The energy equation for a particle in a cubic box of dimensions L_x= L_y = L_z is E_nx_,_{ny, nz} = For a particle in a cubic box, how many degenerate energy levels have energy equal to 14 h²/8 mL²? A) 1 B) 12 C) 6 D) 8 E) 3

Question

Quizplus · Accepted Answer

The equation for the energy levels of a particle in a cubic box is E = h²/8m * (nx² + ny² + nz²). To find the number of energy levels with energy E = 14h²/8m, we need to solve for nx, ny, and nz.

14h²/8m = h²/8m * (nx² + ny² + nz²)
14 = nx² + ny² + nz²

We can use this equation to find all the possible combinations of nx, ny, and nz that satisfy it. Since the energy levels are degenerate, any combination of nx, ny, and nz that satisfies the equation will give an energy level of 14h²/8m.

One way to approach this is to list out all the possible values of nx, ny, and nz that satisfy the equation:

nx=0, ny=0, nz=±√14
nx=0, ny=±√2, nz=±√11
nx=±√2, ny=0, nz=±√11
nx=±√2, ny=±√2, nz=±√8
nx=±√3, ny=±√3, nz=±√8
nx=±√4, ny=0, nz=±√10

In total, there are 6 unique combinations of nx, ny, and nz that satisfy the equation, so the answer is (C) 6.

The Energy Equation for a Particle in a Cubic Box