numerical solutions to laplace’s and schrodinger’s equation
TRANSCRIPT
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Numerical Solutions to Laplace’s and Schrodinger’s
EquationShangyu Jiang
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Numerical Solutions to Laplace’s and Schrodinger’s
EquationShang
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IntroductionPDE’s are very hard to solve analytically. How do we solve them on a computer?
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Gauss’s Law
Also,
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Poisson’s Equation
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Laplace’s Equation
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How to compute this numerically?Recall the Taylor expansion of V(x+h) around x:
Similarly,
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Combining and adding, we get
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Method of Relaxation(Jacobi)
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Example: Griffiths Ex. 3.4• V = 0, y = 0
• V = 0, y = a
• V = V_0, x = b
• V = V_0, x = -b
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Now that we have potential, we can calculate the electric field:
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But how do we know this is right?Analytical solution:
Pretty hard to solve!
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Analytical Solution
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Analytical Calculated
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Analytical Calculated
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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!)
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Analytical Calculated
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Analytical Calculated
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Cross-section of potential
Analytical Calculated
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We can also get crazy:
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Another crazy function
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ImprovementsGauss-Seidel: Use updated values of nearest neighbors.
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Overrelaxation
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Multigrid
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This can be generalized to solve poisson’s equation:
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A single point charge
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Cross-section of point charge potential
A plot of 1/r
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Two positive charges close together
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Two positive charges far apart
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A dipole
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Dipole Electric Field
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Schrodinger’s Equation
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Schrodinger’s Equation
Let’s set m = ħ = 1:
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This is analogous to Poisson’s Equation:Compare
And
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Suggests an analogous iterative method:
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We specify V. But what to do with E?We can use
By noting that
And
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The and are integrals that we can treat as sums.
Use the Taylor expansion we did before for the term:
d = 2
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An example: 2d infinite square well
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Calculated vs. Analytical
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Some excited states
First excited, Nx = 1, Ny = 2
Nx = 2, Ny = 3
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Other examples?https://ide.c9.io/fordhamdining/seminar-presentation
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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
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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.
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Look at my code (optional)https://github.com/shangprograms
More about me: (also optional)
http://www.columbia.edu/~sj2850/
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Thank you for watching!
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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.