Abstract—We study the dynamically shifted quantum oscillator in one dimension, whose detailed examination reveals some very interesting and unexpected results. While several articles have been written on the classical dynamically shifted oscillator, the quantum version is somewhat more difficult since it defies solution by standard algebraic and analytic techniques. In this work, we show that the analytic approach to this problem results in a trivial, non-normalizable, and thus nonphysical solution. Therefore, we undertake a numerical approach and find two strikingly different results. First, the energy eigenvalues rise when the dynamic shift is positive, which is almost intuitive. The second result is surprising as the system shows degeneracy in one dimension despite the fact that this potential violates two of the three required criteria described previously by other investigators. This work, therefore, opens a door to finding multiple classes of one-dimensional degenerate systems with continuous, symmetric potentials.

Index Terms—Degeneracy, one dimension, quantum, dynamically shifted, oscillator.

Pirooz Mohazzabi is with Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, WI 53141, USA (email: mohazzab@uwp.edu).
G. Clark Alexander is with Department of Mathematical Sciences, DePaul University, Chicago, IL 60604, USA.

Cite: Pirooz Mohazzabi, G. Clark Alexander, "Degeneracy in One Dimensional Quantum Systems: Dynamically Shifted Oscillator," International Journal of Applied Physics and Mathematics vol. 7, no. 2, pp. 79-86, 2017.