University of Khartoum

The Complex Quantum Harmonic Oscillator Model

The Complex Quantum Harmonic Oscillator Model

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Title: The Complex Quantum Harmonic Oscillator Model
Author: Arbab, Arbab I.
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., az, ¯az and a¯z, ¯a¯z. The Hamiltonian of the system is Hz¯z =(¯azaz +1)+ωLz, where ω is the oscillator frequency and Lz = 2 (¯a¯za¯z −¯azaz) 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 ∗ = n 2n±1 e. These degenerate wave functions are eigenstates of the orbital angular momentum with eigenvalues n and −n , where h=2π is the Planck’s constant and n=1, 2, . . . . The two wave functions are degenerate with energy En = (n+1) ω. 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., In ∝ ne √ B.
URI: http://khartoumspace.uofk.edu/handle/123456789/17557
Date: 2015-12-13


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