Abstract
We have formulated a model of a complex (two-dimensional) quantum harmonic oscillator. All dynamical physical variables are expressed in terms of the creation and annihilation operators, viz., $a_{z}, \bar{a}_{z}\ {\rm and} \ a_{\bar{z}}, \bar{a}_{\bar{z}} $. The Hamiltonian of the system is $H_{z\bar{z}} = (\bar{a}_{z}a_{z} + 1) + \omega L_{z} $, where ω is the oscillator frequency and $L_{z} = \frac{\hbar }{2}\,(\bar{a}_{\bar{z}} a_{\bar{z}}-\bar{a}_{z}a_{z}) $ is the orbital angular momentum. The oscillator is found to be described by a conserved orbital angular momentum (Lz) besides energy. While the ground-state wave function is real, all excited states are complex and degenerate. The oscillator in these states carry a quantum of charge of $e^{{\ast}} = \frac{n}{2n\pm1} e $. These degenerate wave functions are eigenstates of the orbital angular momentum with eigenvalues nℏ and −nℏ, where h=2πℏ is the Planck's constant and n=1, 2, … . The two wave functions are degenerate with energy En=(n+1)ℏω. The comparison with Landau level reveals that in the presence of the magnetic field, B, where ω is equal to the cyclotron frequency, the current moment is quantized and is proportional to the square root of the magnetic field, i.e., ${\cal I}_{n}\propto n e \sqrt{B} $.