Question

Particle in a 1-D box can be thought of confining the particle moving a long a

string, confined by the two ends of the string. 2-D PIB confines the particle's

motion on a plane, confined by the rectangular perimeter. 3-D PIB confines

the particle's motion in a real box. For those three cases, we can find geometry

analogues in real life. Now, let's tackle a hypothetical 4-D PIB model. The

four dimensions have variable x, y, z, and w, respectively (such as x for 1-D

PIB and x, y for 2-D PIB). The infinite potential energy walls are at (0, a), (0,

b), (0, c), and (0, d) in the four dimensions.

(a) Please write down the Schrodinger equation for this 4-D PIB model and

specify the boundary conditions.

(b) Without solving the differential equations, directly write down the nor-

malised eigen wavefunctions and eigenenergies of the 4-D PIB, based on

your knowledge of 1-D and 2-D PIB.

(c) For the special case a = b = c = d, what is the degeneracy of the first-

excited-state (the state with the second lowest energy)?

Answer 1