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

Biswanath Rath, Pravanjan Mallick

Published Jan 25, 2015• 1 citation

Abstract

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.

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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

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Choice of a metric for the non-Hermitian oscillator

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arXiv:quant-ph/0611150Journal of Physics A Mathematical and TheoreticalDec 12, 200687 citations

Response to “Comment on ‘Quantization of the damped harmonic oscillator’” [J. Math. Phys. 60, 094101 (2019)]

M. Serhan, M. Abusini, Ahmed Al-Jamel +2 more

arXiv:1903.00352Journal of Mathematical PhysicsSep 1, 20194 citations

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

Biswanath Rath, Pravanjan Mallick

Published Jan 25, 2015• 1 citation

OpenAlex

Abstract

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.

Related papers

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

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

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

Biswanath Rath

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.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.

May 19, 20150 citations

Choice of a metric for the non-Hermitian oscillator

Dibwe Pierrot Musumbu, H. B. Geyer, W. D. Heiss

The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian -symmetric part, can be transformed into a Hermitian Hamiltonian. This is achieved by introducing a metric which, in general, renders other observables such as the usual momentum or position as non-Hermitian operators. The metric depends on one real parameter, the full range of which is investigated. The explicit functional dependence of the metric and each associated Hamiltonian is given. A specific choice of this parameter determines a specific combination of position and momentum as being an observable; this can be in particular either standard position or momentum, but not both simultaneously. Singularities of the metric are explored and their removability is investigated. The physical significance of these findings is discussed.The harmonic oscillator Hamiltonian, when augmented by a non-Hermitian -symmetric part, can be transformed into a Hermitian Hamiltonian. This is achieved by introducing a metric which, in general, renders other observables such as the usual momentum or position as non-Hermitian operators. The metric depends on one real parameter, the full range of which is investigated. The explicit functional dependence of the metric and each associated Hamiltonian is given. A specific choice of this parameter determines a specific combination of position and momentum as being an observable; this can be in particular either standard position or momentum, but not both simultaneously. Singularities of the metric are explored and their removability is investigated. The physical significance of these findings is discussed.

arXiv:quant-ph/0611150Journal of Physics A Mathematical and TheoreticalDec 12, 200687 citations

Response to “Comment on ‘Quantization of the damped harmonic oscillator’” [J. Math. Phys. 60, 094101 (2019)]

M. Serhan, M. Abusini, Ahmed Al-Jamel +2 more

This is a response to a recently reported comment [F. M. Fernández, J. Math. Phys. 60, 094101 (2019)] on paper [M. Serhan et al., J. Math. Phys. 59, 082105 (2018)] regarding the quantization of the damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be first achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can be reached using the symmetrization rule. The canonical quantization on the resulted Hermitian Hamiltonian yields the real energy eigenvalues as exactly given by Eq. (35) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)].This is a response to a recently reported comment [F. M. Fernández, J. Math. Phys. 60, 094101 (2019)] on paper [M. Serhan et al., J. Math. Phys. 59, 082105 (2018)] regarding the quantization of the damped harmonic oscillator using a non-Hermitian Hamiltonian with real energy eigenvalues. We assert here that the calculation of Eq. (29) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)] is incorrect, and thus the subsequent steps via the Nikiforov-Uvarov method are affected, and the energy eigenvalues should have been complex. However, we show here that the Hermiticity of the Hamiltonian should be first achieved to make the correct transition from classical Hamiltonian to quantum counterpart, and this can be reached using the symmetrization rule. The canonical quantization on the resulted Hermitian Hamiltonian yields the real energy eigenvalues as exactly given by Eq. (35) of Serhan et al. [J. Math. Phys. 59, 082105 (2018)].

arXiv:1903.00352Journal of Mathematical PhysicsSep 1, 20194 citations

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)

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.

(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.