Hello Bonjour, Kaixo, Aloha! Welcome to the fifth homework session of
Statistical Mechanics: Algorithms and Computations from the Physics Department of Ecole normale
supérieure. This week, we have made our first step into
quantum physics, we use consistently the two languages of wavefunctions and energy eigenvalues
on the one hand, and density matrices and path integrals
on the other hand. Now, there many things to be consolidated.
We need to see, first of all, in a simple example, how to calculate the density matrix
from the exact solution of the Schrödinger equation. The probability pi(x), for the particle to be at x, then provides a benchmark for all our future caculations. In the second step, we will tackle the matrix
squaring procedure, discretized, and in a homogeneous space.
It will allow you to determine pi(x) now without having to resort to something which
is close to the Schrödinger evolution, that is, in the iteration of the algorithm,
by decreasing the temperature. Finally, in the third step, we will have in
fun in firing up the naive Quantum Monte Carlo algorithm, which allows you in a third way to determine pi(x), now by sampling in a histogram the
position x of the particle. All in all, for those of you who already know
Quantum Mechanics, but also for those of you who do not know
anything about Quantum Mechanics, you will learn three different recipes to
study the evolution of such systems.