The Bicomplex Quantum Harmonic Oscillator

Details The Bicomplex Quantum Harmonic Oscillator The Bicomplex Quantum Harmonic Oscillator

2010-01-07

arxiv.org/abs/1001.1149v3

Il Nuovo Cimento 125 B (2010) 1173-92

Physics

Mathematical Physics

Figures suppressed. To appear in Nuovo Cimento B

Scientific paper

10.1393/ncb/i2010-10927-x

The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero divisors. Starting with the commutator of the bicomplex position and momentum operators, and adapting the algebraic treatment of the standard quantum harmonic oscillator, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. We construct an infinite-dimensional bicomplex module from these eigenkets. Turning next to the differential equation approach, we obtain coordinate-basis eigenfunctions of the bicomplex harmonic oscillator Hamiltonian in terms of hyperbolic Hermite polynomials.

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