1. Simulating Electron Dynamics in 1D
Stephen Blama
Towson University
Capstone Project
Fall 2013 to Spring 2014

2. Goals Study some simple numerical methods
Develop a numerical method to simulate dynamics of a quantum particle (electron) in 1D
Generate code (in Matlab) to implement numerical methods

3. Motivation
Pedagogical: to visualize quantum behavior of particles and the form of the wavefunction
Practical: there are many real world applications of numerical simulation
Electron microscopy
Chemical analysis of atoms/molecules

4. Outline 1. Develop Numerical Methods 2. Implement Methods
Start with simple methods (Euler)
Test
Develop basic Crank-Nicolson
Modify into more stable method
2. Implement Methods
Free space propagation
Interaction with simple potentials
Momentum/energy space transformations

5. Numerical Methods Motivation
Not every problem can be solved analytically
Can be used to approximate complicated equations quickly
Fundamentally how computers solve problems

6. Continuous to Discrete
Computers can only hold a finite number of discrete values
Continuous functions are represented by lists of values
Time and space (independent variables) are divided into fixed intervals (steps: )

7. Finite Difference Methods
Use terms of Taylor series expansion to approximate derivatives
Allow the series to be centered about the
current value and evaluate the
function at or

8. Uncertainty All numerical approximations have uncertainty
Usually given in “Big O” notation
Finite different methods: usually
Gives an upper bound on uncertainty
f(x) = O(g(x)) means that there is a constant, c,
such that c*g(x) is always greater than f(x)
Example: suppose you have a function
that is directly proportional to
The uncertainty would be given by

9. Forward Euler Method First order method (uncertainty proportional to )
Good for approximating first derivatives
Derived from Taylor series by keeping up to first order term

10. Central Difference Method
2nd order method
Use for estimating second derivatives
Usually as high as you need to go in physics
Derived by Averaging Taylor series for forward and backward steps
Eliminates first derivative from result
more accurate than Euler, but
need to supply two initial conditions

11. Hooke’s Law Example with First Order Method
F = -kx ,or more formally:
Analytic Solution:
Euler Method: approximate position and
velocity with first order approximations

12. Numeric vs. Analytic Solution (1st Order)
Numeric solution is unstable
Amplitude of numeric solution increases
With every cycle
Uncertainty oscillates
Maximum at peaks where functions changes
rapidly
Minimum when function approximately linear

13. Numeric vs. Analytic Solution (2nd Order)
Now try central difference method
Very stable
Uncertainty not actually flat
Oscillates and grows as in Euler, but
grows very slowly

14. Transition to Quantum Mechanics
Now, let’s try to find a way to model quantum behavior
Dynamics in quantum mechanics is determined by Schrodinger Equation
Particles have wavelike properties, and are described by wavefunctions
There are very few analytic solutions to the Schrodinger Equation
We must find numerical solutions

15. Numerical Solutions to SE
is a function of space and time, so you need to index both variables
The SE is second order in space and first order in time, you may be temped to try using the central difference and Euler methods
But this doesn’t work, solution blows up
A stable method that does work is called the Crank-Nicolson method

16. Crank-Nicolson Method
Works by averaging the Hamiltonian for the current and next times
Solve by getting all and all on one side
Let and

17. Crank-Nicolson Method
This can be solved numerically using matrices (tridiagonal)

18. Comments on Crank-Nicolson
The method derived on the previous slide is very stable and agrees with the analytic result for free space propagation
However, this method becomes somewhat unstable when simulating a particle interacting with potential barriers
Probability density does not remain constant
Now, we will modify our method to make it more stable

19. Cayley Form
The crux of the Crank-Nicolson method is that it averages the Hamiltonian for the current and next times
Let’s try a slightly different method
When we solve the SE, we usually solve the time-independent form and tack on time dependence at the end
The exponential is sometimes called the propagator
Numerically,

20. Cayley Form Continued Notice that we can write
Using a Simple Taylor Approximation:

21. Cayley Form Continued
Writing the previous result in terms of the propagator or
This is like taking a half step forward from and a half step back from
We replace E with the Hamiltonian operator to get a new set of tridiagonal matrices
Notice that left and right hand sides are complex conjugates of each other, only need to find one matrix explicitly

22. Cayley Form Continued
We have
Let and

23. Cayley Form Continued Matrix for new method
This numerical solution is generally stable, and easier to implement, since you only need to construct one matrix explicitly

24. Simulating Electron Motion
Now we have a numerical solution for the Schrodinger Equation
We need to review units suitable for quantum mechanics
We need to choose a form for our initial wavefunction
Something with a known analytic solution so we can compare to numerical solution

25. Quantum Scale SI units are far too large to study quantum effects
We write everything in terms of electron volts, angstroms, and femtoseconds

26. Gaussian Wavefunction
Let’s choose an initial form for the
Wavepacket
Let’s use a Gaussian form
Familiar bell shaped curve
Easy to manipulate initial amplitude, width

27. Analytic Solution Gaussian Wavefunction
Start with
Normalize
Fourier transform to momentum space

28. Analytic Solution Continued
Fourier transform back to position space adding time dependence
Note that energy depends on momentum
And after hours of Fourier transforms and simple algebra mistakes…

29. Comparison of Numerical and Analytic Solutions

30. Potential Barriers
Quantum level: don’t talk about forces, talk about potential energy
Part of wavepacket usually reflected, and part always transmitted through a barrier (tunneling)
Can measure how much transmitted and reflected

31. Transmission and Reflection Coefficients
Take ratio transmitted and reflected waves to incident wave
Analytic solution for square well/barrier

32. Simple Potential Barrier
Reflection Coefficients
Analytic:
Numerical:
Transmission Coefficients
Analytic:
Numerical:

33. Simple Potential Well Reflection Coefficients Analytic: 0.0531
Numerical:
Transmission Coefficients
Analytic:
Numerical:

34. Multiple Barriers

35. Energy Space is a function of space and time
We can transform it to a function of space and energy through a Fourier transform
Lets you see how energy distributed over space

36. Energy Space
In this simulation, we perform a Fourier transform each time after we propagate the electron through a barrier
Note how the plane wave components change after interacting with the barrier

37. Infinite Square Well Review
Particle bounded by infinitely high potential barriers
Particle will bounce back and forth forever
Probability outside walls is zero

38. Implicit Boundary Conditions
The simulated space is necessarily finite
Algorithm simulates behavior for any realistic values of position
Wavefunction undefined outside finite space
Inadvertently create infinite square well

39. Boundary Behavior

40. Absorbing Potential
Can artificially decrease probability particle exists
Construct an imaginary (or complex) potential
Probability density decreases as particle moves through potential
Can’t completely destroy particle

41. Free Space Propagation with Absorbing Potential
Potential is of form

42. Summary
Numerical methods can be used to accurately model physics – classical and non-classical
The Crank-Nicolson method is an accurate method of modeling quantum behavior
Through free space, potential barriers, in other bases (momentum, energy, etc.)
Numerical computation can have unexpected results
Such as implicit boundary conditions

43. Future Research Continue with 1D Move to 2D/3D
Increases accuracy of propagation through barriers—transmission and reflection coefficients
Use energy space transformation to build library of plane wave signatures for various potentials
Move to 2D/3D
More complicated
Can’t use Crank-Nicolson anymore
Too computationally demanding
Matrix of ~1000 elements to matrix of ~1000x1000x1000 elements for 3D!
Study Taylor series approximations