Applied quantum mechanics

The 10 hours will be devoted to build from scratch computer programs to solve single particle quantum mechanical problems. The main objective is that the student acquires minimal techniques to confront the solution of simple quantum mechanical numerical problems.

Lecture 1

Solution of the time independent Schrödinger equation in 1D. We will use a simple tridiagonal discretization of the laplacian operator. The accuracy of the method will be explored. The main focus will be on obtaining the eigenstates and eigenvalues of the 1D harmonic oscillator. (Sample from H. Albalad)

Lecture 2

Time evolution of a wave packet using the eigenbasis computed in Lecture 1. (Sample from P. Sopeña)

Lecture 3

Solution of the time dependent Schrödinger equation using a Crank-Nicolson scheme together with the tridiagonal Hamiltonian of lecture 1. No need for a confining potential.

Lecture 4

Application of the code developed in 3, e.g. a) broadening of a gaussian wave packet in free space, b) Gaussian wave packet smashed into a barrier, c) time dependent Hamiltonians, d) solutions to the Gross-Pitaevskii equation by imaginary time evolution.

Lecture 5

Bright solitons. Study of the behavior of bright solitons using a variant of the code developed in 3. (images from M. Checa and D. Lopez)