COMSOL 6.4 - Schrodinger Equation Interface

Schrodinger Equation Interface — Creating It Step by Step

The following steps show how to define the interface using the implementation defined in the previous section.

From the menu (Windows) or the menu (the cross-platform version), choose . In the window box, click and then select the checkbox if not selected already.

From the menu, choose (

). On the page, click the button (

). The window replaces the window on the COMSOL Desktop.

In the toolbar, click (

) (or right-click the root node () and select ).

Go to the window for . In the section:

In the field enter Schrodinger Equation. Click the button (

) to update the node in the .

Select the checkbox and replace the default with SchrodingerEq.

In the field enter scheq.

If you have a custom for the interface, click the button to locate the icon file. The default is to use the physics.png icon (

The interface should support all space dimensions except 0D. Under keep the defaults for the , which already excludes 0D.

In the list, select the default study types ( and ) and click the button (

) underneath the list. Click the button (

) and select from the list. Click .

In the section, confirm that is selected from the list. This means that the equations in the physics interface apply to the domains in the geometry, which is the case for most physics. Leave the default setting for the list as .

It is good practice to save the physics use interface after completing some steps. From the menu, choose and create a physics interface file, SchrodingerEquation.mphphb in the default location. Click .

This concludes the initial steps to set up the structure for the physics interface. The next steps add equations, boundary conditions, and variables.

Adding Features

First declare the dependent variable Ψ:

Right-click the node and from the menu select (

). Or, in the toolbar, click the button with the same name.

In the window locate the section.

Keep the default for the list ().

Click the button (

) to open the dialog, and then select . Click .

In the field enter psi.

In the field enter Wave function.

In the field enter \psi (the LaTeX syntax for the Greek letter ψ).

Keep the default (), because Ψ is a scalar field.

In the section, both checkboxes are selected by default. Keep these settings.

You also need to add a node in the Schrödinger equation domain feature to create the shape function (element type) for the dependent variable in the domain (see Step 16 below).

Add a Schrodinger Equation Model Domain Feature

Next add the Schrödinger equation in a domain feature:

Right-click the node and from the menu select (

In the window locate the section.

In the field enter Schrodinger equation model. Click the button (

) to update the node in the .

Select the checkbox and replace the default with SchrodingerEqu.

In the field enter schequ.

In the section, keep the default setting () for the and lists. By selecting you can restrict the feature to a subset of the allowed space dimensions or study types for the physics.

In the section, confirm that is displayed in the list. Keep the default setting (), and the default ().

Under , keep the default settings () for the and lists. Also keep the default as .

Under click to select the checkbox to make this the default feature for all domains.

Right-click the node and from the menu select (

In the window locate the section.

In the field enter mu.

In the field enter Reduced mass.

In the field enter \mu (LaTeX syntax for the Greek letter μ).

If desired, here and elsewhere, add a tooltip in the field.

Click the button (

) to open the dialog, and then select . Click .

Keep the default settings in the () and () lists for a basic scalar quantity. Also leave the setting to for an entry of a general scalar number.

In the field enter me_const (the electron mass, which is a built-in physical constant) in the field.

Right-click the node and select (

This user input then becomes available as a variable schequ.mu in, for example, the predefined expressions for results evaluation. You can also refer to mu directly in the weak equation. Keep all other default settings for this node.

Right-click the node and from the menu select (

In the window, locate the section.

In the field enter V.

In the field enter Potential energy.

In the field enter V.

Click the button (

) to open the dialog, and then select . Click .

Keep the default settings in the () and () lists for a basic scalar quantity. Also leave the setting to for an entry of a general scalar number.

Use the default value 0 in the field.

Right-click the node and select (

This user input then becomes available as a variable schequ.V in, for example, the predefined expressions for results evaluation. You can also refer to V directly in the weak equation. Keep all other default settings for this node.

Right-click the node and from the menu select (

). This is where you specify the equation for the domains in the physics interface.

In the window locate the section. In the field, enter -hbar_const^2/(2*mu)*∇psi·test(∇psi)-(V-lambda)*psi·test(psi)

This expression implements the Schrödinger equation formulation for this interface. Here, lambda is the eigenvalue. Press Ctrl+Space to get lists of the supported operations, including any special characters. See Tensor Operators and Other Operators for the keyboard entries to create the del operator (∇) and the dot product (·).

In the section keep the default setting (), which means it inherits the selection from the top node. Also keep the default setting () to use the selected domains as the output. Keep all other defaults.

Right-click the node and from the menu select (

In the window, locate the section. From the list, choose . Click the button (

) to open the dialog, and then select . Click . Lagrange elements are the most widely used and are suitable for this physics interface. The default shape-function order becomes 2. Keep the other defaults.

Add a Default Zero Probability Boundary Condition

Add the default boundary condition:

Right-click the (Physics Interface 1) node and from the menu select (

In the window locate the section.

In the field enter Zero probability. Click the button (

) to update the node in the .

Select the checkbox and replace the default with ZeroProb.

In the field enter zpb.

Keep all the defaults for the , , and sections.

In the section, select the checkbox. Keep the as to make this a default condition on exterior boundaries only.

Right-click the node and from the menu select (

In the window locate the section. Enter 0-psi in the field to make Ψ = 0 on the boundary (this expression is set equal to zero by the constraint).

In the section, from the list select . Click the button (

) to open the dialog, and then select to declare the correct dimension for the constrained variable Ψ. Click . Keep all other default settings.


	Click the Schrodinger Equation node. In the Settings window under Default Features, note that the Schrodinger Equation Model and the Zero Probability boundary condition are listed.

Add a Wave Function Value Boundary condition

Add the Wave Function Value boundary condition Ψ = Ψ0:

Right-click the node and from the menu select (

In the window locate the section.

In the field enter Wave function value. Click the button (

) to update the node in the .

Select the checkbox and replace the default with WaveFunc.

In the field enter wvfcn.

Keep all the defaults for the , , and sections.

Right-click the node and from the menu select (

In the window locate the section.

In the field enter psi0.

In the field enter Wave function value.

In the field enter \psi_0. This is for the symbol Ψ0.

Click the button (

) to open the dialog, and then select . Click .

Keep the default settings in the () and () lists for a basic scalar quantity. Also leave the setting to for an entry of a general scalar number.

Keep the at 0.

Right-click the node and from the menu select (

In the window locate the section. Enter par.psi0-psi in the field to make Ψ = Ψ0 on the boundary. The par prefix indicates a local parameter scope and is necessary in order to refer to a user input that is not defined as a variable.

In the section, from the list select . Click the button (

) to open the dialog, and then select to declare the correct dimension for the constrained variable Ψ. Click .

This feature uses a user input that should appear in a section in the window. The constraint automatically adds an extra section for enabling weak constraints and set the type of constraint, but it is necessary to specify a section for the user input.

Right-click the node and from the menu select (

In the window locate the section. In the field enter WaveFunc_section and in the field enter Wave function.

Under the list, click the button (

) and from the list select . Click .

In the section, from the list select . See User Input Group GUI Options for more information.

The remaining boundary condition, Axial Symmetry, appears automatically on symmetry boundaries in axisymmetric models.

Adding an Additional Variable

Add the probability density function ⏐Ψ⏐2 as a variable that is available in the model and for plotting (when the checkbox is selected in the section, which is the default setting) or evaluating:

Right-click the node and from the menu select (

In the window locate the section.

In the field enter probdens.

In the field enter Probability density function.

In the field enter {\mid\psi\mid}^2 to display ⏐Ψ⏐2.

For the list, keep the default setting ().

Click the button (

) to open the dialog, and then select . Click .

Keep all the default settings in the section. The list is set to for this variable to always appear as a predefined expression in plots.

Right-click the node and select (

In the window locate the section. Enter abs(psi)^2 in the field.

Defining Default Plots and Default Plot Quantities

Define a default 2D plot group for plotting the probability density function and make the probability density function the default scalar quantity for user-defined plots:

Right-click the node and select (

Right-click the (

) node and select (

In the window for in the field, select the checkbox and enter Probability density, and then in the field under , enter probdens.

Right-click the node and select (or press F2). In the dialog, in the field enter Probability Density. Click .

Right-click node and select (

Right-click the node and select (

In the window, in the field enter probdens. In the field enter Probability density. This makes the probability density function the default plot for all scalar plots.

Making a Quantum Mechanics Physics Area

To add the Schrodinger Equation interface to a new physics area for Quantum Mechanics under the Mathematics branch:

Under right-click (

) and select (

In the window locate the section.

In the field enter QuantumMechanics.

In the field enter Quantum Mechanics.

The default is physics.png (

), which is appropriate for this physics. Otherwise, click to use another icon.

In the field enter 10 to make the area appear last in the list under (the higher the weight, the lower position the physics area gets in the tree of physics interfaces).

Under click the node. Click the button (

). This moves the new physics area under the node.

Click the node. Locate the section and click to expand it.

Expand the branch and click . Click the button (

). This places the Schrodinger Equation interface under the physics area.

Save the file that contains the physics interface as SchrodingerEquation.mphphb.

This completes the definition of this Schrodinger Equation interface. Save the file that contains the physics interface.

The final node sequence in the Physics Builder tree should match this figure: