The energy equation for a particle in a cubic box of dimensions Lx = L_y = Lz is
Enx_, _{ny, nz} =
For a particle in a cubic box, how many degenerate energy levels have energy equal to 14 h²/8 mL²?

Question

The energy equation for a particle in a cubic box of dimensions Lx = L_y = Lz is
Enx_, _{ny, nz} =

The energy equation for a particle in a cubic box of dimensions Lx = L<sub>y</sub> = Lz is Enx<sub>,</sub> <sub>ny, nz</sub> = For a particle in a cubic box, how many degenerate energy levels have energy equal to 14 h<sup>2</sup>/8 mL<sup>2</sup>? A) 1 B) 12 C) 6 D) 8 E) 3

For a particle in a cubic box, how many degenerate energy levels have energy equal to 14 h²/8 mL²?

Quizplus · Accepted Answer

According to the equation, the energy levels are given by:
Enx,nynz = ((h^2)/(8mL^2))(nx^2 + ny^2 + nz^2)
To find the number of degenerate levels at energy 14h^2/8mL^2, we set the equation equal to this energy and solve for nx, ny, and nz:

(nx^2 + ny^2 + nz^2) = 14(L^2/h)^2

This is the equation of a sphere in 3D space with radius r = sqrt(14(L^2/h)^2). The degeneracy is simply the number of integer solutions (nx, ny, nz) that lie on this sphere.

Using a formula from number theory, we can determine that the number of integer solutions for any sphere with radius r is given by:

# of solutions = 2((r+3)/2)^2 - 1

Plugging in r = sqrt(14(L^2/h)^2), we get:

# of solutions = 6

Therefore, there are 6 degenerate energy levels at the given energy.

The Energy Equation for a Particle in a Cubic Box