Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSolution of the time dependent Schrodinger equation leading to Fowler-Nordheim field emission
115
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We solve the time-dependent Schrodinger equation describing the emission of electrons from a metal surface by an external electric field $E$, turned on at $t=0$. Starting with a wave function $psi(x,0)$, representing a generalized eigenfunction when $E=0$, we find $psi(x,t)$ and show that it approaches, as $ttoinfty$, the Fowler-Nordheim tunneling wavefunction $psi_E$. The deviation of $psi$ from $psi_E$ decays asymptotically as a power law $t^{-frac32}$. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds.
rate research
Read More
In this paper, classical small perturbations against a stationary solution of the nonlinear Schrodinger equation with the general form of nonlinearity are examined. It is shown that in order to obtain correct (in particular, conserved over time) nonzero expressions for the basic integrals of motion of a perturbation even in the quadratic order in the expansion parameter, it is necessary to consider nonlinear equations of motion for the perturbations. It is also shown that, despite the nonlinearity of the perturbations, the additivity property is valid for the integrals of motion of different nonlinear modes forming the perturbation (at least up to the second order in the expansion parameter).
We investigate enhanced field emission due to a continuous or pulsed oscillating field added to a constant electric field $E$ at the emitter surface. When the frequency of oscillation, field strength, and property of the emitter material satisfy the Keldysh condition $gamma<1/2$ one can use the adiabatic approximation for treating the oscillating field, i.e. consider the tunneling through the instantaneous Fowler-Nordheim barrier created by both fields. Due to the great sensitivity of the emission to the field strength the average tunneling current can be much larger than the current produced by only the constant field. We carry out the computations for arbitrary strong constant electric fields, beyond the commonly used Fowler-Nordheim approximation which exhibit in particular an important property of the wave function inside the potential barrier where it is found to be monotonically decreasing without oscillations.
We present a non-perturbative solution of the Schrodinger equation $ipsi_t(t,x)=-psi_{xx}(t,x)-2(1 +alpha sinomega t) delta(x)psi(t,x)$, written in units in which $hbar=2m=1$, describing the ionization of a model atom by a parametric oscillating potential. This model has been studied extensively by many authors, including us. It has surprisingly many features in common with those observed in the ionization of real atoms and emission by solids, subjected to microwave or laser radiation. Here we use new mathematical methods to go beyond previous investigations and to provide a complete and rigorous analysis of this system. We obtain the Borel-resummed transseries (multi-instanton expansion) valid for all values of $alpha,omega,t$ for the wave function, ionization probability, and energy distribution of the emitted electrons, the latter not studied previously for this model. We show that for large $t$ and small $alpha$ the energy distribution has sharp peaks at energies which are multiples of $omega$, corresponding to photon capture. We obtain small $alpha$ expansions that converge for all $t$, unlike those of standard perturbation theory. We expect that our analysis will serve as a basis for treating more realistic systems revealing a form of universality in different emission processes.
Using a recently developed technique to solve Schrodinger equation for constant mass, we studied the regime in which mass varies with position i.e position dependent mass Schrodinger equation(PDMSE). We obtained an analytical solution for the PDMSE and applied our approach to study a position dependent mass $m(x)$ particle scattered by a potential $mathcal{V}(x)$. We also studied the structural analogy between PDMSE and two-level atomic system interacting with a classical field.
We obtain time dependent $q$-Gaussian wave-packet solutions to a non linear Schrodinger equation recently advanced by Nobre, Rego-Montero and Tsallis (NRT) [Phys. Rev. Lett. 106 (2011) 10601]. The NRT non-linear equation admits plane wave-like solutions ($q$-plane waves) compatible with the celebrated de Broglie relations connecting wave number and frequency, respectively, with energy and momentum. The NRT equation, inspired in the $q$-generalized thermostatistical formalism, is characterized by a parameter $q$, and in the limit $q to 1$ reduces to the standard, linear Schrodinger equation. The $q$-Gaussian solutions to the NRT equation investigated here admit as a particular instance the previously known $q$-plane wave solutions. The present work thus extends the range of possible processes yielded by the NRT dynamics that admit an analytical, exact treatment. In the $q to 1$ limit the $q$-Gaussian solutions correspond to the Gaussian wave packet solutions to the free particle linear Schrodinger equation. In the present work we also show that there are other families of nonlinear Schrodinger-like equations, besides the NRT one, exhibiting a dynamics compatible with the de Broglie relations. Remarkably, however, the existence of time dependent Gaussian-like wave packet solutions is a unique feature of the NRT equation not shared by the aforementioned, more general, families of nonlinear evolution equations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
