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.