Open Physics

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

Published Jan 1, 2016• 0 citations

Abstract

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.

DOI: 10.1515/phys-2016-0054

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Open Physics

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

Published Jan 1, 2016• 0 citations

Publisher

Abstract

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.

DOI: 10.1515/phys-2016-0054

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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

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.

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A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harmonic oscillator, and L is its angular momemtum vector. It is shown that H can be written in the form h/ω(2ν°ν+λ°⋅λ+3/2), where ν° and ν are raising and lowering (boson) operators for ν°ν, which has nonnegative integer eigenvalues k; and λ° and λ are raising and lowering operators for λ°⋅λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form h/ω(2k+l+3/2), previously obtained only by working in the coordinating the operators ν° and λ° to a ’’vacuum’’ vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν,ν°, λ, and λ°, in order to clarify their physical interpretation.A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harmonic oscillator, and L is its angular momemtum vector. It is shown that H can be written in the form h/ω(2ν°ν+λ°⋅λ+3/2), where ν° and ν are raising and lowering (boson) operators for ν°ν, which has nonnegative integer eigenvalues k; and λ° and λ are raising and lowering operators for λ°⋅λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form h/ω(2k+l+3/2), previously obtained only by working in the coordinating the operators ν° and λ° to a ’’vacuum’’ vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν,ν°, λ, and λ°, in order to clarify their physical interpretation.

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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)}})

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 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.

(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.