Numerical Solutions to Laplace’s and Schrodinger’s

EquationShangyu Jiang

Numerical Solutions to Laplace’s and Schrodinger’s

EquationShang

IntroductionPDE’s are very hard to solve analytically. How do we solve them on a computer?

Gauss’s Law

Also,

Poisson’s Equation

Laplace’s Equation

How to compute this numerically?Recall the Taylor expansion of V(x+h) around x:

Similarly,

Combining and adding, we get

Method of Relaxation(Jacobi)

Example: Griffiths Ex. 3.4• V = 0, y = 0

• V = 0, y = a

• V = V_0, x = b

• V = V_0, x = -b

Now that we have potential, we can calculate the electric field:

But how do we know this is right?Analytical solution:

Pretty hard to solve!

Analytical Solution

Analytical Calculated

Analytical Calculated

Another: Griffiths 3.3• V = 0, y = 0

• V = 0, y = a

• V = V_0, x = 0

• V → 0, x → ∞

Analytical Solution: (also pretty hard!)

Analytical Calculated

Analytical Calculated

Cross-section of potential

Analytical Calculated

We can also get crazy:

Another crazy function

ImprovementsGauss-Seidel: Use updated values of nearest neighbors.

Overrelaxation

Multigrid

This can be generalized to solve poisson’s equation:

A single point charge

Cross-section of point charge potential

A plot of 1/r

Two positive charges close together

Two positive charges far apart

A dipole

Dipole Electric Field

Schrodinger’s Equation

Schrodinger’s Equation

Let’s set m = ħ = 1:

This is analogous to Poisson’s Equation:Compare

And

Suggests an analogous iterative method:

We specify V. But what to do with E?We can use

By noting that

And

The and are integrals that we can treat as sums.

Use the Taylor expansion we did before for the term:

d = 2

An example: 2d infinite square well

Calculated vs. Analytical

Some excited states

First excited, Nx = 1, Ny = 2

Nx = 2, Ny = 3

Other examples?https://ide.c9.io/fordhamdining/seminar-presentation

Differences between Laplace and Schrodinger Laplace

• V is known on boundaries

• Want to find V(x,y)

• Convergence to a single, unique solution

• Initial guess does not matter (for convergence)

Schrodinger

• V is known everywhere

• Want to find ψ and E

• Multiple solutions (excited states)

• Initial guess matters

Summary• The Laplace, Poisson, and Schrödinger equations can be discretized using a

Taylor Expansion

• This discretization suggests an iterative numerical method that can find solutions to these equations.

• Computation time scales quickly with number of grid points, so it is important to find optimization techniques.

Look at my code (optional)https://github.com/shangprograms

More about me: (also optional)

http://www.columbia.edu/~sj2850/

Thank you for watching!

SourcesD. Griffiths. Introduction to Electrodynamics. 2013.

V. Igorevich. Lectures on partial differential equations. 2004.

D. Robertson. Relaxation Methods for Partial Differential Equations: Applications to Electrostatics. 2010.

D. Schroeder. The variational-relaxation algorithm for finding quantum bound states. 2017.