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Hello Bonjour, Kaixo, Aloha!

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Welcome to the fifth homework session of
Statistical Mechanics: Algorithms and Computations

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from the Physics Department of Ecole normale
supérieure.

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This week, we have made our first step into
quantum physics, we use consistently the two

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languages of wavefunctions and energy eigenvalues
on the one hand,

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and density matrices and path integrals
on the other hand.

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Now, there many things to be consolidated.
We need to see, first of all, in a simple example,

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how to calculate the density matrix
from the exact solution of the Schrödinger equation.

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The probability pi(x), for the particle to be at x,

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then provides a benchmark for all our future caculations.

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In the second step, we will tackle the matrix
squaring procedure, discretized,

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and in a homogeneous space.
It will allow you to determine pi(x)

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now without having to resort to something which
is close to the Schrödinger evolution,

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that is, in the iteration of the algorithm,
by decreasing the temperature.

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Finally, in the third step, we will have in
fun in firing up the naive Quantum Monte Carlo algorithm,

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which allows you in a third way to determine

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pi(x), now by sampling in a histogram the
position x of the particle.

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All in all, for those of you who already know
Quantum Mechanics,

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but also for those of you who do not know
anything about Quantum Mechanics,

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you will learn three different recipes to
study the evolution of such systems.