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Brazilian Journal of Physics

Non-Hermitian PT Symmetric Hamiltonian with Position-Dependent Masses: Associated Schrödinger Equation and Finite-Norm Solutions

Fernando Dantas Nobre, M. A. Rego-Monteiro

Published Nov 7, 2014• 12 citations
Publisher
DOI
10.1007/s13538-014-0277-8

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A selection rule for transitions in PT-symmetric quantum theory

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Carl Bender and collaborators have developed a quantum theory governed by Hamiltonians that are PT-symmetric rather than Hermitian. To implement this theory, the inner product was redefined to guarantee positive norms of eigenstates of the Hamiltonian. In the general case, which includes arbitrary time-dependence in the Hamiltonian, a modification of the Schrödinger equation is necessary as shown by Gong and Wang to conserve probability. In this paper, we derive the following selection rule: transitions induced by time dependence in a PT-symmetric Hamiltonian cannot occur between normalized states of differing PT-norm. We show three examples of this selection rule in action: two matrix models and one in the continuum.

AIP Advances•Aug 1, 2017•6 citations
Jia and Dutra ( 2006 J. Phys. A: Math. Gen. 39 11877 ) have considered the one- dimensional non-Hermitian complexified potentials with real spectra in the context of position-dependent mass in the Dirac equation. In their second example, a smooth step shape mass distribution is considered and a non-Hermitian non- PT -symmetric Lorentz vector potential is obtained. They have mapped this problem into an exactly solvable Rosen–Morse Schrödinger model and claimed that the energy spectrum is real. The energy spectrum they have reported is pure imaginary or at best forms an empty set. Their claim on the reality of the energy spectrum is fragile, therefore.
Carl Bender and collaborators have developed a quantum theory governed by Hamiltonians that are PT-symmetric rather than Hermitian. To implement this theory, the inner product was redefined to guarantee positive norms of eigenstates of the Hamiltonian. In the general case, which includes arbitrary time-dependence in the Hamiltonian, a modification of the Schrödinger equation is necessary as shown by Gong and Wang to conserve probability. In this paper, we derive the following selection rule: transitions induced by time dependence in a PT-symmetric Hamiltonian cannot occur between normalized states of differing PT-norm. We show three examples of this selection rule in action: two matrix models and one in the continuum.