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arXiv:0908.2341 · Symmetry Integrability and Geometry Methods and Applications

Comment on ''Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty''

Bijan Kumar Bagchi

Published Sep 17, 2009• 2 citations
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Abstract

We demonstrate that the recent paper by Jana and Roy entitled "Non-Hermitian quantum mechanics with minimal length uncertainty" [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions.We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354].In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former.

DOI
10.3842/sigma.2009.089

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Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty

Tapan Jana

We study non-Hermitian quantum mechanics in the presence of a minimal length.In particular we obtain exact solutions of a non-Hermitian displaced harmonic oscillator and the Swanson model with minimal length uncertainty.The spectrum in both the cases are found to be real.It is also shown that the models are η pseudo-Hermitian and the metric operator is found explicitly in both the cases.

arXiv:0908.1755•Symmetry Integrability and Geometry Methods and Applications•Aug 12, 2009•8 citations

Reply to Comment on "Non Hermitian Quantum Mechanics with Minimal Length Uncertainty", arXiv:0908.2341

Tapan Jana, Pinaki Roy

It is shown that the results of ref [1] are consistent.

arXiv:0910.5601•Oct 29, 2009•0 citations

Is Pseudo-Hermitian Quantum Mechanics an Indefinite-Metric Quantum Theory?

Mostafazadeh, A

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Comment on "Reply to Comment on Time-dependent Quasi-Hermitian Hamiltonians and the Unitary Quantum Evolution"

Mostafazadeh, Ali

I point out that if one defines the operator $U_R(t)$ as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of $U_R(t)$ as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator $H$ but an operator $H'$ which differs from $H$ if the metric operator is time-dependent. In the latter case this effective Hamiltonian $H'$ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.

arXiv:0711.1078•Nov 7, 2007•4 citations

Reply to Comment on ``Time-dependent quasi-Hermitian Hamiltonians and the unitary quantum evolution''

Miloslav Znojil

In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.

arXiv:0711.0514•Nov 5, 2007•6 citations

The construction of a general inner product in non-Hermitian quantum theory and some explanation for the nonuniqueness of the C operator in PT quantum mechanics

Frieder Kleefeld

Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.

arXiv:0906.1011•Jun 4, 2009•10 citations
We demonstrate that the recent paper by Jana and Roy entitled "Non-Hermitian quantum mechanics with minimal length uncertainty" [SIGMA 5 (2009), 083, 7 pages, arXiv:0908.1755] contains various misconceptions.We compare with an analysis on the same topic carried out previously in our manuscript [arXiv:0907.5354].In particular, we show that the metric operators computed for the deformed non-Hermitian Swanson models differs in both cases and is inconsistent in the former.
I point out that if one defines the operator UR(t)U_R(t)UR​(t) as done by M. Znojil in his reply [arXiv:0711.0514v1] to my comment [arXiv:0711.0137v1] and also accepts the validity of the defining relation of UR(t)U_R(t)UR​(t) as given in his paper [arXiv:0710.5653v1], one finds that the time-evolution of the associated quantum system is not governed by the Schrödinger equation for the Hamiltonian operator HHH but an operator H′H'H′ which differs from HHH if the metric operator is time-dependent. In the latter case this effective Hamiltonian H′H'H′ is not observable. This is consistent with the conclusions of my paper [Phys. Lett. B 650, 208 (2007), arXiv:0706.1872v2] which allow for unitary time-evolution generated by unobservable Hamiltonians.
In his fresh "Comment" (arXiv:0711.0137v1), A. Mostafazadeh reacts on my very recent letter (arXiv:0710.5653v1) where I tried to clarify certain misunderstandings which occurred in A. M., Phys. Lett. B \textbf{650}, 208 (2007) [arXiv:0706.1872v2, "Paper"]. As long as the "Comment" offers a new support of the original assertions made in the "Paper", I feel obliged to re-clarify the matter by extending my argumentation. I insist that it is possible to escape the main conclusion of the "Paper", indeed. In particular, I point out a gap in the new calculations in "Comment", add a few remarks on the notation and reconfirm that the unitarity of the time-evolution DOES NOT require the time-independence of the metric operator.
Most recently it has been observed e.g. by Bender and Klevansky (arXiv:0905.4673 [hep-th]) that the C-operator related to a PT-symmetric non-Hermitian Hamilton operator is not unique. Moreover it has been remarked by Shi and Sun (arXiv:0905.1771 [hep-th]) very recently that there seems to exist a well defined inner product in the context of the Hamilton operator of the PT-symmetric non-Hermitian Lee model yielding a different C-operator as compared to the one previously derived by Bender et al.. The puzzling observations of both manuscripts are reconciled and explained in the present manuscript as follows: the actual form of the metric operator (and the induced C-operator) related to some non-Hermitian Hamilton operator constructed along the lines of Shi and Su depends on the chosen normalization of the left and right eigenvectors of the Hamilton operator under consideration and is therefore ambiguous. For a specific PT-symmetric 2x2-matrix Hamilton operator it is shown that - by a suitable choice of the norm of its eigenvectors - the metric operator yielding a positive semi-definite inner product can be made even independent of the parameters of the considered Hamilton operator. This surprising feature makes in turn the obtained metric operator rather unique and attractive. For later convenience the metric operator for the Bosonic and Fermionic (anti)causal harmonic oscillator is derived.