Abstract
We present a complete energy and wavefunction analysis of a Harmonic oscillator with simultaneous non-hermitian transformations of co-ordinate
1 Introduction
In physics the generation of a new Hamiltonian typically relates to a transformation of co-ordinate
or momentum
or both. Here x, p are related to the original Hamiltonian: H(x, p) and x̄,p̄ are related to the new Hamiltonian H(x̄,p̄. This type of transformation is well known in classical physics (canonical transformation) [1]. However in quantum physics one has to be careful about the commutation relation: as the commutation relation invariance results in iso-spectral behaviour of the Hamiltonian [2]. Mathematically
where h is Plank’s constant. Of interest in this commutation relation is that one can have simultaneous transformations of co-ordinate and momentum. First the Hamiltonian must be written either in the momentum dimension or co-ordinate dimension. So momentum has to be defined in such a way that from the dimension point of view both co-ordinate and momentum are equally acceptable. In order to give an example of this we consider an exactly solvable model. The widely used exactly solvable model is the Harmonic Oscillator(HO) [1, [2], abd this plays a major role in understanding the limitations of various approximation methods such as the variational method, W.K.B. method, perturbation method, etc. Hence, for simplicity, we address the above commutation relation using HO as an example [2]. The Hamiltonian in old co-ordinate and momentum is written as
Here, m is mass and w0 stands for the frequency of oscillation. The above expression can be written as a momentum base relation as
This implies that the dimension of p remains the same as that of mw0x. Here one can introduce new momentum as [3]
where β is a simple numerical constant which can be varied arbitrarily in order to generate a large number of systems. Similarly one can write the Hamiltonian of the Harmonic Oscillator in co-ordinate base as
In this case one can introduce a new co-ordinate x̄ as
It is seen that the commutation relation is not invariant. However in the later part of the study we will consider its appropriate form [4–6]. Above, λ is a dimensionss parameter like β, which allows a large class of Hamiltonians to be generated using different numerical values. It is clear from the above two transformations that λ, β are dimen-sionless numbers. Further introducing “i” (complex factor ) in the above transformation allows us to study the HO in complex space [4–6]. This is relevant due to the recent experiment in parity (P) - time (T) meta material [7]. Mathematically, the complex nature of transformation means that it must satisfy the following PT condition [8]
So that the product x̄p̄ appearing in above commutation relation becomes PT invariant. Here P stands for parity transformation i.e under P ; x → -x; p → -p. Similarly under time reversal T ; x → x; p → -p ; i → -i. In other words one is likely to study the new Hamiltonian under complex transformation i.e under the PT transformations. We also note that when the co-ordinate and momentum simultaneously under go non-Hermitian transformations [4–6], its energy eigenvalue can be iso-spectral to the original Harmonic oscillator. Under iso-spectral conditions, the wavefunction of the transformed Hamiltonian differs drastically from that of the original Hamiltonian [4–6]. However, a complete picture on wavefunction is still in need of further study. Further, no such explicit calculations on wavefunctions are available at present [4–6]. Hence the aim of this paper is to present a complete picture of wavefunction and energy under iso-spectral conditions using perturbation theory under the PT transformation as discussed above.
2 Energy levels and Wavefunction of Simple Harmonic Oscillator (SHO)
We consider the case m = w0 = h̄ = 1 and write the Hamiltonian of SHO [1, [2] as
whose exact energy eigenvalues is [2]
and corresponding wavefunction is [2]
where Hn(x) is the Hermite polynomial.
3 SHO under non-Hermitian transformation of co-ordinate (x) and momentum (p)
Consider the non-Hermitian transformations of x and p as [4–6]
and
In this transformation we note that the transformed co-ordinate and momentum preserve the commutation relation [4–6] i.e.
Now the new Hamiltonian with transformed x and p becomes non-Hermitian in nature and is
4 Second Quantization and Hamiltonian
In order to solve the above Hamiltonian, we use the second quantization formalism as [2]
and
where the creation operator, a+ and annhilation operator a satisfy the commutation relation
and ω is an unknown parameter. The Hamiltonian can be written as
where
and
4.1 Zero Energy Correction Method (Case Study for U=0)
Now we solve the the eigenvalue relation:
using perturbation theory as follows. Here we express
The zeroth order energy
where
and
The energy correction terms will give zero contribution if the parameter is determined from non-diagonal terms of HN[4]
Let the coefficient of a2 be zero [4] i.e.
which leads to( considering positive sign)
In this case,
Now the perturbation correction term is
In this case one will notice that
Hence it follows that all orders of energy corrections will be zero. Let us consider explicitly corrections up to third order using a standard perturbation series given in literature [2, 4, 9–14], which can be written as
or
Here second order correction is zero due to
with
Form the above, analysis it is seen that the total energy of the Harmonic oscillator and the Harmonic oscillator with non-hermitian transformation remains the same.
4.2 Corresponding Wavefunction using Perturbation Theory
Here we find the wavefunction as
where
The normalization condition here can be written as [9–12]
and so also the eigenvalue relation
4.3 Zero Energy Correction Method (Case Study for V=0)
Let the coefficient of (a+)2 be zero [4] i.e.
which leads to
In this case, ω is calculated using similarity transformation [6] and remains the same as ω2· Now the perturbation term becomes
In this case we note that
or
Here second order correction is zero due to
which is the same as the energy level of harmonic oscillator as given in Eq. (2) and
4.4 Corresponding Wavefunction using Perturbation Theory
Here we consider the wavefunction as
In its compact form
Here we note that for x → ∞ i.e.
and
In this case, the normalization condition can be written as [9–12]
and so also the eigenvalue relation
5 Com parision with Similarity Transformation using Lie-algebra [5]
In the above we note that two different frequencies( w1, w2)corresponds to the same energy eigenvalue. Now we compare our results with that of Zhang et al. [5] using Lie-algebra as follows.Previous authors consider the Hamiltonian
having energy eigenvalue
Case-I
In this case we have the Hamiltonian
Now comparing we get
Hence the
which is the same result as given earlier using perturbation theory.
Case-II
In this case we have the Hamiltonian
Now comparing we get
Hence the
which, once again the same as given earlier using perturbation theory. We see that the results of Lie-algebra match those of perturbation theory for energy level calculation.
6 Comparision with Similarity Transformation [6]
It is worth mentioning that Fernandez [6] has calculated a groundstate wave function of this oscillator using similarity transformation.The explict expression for the wave function [6] is
It is easy to check that our result for H = 0 remains the same as that of Fernandez [6]. However, we do not have literature for further comparison.
7 Conclusion
In this paper, we suggest a simpler procedure for calculating energy levels and wave function of the non-Hermitian harmonic oscillator under simultaneous transfromation of co-ordinate and momentum using perturbation theory. Further more the energy eigenvalue calculated using perturbation theory matches that of the Lie algebric method [5]. At this point it is necessary to mention that the ground state wave function calculated by Fernandez [6] refers to the zeroth order wave function as reflected in VI(b). However we present a complete picture on wave function. In all cases, we show that the energy levels remain the same as for the simple Harmonic oscillator. If the parameter ω is determined using variational principle [13] i.e.
This is directly due to non-commuting operators i.e. [H, HD] ≠ 0 corresponding to different wave functions. Further,one can use nonlinear perturbation series [12] and conclude that if the parameter ω is determined using the condition
Acknowledgement
We thank Prof. A. Khare for suggesting improvements and Prof. M. Znojil for critical reading of the manuscript.We also thank both the Referees for suggesting improvements to the manuscript.
References
[1] Biswas S.N., Classical Mechanics, Ist ed,Books and Allied (P) Ltd Calcutta, India, 1998Search in Google Scholar
[2] Schiff L.I., Quantum Mechanics, 3rd ed, McGraw-Hill, Singapore, 1985Search in Google Scholar
[3] Ahmed Z., Pseudo-Hermiticity of Hamiltonians under gauge-like transformation :real spectrum of non-Hermitian Hamiltonians, Phys. Lett., 2002, A 294, 287-29110.1016/S0375-9601(02)00124-XSearch in Google Scholar
[4] Rath B. and Mallick P., Zero energy correction method for non-Hermitian Harmonic oscillator with simultaneous transformation of co-ordinate and momentum. arxiv:1501.06161(quant-ph)Search in Google Scholar
[5] Zhang H.B., Jiang G.Y. and Wang G.C., Unified algebric method to non-Hermitian systems with Lie algebric linear structure, J. Math. Phys., 2015, 56, 072103 (This paper also reflects iso-spectral condition as reported in [4])10.1063/1.4926354Search in Google Scholar
[6] Fernandez F.M., Non-Hermitian Hamiltonians and Similarity transfromation, arxiv:1502.02694[quant-ph];Int. J. Theo. Phys., 2015,55,843-850 (This paper uses similarity transformation to reflect iso-spectral condition reported in [4])10.1007/s10773-015-2724-xSearch in Google Scholar
[7] Feng L., Xu X.L., Fegadolli W.S., Lu M.H/, Oliveira J.E.B., Almeida V.R., Chen Y.F. and Scherer A., Experimental demonstration of a unidirectional reflectionless parity time metamaterial at optical frequencies. Nature Materials, 2013, 12, 108-11310.1038/nmat3495Search in Google Scholar PubMed
[8] Bender C.M. and Boettcher S., Real Spectra in Non-Hermitian Hamiltonians Having PJ Symmetry. Phys. Rev. Lett, 1998, 80(24),5243-524610.1103/PhysRevLett.80.5243Search in Google Scholar
[9] Zettili N., Quantum Mechanics:Concepts and applications, 2nd ed, John Wiley, New York, 2001Search in Google Scholar
[10] Landau L.D. and Lifshtiz E.M., Quantum Mechanics, 3rd ed, Elsevier, Amsterdam, 2011.Search in Google Scholar
[11] Rath B., A new approach on wave function and energy level calculation through perturbation theory, J. Phys. Soc. Jpn, 1998, 67(9), 3044-304910.1143/JPSJ.67.3044Search in Google Scholar
[12] Rath B., Case study of the convergency of nonlinear perturbation series: Morse-Feshbach nonlinear series. Int. J. Mod. Phys, 1999, A14(13), 2103-211510.1142/S0217751X99001068Search in Google Scholar
[13] Rath B., Second quantization, variational principle and perturbation theory for the anharmonic oscillator. Eur. J. Phys, 1990, 11, 184-18510.1088/0143-0807/11/3/110Search in Google Scholar
[14] Rath B., Energy level calculation through perturbation theory. Phys. Rev 1990, A 42(5), 2520-252310.1103/PhysRevA.42.2520Search in Google Scholar PubMed
© 2016 Biswanath Rath and P. Mallick
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.