Abstract:

We have formulated a model of a complex (twodimensional) 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 groundstate 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. 