We notice that PT symmetric non-Hermitian one dimensional simple Harmonic Oscillator under simultaneous transformation of co-ordinate and momentum with proper choice of positive oscillating frequency can reflect negative spectrum with well behaved wave function in real space. We also present a suitable comuter programme to realise the negative spectrum directly. PACS(2008) : 03.65.Db
We notice that PT symmetric non-Hermitian one dimensional simple Harmonic Oscillator under simultaneous transformation of co-ordinate and momentum with proper choice of positive oscillating frequency can reflect negative spectrum with well behaved wave function in real space. We also present a suitable comuter programme to realise the negative spectrum directly. PACS(2008) : 03.65.Db
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.
We present a complete analysis on energy and wave function of Harmonic oscillator with simultaneous non-hermitian transformation of co-ordinate ((x→(1+βλ)(x+iλp) and momentum (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,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.
We propose zero energy correction method for non-Hermiition Harmonic oscillator under simultaneous transformation of co-ordinate (x→(1+βλ)(x+iλp) and momentum (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) 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.
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→(1+βλ)(x+iλp)) and momentum ( p → ( p + i β x ) ( 1 + β λ ) ) (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.
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.
Negative-energy PT-symmetric Hamiltonians
The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian H=p2+x2(ix)ε has real, positive, and discrete eigenvalues for all ε≥0. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues En=2n+1 (n=0, 1, 2, 3, ...) at ε=0. However, the harmonic oscillator also has negative eigenvalues En=−2n−1 (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian H=p2+x2(ix)ε also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, ε lies in the range $(4N-6)/3