Introduction to Quantum Mechanics · Problem 4.2

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · infinite_cubical_well

Problem

Use separation of variables in cartesian coordinates to solve the infinite cubical well (or "particle in a box"): V(x,y,z)={0,x,y,z all between 0 and a;∞,otherwise.V(x, y, z) = \begin{cases} 0, & x, y, z \text{ all between 0 and } a; \\ \infty, & \text{otherwise.} \end{cases} (a) Find the stationary states, and the corresponding energies. (b) Call the distinct energies E1,E2,E3,…E_1, E_2, E_3, \ldots, in order of increasing energy. Find E1,E2,E3,E4,E5E_1, E_2, E_3, E_4, E_5, and E6E_6. Determine their degeneracies (that is, the number of different states that share the same energy). Comment: In one dimension degenerate bound states do not occur (see Problem 2.44), but in three dimensions they are very common. (c) What is the degeneracy of E14E_{14}, and why is this case interesting?

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