Quantum of Potential Energy

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.

Eric Hand

Kyu Hwang Yeon, Shou Zhang, Yong Dae Kim +2 more

The quantum harmonic-parabola system, whose potential energy is the negative harmonic potential, is analyzed and applied to the cases of a quantum well, barrier, and periodic lattices. The eigenstate of the quantum Hamiltonian of the harmonic-parabola system is obtained. It is shown that any function may be expanded in terms of the eigenfunctions of the system in a finite interval, and the propagator of the system can be obtained from the eigenfunctions. The energy eigenvalues, uncertainty, and probability density for the first few states are treated for the infinite harmonic-parabola well. The energy eigenvalues and their enumeration, the energy band, and the probability density for the first few states are obtained for a well composed of the parabola and constant potentials. The transmission coefficients and each potential interval dependence are determined for this well and the barrier of the parabola-constant potential structure. Periodic lattices composed of the parabola potential and parabola-constant potentials are constructed, and their dispersion relations, energy states, and bands are obtained.

Metin Tolan, Joachim Stolze

Hasan Arslan, N. Hulaguhanoglu

The derivation of the virial theorem is presented both in classical and quantum mechanical approach.The kinetic energy and potential energy of the mechanical energy is converted to each other due to the virial theorem.Some of the dierent potentials are considered.For some of these potentials, the wavefunctions and energy eigenvalues of the Schrödinger Equation are derived.