Comparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation

Comparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation Student: Yael Elmatad Overview The goal of this research was to examine the quantum mechanical phenomenon of tunneling with respect to the Schrödinger Equation. A quantum mechanical wave is said to “tunnel” when it travels (propagates) through a classically forbidden region. In a more physical interpretation, it is when the energy (E) of the wave is lower than the potential at a specific point V(x). [A point where V(x) is greater than E is referred to as a classical turning point.] The phenomenon of a tunneling was examined in a double-well potential. Another goal of this project was to compare two methods for approximating the tunneling time of a wave packet. The two methods used were the finite differences (numerical) and WKB (analytical) approximation methods. In order to compare a potential that experienced tunneling and one that did not, the solution to a quadratic potential was also solved. Theory Schrödinger Equation Theory: The one dimensional Schrödinger equation is given as follows: Ĥψ(x,t)=Eψ(x,t) Where Ĥ is the Hamiltonian operator, ψ is the wave function, and E is the energy. In atomic units, the Hamiltonian operator for a particle in a potential is: Ĥ = 2 ∇ + V(x) The time-independent Schrödinger equation (TISE) bounded inside a potential becomes: -φxx(x, 0) + V(x)φ(x,0) = Eφ(x,0) (φ is the time-independent wave function) As stated, this equation is an eigenevalue problem where the operator returns the wave function multiplied by some eigenvalue, E. This amounts to solving for the stationary states (normal modes) of the system by finding the set of eigenvalues and eigenvectors given a known potential V(x). The time dependent Schrödinger equation (TDSE), bounded inside of a potential is: iψt(x, t) + ψxx(x, t) + V(x)ψ(x,t) = 0 (t is time) The TDSE becomes the TISE when t=0. The relationship between the two is: iψt = -Eφ or ψt = -iEφ Since t is 0, ψ = φ (1.1)