The effect of a $\delta$ distribution potential on a quantum mechanical particle in a box

• URL: http://arxiv.org/abs/2311.02611v1
• Date: Sun, 5 Nov 2023 09:56:48 GMT
• Title: The effect of a $\delta$ distribution potential on a quantum mechanical particle in a box
• Authors: Pedro Martins Gir\~ao and Jo\~ao Pedro Nunes
• Abstract summary: 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 $|alpha|toinfty$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the effect of a $\delta$ distribution potential placed at $x_0\geq 0$ and multiplied by a parameter $\alpha$ on a quantum mechanical particle in an infinite square well over the segment $\left[-\,\frac{L}{2},\frac{L}{2}\right]$. We obtain the limit of the eigenfunctions of the time independent Schr\"{o}dinger equation as $\alpha\nearrow+\infty$ and as $\alpha\searrow-\infty$. We see how each solution of the Schr\"{o}dinger equation corresponding to $\alpha=0$ changes as $\alpha$ runs through the real line. When $x_0$ is a rational multiple of $L$, there exist solutions of the Schr\"{o}dinger equation which vanish at $x_0$ and are unaffected by the value of $\alpha$. We show that each one of these has an energy that coincides with the energy of a certain limiting eigenfunction obtained by taking $|\alpha|\to\infty$. The expectation value of the position of a particle with wave function equal to the limiting eigenfunction is $x_0$.

Related papers

• Quantum charges of harmonic oscillators [55.2480439325792]
  We show that the energy eigenfunctions $psi_n$ with $nge 1$ are complex coordinates on orbifolds $mathbbR2/mathbbZ_n$. We also discuss "antioscillators" with opposite quantum charges and the same positive energy.
  arXiv  Detail & Related papers  (2024-04-02T09:16:18Z)
• 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)
• 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)
• Non-perturbative Solution of the 1d Schrodinger Equation Describing Photoemission from a Sommerfeld model Metal by an Oscillating Field [0.0]
  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.
  arXiv  Detail & Related papers  (2022-09-15T19:14:53Z)
• On parametric resonance in the laser action [91.3755431537592]
  We consider the selfconsistent semiclassical Maxwell--Schr"odinger system for the solid state laser. We introduce the corresponding Poincar'e map $P$ and consider the differential $DP(Y0)$ at suitable stationary state $Y0$.
  arXiv  Detail & Related papers  (2022-08-22T09:43:57Z)
• Quantum particle in a spherical well confined by a cone [0.0]
  We consider the quantum problem of a particle in either a spherical box or a finite spherical well confined by a circular cone. The angular parts of the eigenstates depend on azimuthal angle $varphi$ and polar angle $theta$ as $P_lambdam(costheta)rm eimvarphi$.
  arXiv  Detail & Related papers  (2022-07-04T15:32:41Z)
• Quantum-information theory of a Dirichlet ring with Aharonov-Bohm field [0.0]
  Shannon information entropies $S_rho,gamma$, Fisher informations $I_rho,gamma$, Onicescu energies $O_rho,gamma$ and R'enyi entropies $R_rho,gamma(alpha)$ are calculated.
  arXiv  Detail & Related papers  (2022-02-09T19:26:56Z)
• From quartic anharmonic oscillator to double well potential [77.34726150561087]
  It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $Psi_ao(u)$, obtained recently, it is possible to get highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
  arXiv  Detail & Related papers  (2021-10-30T20:16:27Z)
• Superharmonic double-well systems with zero-energy ground states: Relevance for diffusive relaxation scenarios [0.0]
  Relaxation properties of the Smoluchowski diffusion process on a line can be spectrally quantified. A peculiarity of $hatH$ is that it refers to a family of quasi-exactly solvable Schr"odinger-type systems.
  arXiv  Detail & Related papers  (2021-04-24T08:11:54Z)
• 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)
• Scattering data and bound states of a squeezed double-layer structure [77.34726150561087]
  A structure composed of two parallel homogeneous layers is studied in the limit as their widths $l_j$ and $l_j$, and the distance between them $r$ shrinks to zero simultaneously. The existence of non-trivial bound states is proven in the squeezing limit, including the particular example of the squeezed potential in the form of the derivative of Dirac's delta function. The scenario how a single bound state survives in the squeezed system from a finite number of bound states in the finite system is described in detail.
  arXiv  Detail & Related papers  (2020-11-23T14:40:27Z)

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.