Investigating the Behavior of Electrons in Different Quantum Systems

1. Investigating the Behavior of Electrons in Different Quantum Systems Christopher Seymour March 31, 2011

2. Abstract The focus of this project is to investigate the physical behavior of electrons in different quantum systems. There are very few systems that can be solved analytically. Many systems of interest encountered in research and teaching require numerical methods of solutions. We will study the use of stable, norm-preserving method to solve the time-dependent Schrodinger equation to simulate the behavior of quantum systems, including particles in the infinite potential well, finite square well or barrier, Dirac delta function, and hydrogen atoms. Interactions of these systems with laser fields will be simulated. Physical properties such as reaction rates will be investigated.

3. Harmonic Oscillator from pylab import* h=0.01 n=400 g,L=9.8,1 w0=g/L ##Euler x,w,t=pi/2,0,0 xa=[] wa=[] ta=[] for i in range(n): x=x+w*h w=w-w0*sin(x)*h t=t+h xa.append(x) wa.append(w) ta.append(t) #Runge Kutta x,w,t=pi/2,0,0 xb=[] wb=[] tb=[] for i in range(n): a=w b=w-w0*sin(x)*h/2.0 c=b-w0*sin(x)*h/2.0 d=c-w0*sin(x)*h x=x+1/6.0*h*(a+b*2+c*2+d) w=w-w0*sin(x)*h t=t+h xb.append(x) wb.append(w) tb.append(t) figure() plot(ta,xa) plot(ta,xb) xlabel("X") ylabel("Y") show()

4. Time-Independent Schrödinger Equation Ĥ Ψ = EΨ −(ħ^2/2m)* ∇^2 Ψ + V Ψ = EΨ

5. The Wave Function Probability of finding the particle between a and b, at time t = int(^b)(_a) |Ψ(x, t)|^2 dx Normalization int(^infty)(_-infty) |Ψ(x, t)|^2 dx = 1 Properties: Ψ(x, t) must be continuous and differentiable dΨ/dx must be continuous and differentiable

6. Operators tell you to do something to the function that follows x = x, p=-iħ (partial/partial x) Ĥ = p^2 / 2m = -ħ^2 / 2m (partial/partial x)^2 De Broglie p = h / λ = 2 π ħ / λ A spread in wavelength corresponds to a spread in momentum. Expectation Values <x> = int(_-infty)(^infty)Ψ *x Ψ dx <p> = m d<x>/dt = int(_-infty)(^infty)Ψ *x Ψ dx

7. Standard Deviation σj = sqrt(<j^2> - <j>^2) Uncertainty Principle σx σp ≥ ħ/2 The more precisely determines a particle position is, the less precisely its momentum is known, quantitatively.

8. Time-Dependant Schrödinger Equation iħ (partial Ψ)/(partial t) = H Ψ Iħ (partial Ψ)/(partial t) = Ψħ^2/2m)* ∇^2 Ψ + V Ψ

9. Leap Frog Integration x_(i+1) = x_i + v_(i+1/2) Δt v_(i+1/2) = v_(i-1/2) + a_(i) Δt

10. Future Work -Implement leap from algorithm. -Solve hydrogen atom. -Implement a laser interaction. -Look at the physical behavior of the electron based on the rate of transitions between energy levels, called reaction rates.