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

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

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

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

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

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: )

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

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

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

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

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

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

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

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

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

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

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

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

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,

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

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

Cayley Form Continued We have Let and

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

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

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

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

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

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…

Comparison of Numerical and Analytic Solutions

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

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

Simple Potential Barrier Reflection Coefficients Analytic: 0.3210 Numerical: 0.3051 Transmission Coefficients Analytic: 0.6790 Numerical: 0.6949

Simple Potential Well Reflection Coefficients Analytic: 0.0531 Numerical: 0.0447 Transmission Coefficients Analytic: 0.9469 Numerical: 0.9553

Multiple Barriers

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

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

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

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

Boundary Behavior

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

Free Space Propagation with Absorbing Potential Potential is of form

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

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