What are some recommended books on non-standard quantum mechanical models?

[SemM]

By fresh42's invitation, I here attach a paper which is related to the post on non-standard qm-models

https://arxiv.org/pdf/cond-mat/9705290

 

[A. Neumaier]

By fresh42's invitation, I here attach a paper which is related to the post on non-standard qm-models

https://arxiv.org/pdf/cond-mat/9705290

Nonhermitian quantum mechanics is just quantum mechanics with Hamiltonians that are selfadjoint not in the standard inner product but in a nonstandard one.

 

[SemM]

Nonhermitian quantum mechanics is just quantum mechanics with Hamiltonians that are selfadjoint not in the standard inner product but in a nonstandard one.

It's not apparent to me which metric space do they use, and how do they derive physically sound eigenvalues without an inner product.

Thanks!

 

[A. Neumaier]

It's not apparent to me which metric space do they use, and how do they derive physically sound eigenvalues without an inner product.

Actually, I had not looked closely enough at the paper. I thought it was about PT-symmetric quantum mechanics, as in http://aip.scitation.org/doi/pdf/10.1063/1.532860

But the paper is about dissipative quantum mechanics, where the dissipation is modeled by a complex potential, often called an optical potential. In this case, the usual inner product is used, the eigenvalues typically have a nonzero imaginary part related to the decay rate, and eigenvectors are usually no longer orthogonal. Just like in the eigenvalue problem for nonhermitian matrices.

 

[SemM]

Actually, I had not looked closely enough at the paper. I thought it was about PT-symmetric quantum mechanics, as in http://aip.scitation.org/doi/pdf/10.1063/1.532860

But the paper is about dissipative quantum mechanics, where the dissipation is modeled by a complex potential, often called an optical potential. In this case, the usual inner product is used, the eigenvalues typically have a nonzero imaginary part related to the decay rate, and eigenvectors are usually no longer orthogonal. Just like in the eigenvalue problem for nonhermitian matrices.

I get non-standard values when I consider complex ODE's with similarities to the Schrödinger model except for that they are not equaled to EY, but to zero. It appears that Bender uses a deformed potential, H=p2+x2(ix), where that imaginary component is present. That occurs also in rogue wave models, with imaginary components at the momentum term. I am not sure however, when an ODE has realistic properties - though it gives non-trivial eigenvalues, or when it does not.

Bender writes about this:

The complex quantum mechanical Hamiltonian:

H=p2+x2(ix),

was investigated. Despite the lack of conventional Hermiticity the spectrum of H is real and positive for all e >0. As shown in Fig. 11 in this paper and Fig. 1 of Ref. 1, the spectrum is discrete and each of the energy levels increases as a function of increasing e . We will argue below that the reality of the spectrum is a consequence of PT invariance.

Are you aware of a paper that shows various models of ODE's for quantum physical phenomena (Hermitian or not), except the Schrödinger eqn and the Wave eqn?

Thanks

 

[A. Neumaier]

Do you know of a paper that shows various models of ODE's for quantum physical phenomena, except the Schrödinger eqn and the Wave eqn?

Except for problems with a single degree of freedom (or with only spin degree of freedom), one always has PDEs, not ODEs (or ODEs in function spaces, which is the former in disguise).

In nonrelativistic quantum mechanics based on wave functions you always have the Schroedinger equation, just with different Hamiltonians. The conservative case has a selfadjoint $H$ (in the PT symmetric case with respect to a nonstandard inner product), the dissipative case typically has an optical potential. There are also potentials derived by complex scaling (N. Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports 302.5-6 (1998): 212-293.)

But the dissipative case is often better modeled by using density operators/matrices, then the dynamics is that of a Lindblad equation.

 

[SemM]

Except for problems with a single degree of freedom (or with only spin degree of freedom), one always has PDEs, not ODEs (or ODEs in function spaces, which is the former in disguise).

In nonrelativistic quantum mechanics based on wave functions you always have the Schroedinger equation, just with different Hamiltonians. The conservative case has a selfadjoint $H$ (in the PT symmetric case with respect to a nonstandard inner product), the dissipative case typically has an optical potential. There are also potentials derived by complex scaling (N. Moiseyev, Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling, Physics Reports 302.5-6 (1998): 212-293.)

But the dissipative case is often better modeled by using density operators/matrices, then the dynamics is that of a Lindblad equation.

Yes, the Lindblad equation is required to make the Non-Hermitian matrices time-unitary. A Swedish professor developed them: I wonder why he didn't get a Nobel prize for them. I have found this too:

https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.80.5243

Thanks for that reference, I am downloading it.

 

[DrClaude]

You should get your hands on a copy of Moiseyev's book Non-Hermitian Quantum Mechanics.

 

[A. Neumaier]

You should get your hands on a copy of Moiseyev's book Non-Hermitian Quantum Mechanics.

Nice. I didn't know this book. Another useful book on the topic is

V.I. Kukulin, V.M. Krasnopol'sky and I. Horacek,
Theory of Resonances. Principles and Applications,
Kluwer, Dordrecht 1989.

 

[SemM]

Nice. I didn't know this book. Another useful book on the topic is

V.I. Kukulin, V.M. Krasnopol'sky and I. Horacek,
Theory of Resonances. Principles and Applications,
Kluwer, Dordrecht 1989.

Have a look at this too: