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Physical Review A

Complex density probability in non-Hermitian quantum mechanics: Interpretation and a formula for resonant tunneling probability amplitude

Hadas Barkay, Nimrod Moiseyev

Published Sep 10, 2001• 19 citations
Publisher

Abstract

In non-Hermitian quantum mechanics the density probability is a complex function $\ensuremath{\rho}.$ Here we show that when ${\ensuremath{\Psi}}_{\mathrm{left}}\ensuremath{\ne}{\ensuremath{\Psi}}_{\mathrm{right}}$ the spatial phase of $\sqrt{\ensuremath{\rho}}=({\ensuremath{\Psi}}_{\mathrm{left}}{\ensuremath{\Psi}}_{\mathrm{right}}{)}^{1/2}$ and not of ${\ensuremath{\Psi}}_{\mathrm{right}}$ or ${\ensuremath{\Psi}}_{\mathrm{left}}$ provides the phase of the measurable complex transition probability amplitude in scattering experiments, $\mathrm{arg}{t(E)}.$ Using the non-Hermitian representation of quantum mechanics we derived a formula for the resonant tunneling probability amplitude of an electron which is scattered from a general potential. We illustrate the applicability of our formula to a one-dimensional double-barrier potential problem, although the derivation of this formula holds for many-body problems with multidimensional potentials.

DOI
10.1103/physreva.64.044702

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In non-Hermitian quantum mechanics the density probability is a complex function \ensuremathρ.\ensuremath{\rho}.\ensuremathρ. Here we show that when \ensuremathΨleft\ensuremath≠\ensuremathΨright{\ensuremath{\Psi}}_{\mathrm{left}}\ensuremath{\ne}{\ensuremath{\Psi}}_{\mathrm{right}}\ensuremathΨleft​\ensuremath=\ensuremathΨright​ the spatial phase of \ensuremathρ=(\ensuremathΨleft\ensuremathΨright)1/2\sqrt{\ensuremath{\rho}}=({\ensuremath{\Psi}}_{\mathrm{left}}{\ensuremath{\Psi}}_{\mathrm{right}}{)}^{1/2}\ensuremathρ​=(\ensuremathΨleft​\ensuremathΨright​)1/2 and not of \ensuremathΨright{\ensuremath{\Psi}}_{\mathrm{right}}\ensuremathΨright​ or \ensuremathΨleft{\ensuremath{\Psi}}_{\mathrm{left}}\ensuremathΨleft​ provides the phase of the measurable complex transition probability amplitude in scattering experiments, argt(E).\mathrm{arg}{t(E)}.argt(E). Using the non-Hermitian representation of quantum mechanics we derived a formula for the resonant tunneling probability amplitude of an electron which is scattered from a general potential. We illustrate the applicability of our formula to a one-dimensional double-barrier potential problem, although the derivation of this formula holds for many-body problems with multidimensional potentials.
We generally derive complex geometric phases in non-Hermitian quantum mechanics. The formalism of complex geometric phases for non-Hermitian quantum systems is general and can be applied in the cases of cyclic and noncyclic states in non-Hermitian quantum mechanics. As an example, we show the interpretation of complex geometric phases by a generalized density matrix defined on a two-level non-Hermitian quantum system and the two-level Bethe-Lamb model. Also, in the Hermitian case, our theoretical framework of a complex geometric phase reduces to the counterpart of the Pancharatnam phase.
The absolute value squared of the probability amplitude corresponding to the overlap of an initial state with a continuum wave solution to the Schrödinger equation of the problem, has the physical interpretation provided by the Born rule. Here, it is shown that for an open quantum system, the above probability may be written in an exact analytical fashion as an expansion in terms of the non‐Hermitian resonance (quasinormal) states and complex poles to the problem which provides an underlying non‐Hermitian character of the Born rule.
We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square wells or delta functions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processes that have been ignored. However, if they are treated as fundamental theories, the Hilbert-space metric must be changed. In order for the newly defined probability to be conserved, it must differ from the standard one, even at asymptotically large distances from the scattering center, and the mechanism for this is the nonlocality of the new metric, as we show in detail in the model of a single complex delta function. However, properties of distant bound-state systems, which do not interact physically with the non-Hermitian scattering potential, should not be affected. We analyze a model Hamiltonian that supports this contention.
The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time.
A probability density is proposed in complex space. This probability density is defined by replacing xxx with zzz in the Born probability density of quantum mechanics, and it is real valued and non-negative in complex space. In addition, the Eulerian rate equation for the probability density is derived, and the rate of change of the probability density is determined by the total energy density. Furthermore, this complex-extended Born probability density correctly reflects information of the trajectory dynamics for the wave function in complex space.
The times spent by an electron in a scattering event or tunnelling through a potential barrier are investigated using a method based on the absorption probabilities. The reflection and transmission times derived from this method are equal to the local Larmor times if the transmission and reflection probability amplitudes are complex analytic functions of the complex potential. The numerical results show that they coincide with the phase times except as the incident electron energy approaches zero or when the transmission probability is too small. If the imaginary potential covers the whole space the tunnelling times are again equal to the phase times. The results show that the tunnelling times based on absorption probabilities are the best of the various candidates.