Non-perturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field

• URL: http://arxiv.org/abs/2209.07570v2
• Date: Wed, 26 Oct 2022 19:56:57 GMT
• Title: Non-perturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field
• Authors: Ovidiu Costin, Rodica Costin, Ian Jauslin, Joel L. Lebowitz
• Abstract summary: We prove existence and uniqueness of classical solutions of the Schr"odinger equation for general initial conditions. We show that the solution approaches in the large $t$ limit a periodic state that satisfies an infinite set of equations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We analyze non-perturbatively the one-dimensional Schr\"odinger equation describing the emission of electrons from a model metal surface by a classical oscillating electric field. Placing the metal in the half-space $x\leqslant 0$, the Schr\"odinger equation of the system is $i\partial_t\psi=-\frac12\partial_x^2\psi+\Theta(x) (U-E x \cos\omega t)\psi$, $t>0$, $x\in\mathbb R$, where $\Theta(x)$ is the Heaviside function and $U>0$ is the effective confining potential (we choose units so that $m=e=\hbar=1$). The amplitude $E$ of the external electric field and the frequency $\omega$ are arbitrary. We prove existence and uniqueness of classical solutions of this equation for general initial conditions $\psi(x,0)=f(x)$, $x\in\mathbb R$. When the initial condition is in $L^2$ the evolution is unitary and the wave function goes to zero at any fixed $x$ as $t\to\infty$. To show this we prove a RAGE type theorem and show that the discrete spectrum of the quasienergy operator is empty. To obtain positive electron current we consider non-$L^2$ initial conditions containing an incoming beam from the left. The beam is partially reflected and partially transmitted for all $t>0$. For these we show that the solution approaches in the large $t$ limit a periodic state that satisfies an infinite set of equations formally derived by Faisal, et. al. Due to a number of pathological features of the Hamiltonian (among which unboundedness in the physical as well as the spatial Fourier domain) the existing methods to prove such results do not apply, and we introduce new, more general ones. The actual solution exhibits a very complex behavior. It shows a steep increase in the current as the frequency passes a threshold value $\omega=\omega_c$, with $\omega_c$ depending on the strength of the electric field. For small $E$, $\omega_c$ represents the threshold in the classical photoelectric effect.

Related papers

• Vacuum Force and Confinement [65.268245109828]
  We show that confinement of quarks and gluons can be explained by their interaction with the vacuum Abelian gauge field $A_sfvac$.
  arXiv  Detail & Related papers  (2024-02-09T13:42:34Z)
• The effect of a $\delta$ distribution potential on a quantum mechanical particle in a box [0.0]
  We obtain the limit of the eigenfunctions of the time independent Schr"odinger equation as $alphanearrow+infty$ and as $alphasearrow-infty$. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking $|alpha|toinfty$.
  arXiv  Detail & Related papers  (2023-11-05T09:56:48Z)
• Quantum connection, charges and virtual particles [65.268245109828]
  A quantum bundle $L_hbar$ is endowed with a connection $A_hbar$ and its sections are standard wave functions $psi$ obeying the Schr"odinger equation. We will lift the bundles $L_Cpm$ and connection $A_hbar$ on them to the relativistic phase space $T*R3,1$ and couple them to the Dirac spinor bundle describing both particles and antiparticles.
  arXiv  Detail & Related papers  (2023-10-10T10:27:09Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• A Regularized $(XP)^2$ Model [0.5729101515196013]
  We investigate a dynamic model described by the classical Hamiltonian $H(x,p)=(x2+a2)(p2+a2)$, where $a2>0$, in classical, semi-classical, and quantum mechanics. We present three different forms of quantized Hamiltonians, and reformulate them into the standard Schr" odinger equation with $cosh 2x$-like potentials.
  arXiv  Detail & Related papers  (2023-08-18T01:34:57Z)
• Small-time controllability for the nonlinear Schr\"odinger equation on $\mathbb{R}^N$ via bilinear electromagnetic fields [55.2480439325792]
  We address the small-time controllability problem for a nonlinear Schr"odinger equation (NLS) on $mathbbRN$ in the presence of magnetic and electric external fields. In detail, we study when it is possible to control the dynamics of (NLS) as fast as desired via sufficiently large control signals.
  arXiv  Detail & Related papers  (2023-07-28T21:30:44Z)
• Rigorous derivation of the Efimov effect in a simple model [68.8204255655161]
  We consider a system of three identical bosons in $mathbbR3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$.
  arXiv  Detail & Related papers  (2023-06-21T10:11:28Z)
• The Quantum Mechanical Problem of a Particle on a Ring with Delta Well [0.0]
  The problem of a spin-free electron with mass $m$, charge $e$ confined onto a ring of radius $R_0 and with an attractive Dirac delta potential with scaling factor (depth) $kappa$ in non-relativistic theory has closed form analytical solutions. The single bound state function is of the form of a hyperbolic cosine that however contains a parameter $d>0$ which is the single positive real solution of the transcendental equation $coth(d) = lambda d$ for non zero real.
  arXiv  Detail & Related papers  (2022-11-29T12:24:19Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.