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

The energy levels for a particle in a three-dimensional cubic box are described by the equation:\[E_{n_x, n_y, n_z} = \frac{h^2}{8ma^2} \left( n_x^2 + n_y^2 + n_z^2 \right)\]Here, \( n_x, n_y, n_z \) represent the quantum numbers along the x, y, and z axes, respectively, and \( a \) denotes the side length of the cubic box.Degeneracy of an energy level is defined as the count of unique sets of quantum numbers \( (n_x, n_y, n_z) \) yielding the same energy. For the specific energy value of \( \frac{14h^2}{8ma^2} \), the degeneracy is 6, indicating that six different combinations of quantum numbers result in this energy state. Final Answer: \[\boxed{6}\]