Complex Energy of Harmonic Oscillator under Non-Hermitian transformation of momentum with real wave function

Biswanath Rath

Published May 19, 2015• 0 citations

Abstract

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 $(w)$ under some selective choice of $L$ and $W$ .Interestingly for the same values of $L$ and $W$, if the frequency of vibration $w$ in the real wave function is (some how) related as $w=L\pmW$ or $w=W-L$ 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.

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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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Jan 25, 20151 citation

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

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Jul 9, 20150 citations

Complex Energy of Harmonic Oscillator under Non-Hermitian transformation of momentum with real wave function

Biswanath Rath

Published May 19, 2015• 0 citations

OpenAlex

Abstract

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)

under some selective choice of

L

and

W

.Interestingly for the same values of

L

and

W

, if the frequency of vibration

w

in the real wave function is (some how) related as

w=L\pmW

w=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.

Related papers

Negative spectrum in Harmonic oscillator under simultaneous Non-hermitian transformation of co-ordinate and momentum with Real wave function

Biswanath Rath

arXiv:1502.07891Feb 27, 20151 citation

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

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.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.

Jan 25, 20151 citation

Energy and Wave function Analysis on Harmonic Oscillator Under Simultaneous Non-Hermitian Transformations of Co-ordinate and Momentum: Iso-spectral case

Biswanath Rath, Pravanjan Mallick

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.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.

Open PhysicsJan 1, 20160 citations

DEFORMED HARMONIC OSCILLATOR FOR NON-HERMITIAN OPERATOR AND THE BEHAVIOR OF PT AND CPT SYMMETRIES

A. D. Jannussis, Kostas Vlachos, V. Papatheou +1 more

In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator [Formula: see text] where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators [Formula: see text] in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.In the present paper we study the deformed harmonic oscillator for the non-Hermitian operator [Formula: see text] where λ,θ are real positive parameters, since the parameters α,β,m are for the general case complex. For the case α=1,β=1 and mass m real, we find the eigenfunctions and eigenvalues of energy, the coherent states, the time evolution of the operators [Formula: see text] in the Heisenberg picture and the uncertainty relations. In this case the operator ℋ is Hermitian and PT-symmetric. Also for the case m complex α=1,β=1, the operator ℋ is non-Hermitian and no more PT symmetric, but CPT symmetric with real discrete positive spectrum and the CPT symmetry is preserved. In the general case α,β,m complex, for the non-Hermitian operator ℋ, we obtain complex spectrum and for the special values of the complex parameters α,β the spectrum is real discrete and positive and the CPT symmetry is preserved. The general problem of deformed oscillator for non hermitian operators can be applied to the Solid State Physics.

International Journal of Modern Physics BJun 30, 20062 citations

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

Biswanath Rath, Pravanjan Mallick

We propose zero energy correction method for non-Hermiition Harmonic oscillator under simultaneous 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 in place of extending the idea of gaugelike transformation proposed earlier in momentum transformation $(p \rightarrow p+i\beta x)$ 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.We propose zero energy correction method for non-Hermiition Harmonic oscillator under simultaneous 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 in place of extending the idea of gaugelike transformation proposed earlier in momentum transformation $(p \rightarrow p+i\beta x)$ 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.

Jan 25, 20151 citation

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.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, 20150 citations

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)

under some selective choice of

L

and

W

.Interestingly for the same values of

L

and

W

, if the frequency of vibration

w

in the real wave function is (some how) related as

w=L\pmW

w=W-L

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.

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.

We propose zero energy correction method for non-Hermiition Harmonic oscillator under simultaneous 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 in place of extending the idea of gaugelike transformation proposed earlier in momentum transformation

(p \rightarrow p+i\beta x)

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.