Some Aspects of an Infinite N-dimensional Spherical Potential Well

Sami A. Al-Jaber

Hasan Hüseyin Erbil, Hüseyin Ertik, Hüseyin Şirin +1 more

Eugene Stanley

Mark Kac, Carol Thompson

Yuanbei Sun

This paper solves the wavefunctions and energy level formulas of two types of three-dimensional infinite deep cubic potential wells and three-dimensional infinite deep spher-ical potential wells, and studies their degeneracy to elucidate how symmetry governs energy level degeneracy in quantum systems. For the cubic potential well, the potential energy inside the cube is zero, and outside the cube is infinite. Solving the Schrödinger equation shows that the energy is determined by the total of the squares of three quantum numbers (nx, ny, nz). Degeneracy occurs when different combinations of quantum numbers produce the same sum of squares, which reflects cubic symmetry, i.e., the equivalence of the x, y, and z directions, such that permuted states are physically identical. For the spherical potential well, the poten-tial energy inside the sphere is zero, and outside the sphere is infinite. The wavefunction can be devided into angular and radial parts. The energy depends on the radial quantum number n and the angular momentum quantum number l, and degeneracy arises from the magnetic quantum number m, which ranges from −l to l. Since the potential energy of the spherical potential well is independent of direction, it has spherical symmetry, which leads to degen-eracy of magnitude 2l + 1-fold. To summarize, the degeneracy of the cubic potential well originates from its cubic symmetry, while that of the spherical potential well originates from its spherical symmetry. This highlights the central role of symmetry in quantum energy level degeneracy.

Reza Moazzemi

In this paper we calculate the Berry phase for a wave function of a particle in an infinite spherical potential well with adiabatically varying. In order to do this, we need the solutions of the corresponding Schrödinger equation with a time dependent Hamiltonian. Here, we obtain these solutions for the first time. In addition, we calculate the Berry phase in one dimensional case for an infinite potential well. We see that this phase is comparable with the usual dynamical phase but with different sign. Therefore, it is important to take it into account in the quantum mechanical calculations.