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Zero energy correction method for non-Hermitian Harmonic oscillator with simultaneous transformation of co-ordinate and momentum:Wave function analysis under Iso-spectral condition

Biswanath Rath, Pravanjan Mallick

Published Jan 25, 2015• 1 citation
OpenAlex

Abstract

We present a complete analysis on energy and wave function of Harmonic oscillator with simultaneous non-hermitian transformation of co-ordinate ($(x \rightarrow \frac{(x+ i\lambda p)}{\sqrt{(1+\beta \lambda)}}$ and momentum $(p \rightarrow \frac{(p+ i\beta x)}{\sqrt{(1+\beta \lambda)}}$ for getting energy eigenvalue using perturbation theory under iso-spectral condition. Further we notice that two different frequency of oscillation ($w_{1}, w_{2}$)correspond to same energy eigenvalue, which can also be verified using Lie algebraic approach [Zhang this http URL J.Math.Phys 56 ,072103 (2015)]. Interestingly wave function analysis using similarity transformation [F.M. Fernandez, Int. J. Theo. Phys. (2015)(in Press)] refers to a very special case.

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Abstract We present a complete energy and wavefunction analysis of a Harmonic oscillator with simultaneous non-hermitian transformations of co-ordinate ( x → ( x + i λ p ) ( 1 + β λ ) ) $(x \rightarrow \frac{(x + i\lambda p)}{\sqrt{(1+\beta \lambda)}})$ and momentum ( p → ( p + i β x ) ( 1 + β λ ) ) $(p \rightarrow \frac {(p+i\beta x)}{\sqrt{(1+\beta \lambda)}})$ using perturbation theory under iso-spectral conditions. We observe that two different frequencies of oscillation (w 1, w 2)correspond to the same energy eigenvalue, - which can also be verified using a Lie algebraic approach.

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Zero energy correction method for non-Hermitian Harmonic oscillator with simultaneous transformation of co-ordinate and momentum and

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Do we have a general eigenvalue relation in Quantum Mechanics ? No

Biswanath Rath

Here we clearly notice that invariance in commutation relation can be associated with complex energy eigenvalues. Hence present work clearly demonstates that as such we, do not have a general eigenvalue condition in Quantum Physics. Here it is shown that existing all the three eigenvalue conditions namely: Hermiticity, PT symmetry and Pseudo-Hermiticity, are not sufficient enough even to explain the spectra of Harmonic Oscillator under non-Hermitian transformation of momentum, co-ordinate or both, on appearance of complex energy eigenvalues. In fact complex energy eigenvalues become an inherent feature of transformed oscillator, when its frequency of oscillation matches with that of variational based calculation.

•Jul 9, 2015•0 citations
We present a complete analysis on energy and wave function of Harmonic oscillator with simultaneous non-hermitian transformation of co-ordinate ((x→(x+iλp)(1+βλ)(x \rightarrow \frac{(x+ i\lambda p)}{\sqrt{(1+\beta \lambda)}}(x→(1+βλ)​(x+iλp)​ and momentum (p→(p+iβx)(1+βλ)(p \rightarrow \frac{(p+ i\beta x)}{\sqrt{(1+\beta \lambda)}}(p→(1+βλ)​(p+iβx)​ for getting energy eigenvalue using perturbation theory under iso-spectral condition. Further we notice that two different frequency of oscillation (w1,w2w_{1}, w_{2}w1​,w2​)correspond to same energy eigenvalue, which can also be verified using Lie algebraic approach [Zhang this http URL J.Math.Phys 56 ,072103 (2015)]. Interestingly wave function analysis using similarity transformation [F.M. Fernandez, Int. J. Theo. Phys. (2015)(in Press)] refers to a very special case.
Abstract We present a complete energy and wavefunction analysis of a Harmonic oscillator with simultaneous non-hermitian transformations of co-ordinate ( x → ( x + i λ p ) ( 1 + β λ ) ) (x→(x+iλp)(1+βλ))(x \rightarrow \frac{(x + i\lambda p)}{\sqrt{(1+\beta \lambda)}})(x→(1+βλ)​(x+iλp)​) and momentum ( p → ( p + i β x ) ( 1 + β λ ) ) (p→(p+iβx)(1+βλ))(p \rightarrow \frac {(p+i\beta x)}{\sqrt{(1+\beta \lambda)}})(p→(1+βλ)​(p+iβx)​) using perturbation theory under iso-spectral conditions. We observe that two different frequencies of oscillation (w 1, w 2)correspond to the same energy eigenvalue, - which can also be verified using a Lie algebraic approach.
We propose zero energy correction method for non-Hermiition Harmonic oscillator under simultaneous transformation of co-ordinate (x→(x+iλp)(1+βλ)(x \rightarrow \frac{(x+ i\lambda p)}{\sqrt{(1+\beta \lambda)}}(x→(1+βλ)​(x+iλp)​ and momentum (p→(p+iβx)(1+βλ)(p \rightarrow \frac{(p+ i\beta x)}{\sqrt{(1+\beta \lambda)}}(p→(1+βλ)​(p+iβx)​ for getting energy eigenvalue in place of extending the idea of gaugelike transformation proposed earlier in momentum transformation (p→p+iβx)(p \rightarrow p+i\beta x)(p→p+iβx) by Z.Ahmed [Phys.Lett A 294,287 (2002)]. Further energy of non-Hermitian Harmonic oscillator remains the same as that of Harmonic oscillator. PACS: 03. 65 Db Key words.Non-Hermitian Harmonic oscillator, Perturbation theory,Energy level.
For the first time in the literature of Quantum Physics, we present complex energy eigenvalues of non-Hermitian Harmonic Oscillator $H=\frac{(p+iLx)}^{2}}{2} + W^{2} \frac{x^{2}}{2}$ with real wave function having positive frequency of vibration (w)(w)(w) under some selective choice of LLL and WWW .Interestingly for the same values of LLL and WWW, if the frequency of vibration www in the real wave function is (some how) related as $w=L\pmW$ or w=W−Lw=W-Lw=W−L then the same oscillator can reflect either pure positive or negative energy eigenvalues.The real energy levels are in conformity with the perturbative calculation. PACS :03.65.Db;this http URL. Key words: Positive frequency, real wave function, complex energy, real positive energy,negative energy.
The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
Here we clearly notice that invariance in commutation relation can be associated with complex energy eigenvalues. Hence present work clearly demonstates that as such we, do not have a general eigenvalue condition in Quantum Physics. Here it is shown that existing all the three eigenvalue conditions namely: Hermiticity, PT symmetry and Pseudo-Hermiticity, are not sufficient enough even to explain the spectra of Harmonic Oscillator under non-Hermitian transformation of momentum, co-ordinate or both, on appearance of complex energy eigenvalues. In fact complex energy eigenvalues become an inherent feature of transformed oscillator, when its frequency of oscillation matches with that of variational based calculation.