Question

What is the degeneracy of the energy level \( \frac{14h^2}{8ma^2} \) for a particle in a three-dimensional cubic box of edge length \( a \)?

• A. 2
• B. 4
• C. 6
• D. 8

Hint:

For a cubic box, the degeneracy of an energy level corresponds to the number of distinct quantum number combinations that produce the same energy.

Answer 1

The Correct Option is C

For a particle in a three-dimensional cubic box, the energy levels are given by the formula: \[ E_{n_x, n_y, n_z} = \frac{h^2}{8ma^2} \left( n_x^2 + n_y^2 + n_z^2 \right) \] Where \( n_x, n_y, n_z \) are the quantum numbers in the x, y, and z directions, and \( a \) is the length of the edge of the cubic box. The degeneracy of an energy level is determined by the number of different sets of quantum numbers \( (n_x, n_y, n_z) \) that lead to the same energy. For the given energy \( \frac{14h^2}{8ma^2} \), the degeneracy is 6, as there are 6 distinct combinations of quantum numbers that lead to this energy level. Final Answer: \[ \boxed{6} \]