SchrodingerPDEComponent

SchrodingerPDEComponent[vars,pars]

yields a Schrödinger PDE term with model variables vars and model parameters pars.

Details

  • Generates the Schrödinger equation to be solved with NDSolve/NDEigensystem. Given the variables and parameters, a PDE operator is returned. Examples are shown.
  • SchrodingerPDEComponent is the non-relativistic quantum mechanics analogous to Newton's second law and describes the time evolution of a wavefunction in quantum mechanics.
  • SchrodingerPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
  • SchrodingerPDEComponent can be used to model Schrödinger equations with independent variables in units of meter TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF], dependent variable in units of TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]^(-n/2) and time variable in units of TemplateBox[{InterpretationBox[, 1], "s", seconds, "Seconds"}, QuantityTF].
  • Stationary model variables vars are vars={Ψ[x1,,xn],{x1,,xn}}.
  • Time-dependent model variables vars are vars={Ψ[t,x1,,xn],t,{x1,,xn}}.
  • The SchrodingerPDEComponent is based on a kinetic term and a potential term:
  • is the reduced Planck constant in units of TemplateBox[{InterpretationBox[, 1], {"s",  , "J"}, second joules, {"Joules",  , "Seconds"}}, QuantityTF], and is the Schrödinger potential in units of TemplateBox[{InterpretationBox[, 1], "J", joules, "Joules"}, QuantityTF].
  • For this equation, the mass can be isotropic, orthotropic or anisotropic.
  • For a magnetic field interaction, the SchrodingerPDEComponent is:
  • represents the electric charge of the particle in units of coulombs TemplateBox[{InterpretationBox[, 1], "C", coulombs, "Coulombs"}, QuantityTF].
  • is the magnetic vector potential in units of TemplateBox[{InterpretationBox[, 1], {"s",  , "V", , "/", , "m"}, second volts per meter, {{(, {"Volts",  , "Seconds"}, )}, /, {(, "Meters", )}}}, QuantityTF], defined such that , where is the magnetic flux density in units of TemplateBox[{InterpretationBox[, 1], "T", teslas, "Teslas"}, QuantityTF].
  • The magnetic field interaction form assumes Coulomb's gauge-fixing condition: .
  • The units of the Schrödinger PDE terms are joules TemplateBox[{InterpretationBox[, 1], "J", joules, "Joules"}, QuantityTF] times the units of the wavefunction in (TemplateBox[{InterpretationBox[, 1], "J", joules, "Joules"}, QuantityTF])/(TemplateBox[{InterpretationBox[, 1], {{sqrt(, "m", )}}, square root meters, {sqrt(, "Meters", )}}, QuantityTF]^n).
  • The following model parameters pars can be given:
  • parameterdefaultsymbol
    "AzimuthalQuantumNumber"None
  • l
  • "MagneticVectorPotential"0, magnetic vector potential in
    "Mass"1, mass in
    "PlanckConstant", Planck's constant in
    "ReducedPlanckConstant", reduced Planck's constant in
    "SchrodingerPotential"0V, potential in J
    "RegionSymmetry"None
    "ParticleCharge"1, charge in
  • All parameters may depend on any of , and , as well as other dependent variables.
  • Parameters given with units are converted to SI base units.
  • Default values for Planck's and reduced Planck's constants are in SI base units.
  • A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
  • "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
  • dimensionreductiongeneral equation
    1D
    2D
  • With an "AzimuthalQuantumNumber" , the SchrodingerPDEComponent is:
  • The diffusion component affects the meaning of NeumannValue.
  • If the SchrodingerPDEComponent depends on parameters that are specified in the association pars as ,keypi,pivi,, the parameters are replaced with .

Examples

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Basic Examples  (3)

Specify a PDE operator for a particle with mass and potential :

Specify a PDE operator for a particle with mass and potential with a reduced Planck constant :

Define a Hamiltonian with a reduced Planck constant of , a harmonic potential and a mass of :

Find the 10 smallest eigenvalues and eigenfunctions on a refined mesh:

Visualize the eigenfunctions scaled by the reduced Planck constant and offset by the respective eigenvalue:

Scope  (16)

1D  (6)

Define a Schrödinger PDE operator:

Activate the PDE operator:

Define a Schrödinger PDE operator with as the Planck constant:

Define a Schrödinger PDE operator with as the reduced Planck constant:

Define a Schrödinger PDE operator with mass :

Define a Schrödinger PDE operator with a potential :

Define a Schrödinger PDE operator with as the reduced Planck constant, mass , a magnetic vector potential and charge :

2D  (1)

Specify a PDE operator for a particle with mass , reduced Planck constant and a potential in two spatial dimensions:

3D  (2)

Specify a PDE operator for a particle with mass , reduced Planck constant and a potential in three spatial dimensions:

Define a PDE operator for a particle with electric charge and mass in an external magnetic field with a correspondent magnetic vector potential :

Axisymmetric  (3)

Specify an axisymmetric time-independent Schrödinger PDE term with the reduction , for a particle with mass , reduced Planck constant and a potential :

Activate the PDE operator:

Specify an axisymmetric time-independent Schrödinger PDE term with the reduction , for a particle with mass , reduced Planck constant and a potential :

Activate the PDE operator:

Define a PDE operator with a reduced Planck constant , a mass and an azimuthal quantum number :

Replace the parameters in the PDE with , and :

Activate the PDE operator:

Time-Dependent  (1)

Specify a time-dependent Schrödinger PDE term for a particle with mass , reduced Planck constant and a potential :

Material Symmetry  (3)

Define a PDE operator with a potential and with an isotropic mass :

Define a PDE operator with a potential , an orthotropic mass and a Planck constant of 1:

Define a Hamiltonian with a potential , an anisotropic mass and a Planck constant of 1:

Generalizations & Extensions  (1)

Numerically specify a particle with mass and potential :

Applications  (11)

Compute the eigensystem of a time-independent Schrödinger PDE operator with a Planck constant of 1, a mass of 1 and a potential :

Visualize the eigenfunctions and label the eigenvalues:

Find the 10 smallest eigenvalues and eigenfunctions with a reduced Planck constant and a piecewise potential that is for negative values of and for positive values of .

Define the PDE operator:

Plot the potential:

Find the 10 smallest eigenvalues and eigenfunctions on a refined mesh:

Look at the energy eigenvalues:

Visualize the eigenfunctions scaled by and offset by the respective eigenvalue:

Describe a particle confined in a two-dimensional disk of radius , assuming symmetry around the axis that passes through the center of the disk. Assume that the wavefunction for the particle has no polar coordinate dependence.

Set up an axisymmetric Schrödinger PDE operator, with a Planck constant equal to :

Set the radius of the disk:

The potential energy outside the region is infinite, then the wavefunction vanishes at the boundary.

Define a Dirichlet boundary condition:

Solve the eigenvalue problem:

Visualize the probabiity densities:

Consider a model in which the wavefunction can be separated in the following way: , where , and are the radial, azimuthal and height coordinates in a cylindrical coordinate system, respectively. Here, is the azimuthal quantum number.

This particular example considers a particle confined in a torus with infinite potential walls. Given these considerations, it is only necessary to solve for in a cross section of the torus.

Define the major radius of the torus:

Define the region with a minor radius of :

Visualize the mesh as the torus cross section it represents:

Define the boundary condition such that the wavefunction vanishes at the boundary of the torus:

Create a helper function to find six eigenstates for different values of the azimuthal quantum number with a PDE operator such that and for a simplified model and for the particle inside the torus:

It is possible to explore how the energy levels vary for different values of the azimuthal quantum number. To exemplify this, calculate the energy for between 0 and 5.

Calculate the energy eigenvalues for each value of between 0 and 5:

Plot the energy eigenvalues as a function of the azimuthal quantum number:

Observe how the energy levels grow nonlinearly as grows. Also, it can be noticed how the ground state is non-degenerate, while states two and three, as well as four and five, are two-fold degenerate.

To explore the effect of the azimuthal quantum number on the wavefunctions, plot the real and imaginary parts of the total wavefunction . For this purpose, you can choose any value of , for instance, .

Calculate the energy and the part of the wavefunction that depends on and :

Plot the real and imaginary parts of the wavefunction :

Now plot the probability densities for the same value of :

The probability density is independent of the coordinate , as expected.

You can study a quantum billiard system, in particular, a particle confined in a 2D Bunimovich stadium.

Define the region and visualize it:

Discretize the region:

Generate a Schrödinger operator with no potential:

Given that the potential energy outside the region is infinite, the wavefunction vanishes at the boundary.

Define the boundary condition:

Find the first 100 eigenstates:

Visualize the energy levels:

Plot the wavefunctions for different states:

Set up a time-dependent Schrödinger PDE operator with a reduced Planck constant:

Solve the equation with a Gaussian as an initial setting:

Apply Animate to the solution:

Set up a nonlinear, time-dependent Schrödinger PDE operator:

Define boundary and initial conditions:

Compute the solution:

Plot the absolute value of the solution:

Set up a time-dependent Schrödinger PDE operator with a harmonic potential:

Solve the equation with a coherent state as an initial setting:

Animate the probability density with the potential scaled down by a factor of :

Set up a time-dependent Schrödinger PDE operator with a reduced Planck constant and a Gaussian potential :

Specify an initial condition of a wave packet with positive momentum:

Visualize it along with the potential that is being scaled down by a factor to fit in the plot:

Solve the equation with a refined mesh:

Visualize the solution where the vertical and horizontal axes represent time, , and position, , respectively:

Note that at about , the wave packed hits the wall and gets reflected.

Visualize the solution with an animation:

Define an operator for a free particle:

Define a region that defines the double slit and visualize it:

Set up the initial conditions as a wave packet:

Visualize the initial condition:

Solve the PDE:

Visualize the solution:

You can model a spinless charged particle in a magnetic field that is pointing in the direction. The particle is described by an initial wave packet, and the goal is to study the evolution of the probability density under magnetic field interaction. Since this is a charged particle in a magnetic field, one can expect for its probability density to reflect some kind of circular trajectory, similar to the behavior of classical charged particles. For that reason, a very high magnetic flux density has been chosen, such that the cyclotron radius is in the order of angstroms.

Set up the reduced Planck constant, particle's mass, magnetic flux density and the particle's electric charge, respectively:

Define a constant to measure the time in femtoseconds:

The particle's initial condition will be described by a wave packet with a momentum , meaning it would move in a straight line if there were no magnetic field interaction.

Define with units of TemplateBox[{1, {"/", , "Å"}, reciprocal ångström, {1, /, {(, "Angstroms", )}}}, QuantityTF]:

A magnetic flux density is chosen to have a cyclotron radius in the order of angstroms.

Compute the cyclotron radius of the particle:

The magnetic vector potential can be chosen such that :

The work region can be defined as a rectangle of TemplateBox[{InterpretationBox[, 1], "Å", ångströms, "Angstroms"}, QuantityTF].

Define the region:

Since the length units are Angstroms (TemplateBox[{InterpretationBox[, 1], "Å", ångströms, "Angstroms"}, QuantityTF]), you need to take that into account when defining the PDE operator. For that reason, "ScaleUnits" -> {"Meters" -> "Angstroms"} are used in the parameters argument of the SchrodingerPDEComponent.

Define a Schrödinger PDE operator that assumes Coulomb's gauge for the magnetic vector potential:

Set up an initial condition:

Change the units of to be used in the boundary condition:

Solve the problem with a transparent boundary condition with a NeumannValue:

Animate the solution:

Properties & Relations  (2)

Specify an operator for a particle with mass and potential with a reduced Planck constant :

Activate the component:

When parameters are given with units, they are automatically converted to SI base units.

Specify a PDE operator with the electron's mass and the reduced Planck constant :

Inspect the difference between each unit system in the diffusion coefficient. Compute the Planck constant divided by the electron mass:

Convert the result to SI base units:

Neat Examples  (1)

Define a quantum well potential energy as a piecewise function:

Set up the variables and parameters, with the mass and the reduced Planck constant equal to , for a simplified Schrödinger PDE operator:

Use NDEigensystem to solve the time-independent Schrödinger equation in one dimension. The objective of this visualization is to see the effect that the barrier height and the well's length have on the eigenstates. One can see how increasing the well's length decreases the energy for each wavefunction. On the other hand, increasing the barrier height results in higher energy eigenvalues.

Define a function to calculate the eigenstates for any given value of potential barrier, well's width or number of eigenstates:

Set up a Manipulate:

Wolfram Research (2023), SchrodingerPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/SchrodingerPDEComponent.html.

Text

Wolfram Research (2023), SchrodingerPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/SchrodingerPDEComponent.html.

CMS

Wolfram Language. 2023. "SchrodingerPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SchrodingerPDEComponent.html.

APA

Wolfram Language. (2023). SchrodingerPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SchrodingerPDEComponent.html

BibTeX

@misc{reference.wolfram_2024_schrodingerpdecomponent, author="Wolfram Research", title="{SchrodingerPDEComponent}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/SchrodingerPDEComponent.html}", note=[Accessed: 24-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_schrodingerpdecomponent, organization={Wolfram Research}, title={SchrodingerPDEComponent}, year={2023}, url={https://reference.wolfram.com/language/ref/SchrodingerPDEComponent.html}, note=[Accessed: 24-April-2024 ]}