy corresponding to the three-dimensional solution is just the sum of the energies for each of the three one-dimensional solutions:

$E=\frac{\pi^2 \hbar^2}{2mL^2}(n_x^2+n_y^2+n_z^2)$ .

Part A

What is the smallest allowed energy E_0

for a particle in a cubical box?

Express your answer in terms of hbar

, and

ANSWER:

$\frac{3}{2}\frac{{\pi}^{2}{\hbar}^{2}}{mL^{2}}$

When using quantum mechanics to describe large collections of particles, such as the electrons in a piece of metal, it is important to know how many quantum states exist below a certain energy. For the three-dimensional box potential, there is a convenient graphical way to do this. Since there are three different quantum numbers ( n_x

, and

), picture three-dimensional space with each number measured down one of the axes.

The energy is proportional to the sum of the squares of the three numbers, which corresponds to the distance from the origin squared. To ask how many states have energy less than or equal to some number $n_{\rm rs}^2 \pi^2\hbar^2/(2mL^2)$ is equivalent to asking how many states are inside of the sphere (sketched in the figure) defined by $n_x^2+n_y^2+n_z^2 = n_{\rm rs}^2$ . This sphere has radius n_rs

. (The subscript "rs" just stands for "radius of sphere".) Because all of the integers must be positive, you are really only working with one octant (one-eighth) of the sphere.

Part B
If you are dealing with a very large , you can assume that each state (point with integer coordinates) corresponds roughly to one unit of volume inside of the sphere. So, the number of states is approximately equal to the volume of the octant of the sphere. Use this idea to find the number of states with energy less than or equal to $n_{\rm rs}^2 \pi^2\hbar^2/(2mL^2)$ for a large .

Express your answer in terms of n_rs

ANSWER:

$\frac{{\pi}}{6}{\cdot}n_{rs}{^{3}}$

Part C
Suppose that you have particles, each of which has to go into its own state, in the cubical box. Assume that no state is occupied if a lower energy state remains unoccupied. What is the highest energy that one of these particles has? Assume that is large so that the analysis from the previous part applies.

Express your answer in terms of

, and

ANSWER: