Particle in a 3-dimensional box:

The particle in a 3-dimensional box is a commonly used model in quantum mechanics to study the behavior of a particle confined within a finite region. In this model, the particle is assumed to be free to move in three dimensions, but is restricted to the boundaries of the box.

a) The energy levels are evenly spaced:

In a 3-dimensional box, the energy levels of the particle are not evenly spaced. In fact, the energy levels follow a pattern where the energy increases as the quantum numbers associated with each level increase. The energy eigenvalues of a particle in a box can be determined by solving the Schrödinger equation for the system.

b) Ground state energy is non-zero:

The ground state of a particle in a 3-dimensional box refers to the lowest energy level that the particle can occupy. In this state, all three quantum numbers associated with the particle's motion are at their lowest possible values. The ground state energy of a particle in a box is indeed non-zero and it corresponds to the minimum energy that the particle can have while still being confined within the box.

c) The second excited state is 3-fold degenerate:

Degeneracy refers to the phenomenon where multiple states have the same energy. In the case of a particle in a 3-dimensional box, the energy levels can be classified according to the number of nodes along each axis. The second excited state corresponds to having two nodes along one axis and one node along each of the other two axes. This state is indeed 3-fold degenerate because there are three different combinations of quantum numbers that can yield the same energy level.

d) The second excited state is non-degenerate:

Contrary to option d, the second excited state of a particle in a 3-dimensional box is degenerate. As mentioned earlier, this state corresponds to having two nodes along one axis and one node along each of the other two axes. The degeneracy arises from the fact that there are multiple combinations of quantum numbers that satisfy this condition and yield the same energy level.

Therefore, the correct answers for this question are option b) - the ground state energy is non-zero, and option c) - the second excited state is 3-fold degenerate.