We study the massless limit of the Klein-Gordon (K-G) equation in $1+1$ dimensions with static complex potentials in order to provide an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in an active medium. In the case of a dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy dependent. We study also the nonrelativistic Schr\"odinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of the solutions of this Schr\"odinger equation is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. In particular, both equations exhibit a discrepancy between the time-dependent and stationary results; our study attributes this discrepancy to the appearance of time-growing bound eigenstates corresponding to poles in the transmission and reflection amplitudes located in the upper half of the wave-number plane. The omission of these bound states in the expansion in stationary states leads to the observed discrepancy. Furthermore, it was demonstrated that there is a frequency- (energy) -and-size-dependent gain threshold above which this discrepancy appears. This threshold corresponds to the value of the gain at which the first pole crosses the real axis.We study the massless limit of the Klein-Gordon (K-G) equation in $1+1$ dimensions with static complex potentials in order to provide an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in an active medium. In the case of a dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy dependent. We study also the nonrelativistic Schr\"odinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of the solutions of this Schr\"odinger equation is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. In particular, both equations exhibit a discrepancy between the time-dependent and stationary results; our study attributes this discrepancy to the appearance of time-growing bound eigenstates corresponding to poles in the transmission and reflection amplitudes located in the upper half of the wave-number plane. The omission of these bound states in the expansion in stationary states leads to the observed discrepancy. Furthermore, it was demonstrated that there is a frequency- (energy) -and-size-dependent gain threshold above which this discrepancy appears. This threshold corresponds to the value of the gain at which the first pole crosses the real axis.

arXiv:cond-mat/0505469Physical Review BSep 28, 200519 citations

Wave Equations and the Conservation of Energy

A. Lees

NatureMar 1, 19330 citations

Study of the Dirac equation with the position-dependent mass for modified manning Rosen potential using Laplace method

Suparmi Suparmi, L. K. Permatahati, C. Cari

We study the Dirac equation with position dependent mass (PDM) for Modified Manning Rosen potential. The Dirac equation with PDM is solved using Laplace method to obtain the energy equation and the wave function. We calculate the energy eigenvalues and the wave function numerically for several quantum numbers and parameters potential. The results showed that the energy eigenvalue increases as the quantum number and the parameters potential increase. The variation of the parameter potential gives the different of waveforms in each state.We study the Dirac equation with position dependent mass (PDM) for Modified Manning Rosen potential. The Dirac equation with PDM is solved using Laplace method to obtain the energy equation and the wave function. We calculate the energy eigenvalues and the wave function numerically for several quantum numbers and parameters potential. The results showed that the energy eigenvalue increases as the quantum number and the parameters potential increase. The variation of the parameter potential gives the different of waveforms in each state.

AIP conference proceedingsJan 1, 20231 citation

The Schrödinger Wave Equation

John C. Morrison

Elsevier eBooksJan 1, 20150 citations

Deriving the Schrödinger Equation

Steven M. Pascal

The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .

CRC Press eBooksJul 12, 20230 citations

The Schrödinger wave equation

John Morrison

Elsevier eBooksOct 16, 20200 citations

We study the massless limit of the Klein-Gordon (K-G) equation in

1+1

dimensions with static complex potentials in order to provide an alternative, but equivalent, representation of plane electromagnetic (em) wave propagation in an active medium. In the case of a dispersionless em medium, the analogy dictates that the potential in the K-G equation is complex and energy dependent. We study also the nonrelativistic Schr\"odinger equation with a potential that has the same energy dependence as that of the K-G equation. The behavior of the solutions of this Schr\"odinger equation is compared with those found elsewhere in the literature for the propagation of electromagnetic plane waves in a uniform active medium with complex dielectric constant. In particular, both equations exhibit a discrepancy between the time-dependent and stationary results; our study attributes this discrepancy to the appearance of time-growing bound eigenstates corresponding to poles in the transmission and reflection amplitudes located in the upper half of the wave-number plane. The omission of these bound states in the expansion in stationary states leads to the observed discrepancy. Furthermore, it was demonstrated that there is a frequency- (energy) -and-size-dependent gain threshold above which this discrepancy appears. This threshold corresponds to the value of the gain at which the first pole crosses the real axis.