Question

An electron is confined in a three-dimensional infinite potential well. The sides parallel to the x-, y-, and z-axes are of length L each.

(a) Write the appropriate Schrodinger equation.

(b) Write the time-independent wave function corresponding to the state of the lowest possible energy.

(c) Give an expression for the number of states, N, having energy less than some given E. Assume N >> 1

Answer 1

(a) The Schrodinger equation is

(b) By separation of variables, we can take that the wave function to be the product of three wave functions each of a one-dimensional infinite well potential. The wave function of the lowest energy level is

The corresponding energy is E₁₁₁ = 3h²r²/2mL².

(c) For a set of quantum numbers n_x, n_y, n_z for the three dimensions, the energy is

Hence the number N of states whose energy is less than or equal to E is equal to the number of sets of three positive integers n_x, n_y, n_z satisfying the inequality

\(n_X^2 + n_y^2 + n_z^2 \le \frac{2mL^2}{h^2\pi^2} E\)

Consider a Cartesian coordinate system of axes n_x, n_y, n_z. The number N required is numerically equal to the volume in the first quadrant of a sphere of radius (2mL²E/h²π²)^1/2, provided N ≥ 1. Thus