Numerical Analysis of the Time Independent Schrodinger Equation

Numerical solutions to the Time Independent Schrodinger Equation (TDSE) were analyzed using the open source programming language python and using various numerical schemes to compare accuracy of solutions in space, time, and energy. The methods involved were Euler, fourth order Runge-Kutta (RK4), second order Runge-Kutta (RK2), and leapfrog. Furthermore, the time independent solutions were then plotted using the principle of superposition to observe how these solutions evolved over time using separation of variables techniques in partial differential equations. 1 The Schrodinger Equation The TDSE is given by ι~ ∂ ∂t Ψ = ĤΨ Using seperation of variables techniques, this can be reduced to two ordinary differential equations in both space and time which yields a time solution of h(t) = e− ιEt ~ And the time independent equation now looks like ĤΨ = EΨ Where E is the energy of the particle in question, and Ĥ is the Hamiltonian operator which is the total energy of the system, with potential energy V and kinetic energy K, where V is typically dependent on position and K is dependent on the mass and momentum of the particle as K = p 2m Where p is the momentum and m is the mass of the particle. Expanding the Hamiltonian gives. (K + V )Ψ = p 2m Ψ + V (x)Ψ = EΨ Essentially, this equation is simply demonstrating that the total energy of a particle is equal to its energy. We can all agree on this, and if this were as deep as the iceberg went, this would be a trivial problem in physics. What makes this a non-trivial problem is how we relate quantum mechanics to classical physical observables. In classical mechanics, if we are given the initial position and momentum of a classical particle, we know all there is to know about the particle. However, in quantum mechanics, position and momentum are no longer functions of time. They are instead operators, which operate on the wavefunction Ψ. By taking an ordinary plane wave, it can be demonstrated (Heuristically) that momentum operates on the wavefunction Ψ as pΨ = −ι~ ∂ ∂x Ψ