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Hello, Bonjour, Sabaidee, Stravo!

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Welcome to the the fifth week of
Statistical Mechanics: Algorithms and Computations

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from the Physics Department of Ecole normale supérieure.

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For four weeks now, we have concentrated on
classical Statistical Mechanics,

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and from the equiprobability principle,
we have just arrived at the Boltzmann distribution.

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Time has come to pay a three-week visit to
the world of quantum physics,

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the world of wave functions, and of the Schrödinger equation.

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We will go even farther, into the world of quantum
statistical mechanics,

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where we have at the same time the quantum wave functions
and the Boltzmann distribution of thermal equilibrium.

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In this lecture, lecture 5, we introduce to
one of the basic models in quantum physics,

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namely a particle in a harmonic potential,
described by energy levels and wave functions

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that we know exactly. Here is the groundstate
wavefunction

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of the particle, at energy E = 1/2.

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The square of the wavefunction gives
the probability for the particle to be at position x...

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And here is the first excited state, with energy
3/2: the second excited state with energy 5/2

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and so on and so on... Wait a few moments
to create all these states by yourself!

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At a given temperature, these energy levels
are subjet to the equiprobability principle

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and to the Boltzmann distribution. In the
lecture, in just a few moments, we will discuss

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exactly how this works, and this will lead
us very quickly to the density matrix

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and the celebrated Feynman path integral that
describes the spread of the wavefunctions

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through the fluctuations of a path.

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We all know that at high temperature, the
world is not really governed

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by quantum physics. The essence of our approach
to quantum statistical mechanics is

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a certain transformation called the Trotter decomposition,

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that iteratively brings us from the semiclassical
world at high temperature

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down to the full quantum world
at low-temperatures.

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How this works exactly will be explained in
this week's tutorial.

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 We will also discuss the time evolution
and program the quantum equivalent of

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molecular dynamics for simple quantum systems.

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This week's homework session will be again all
about practical computing:

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You will take your first steps in computing wavefunctions
and in Quantum Monte Carlo,

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for the harmonic oscillator.

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The quantum mechanical harmonic oscillator
describes a particle of mass m in a potential

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1/2 m omega^2 x^2 governed by the Schrödinger equation.

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Let us simplify this equation by putting

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Planck's constant h_bar = 1, the mass
of the particle = 1 and the oscillator constant

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omega = 1. This is not a restriction, and
Michael, Alberto, Vivien, and I will present

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the most important equations both with and
without the constants.

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We arrive at the time-independent Schrödinger
equation where H is the Hamilton operator

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or the Hamiltonian. Its solutions, we saw
them before, are the ground state wavefunction

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of energy 1/2, the first excited state of
energy 3/2, the second excited state of energy

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5/2, and so on, and so on...

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These wave functions were produced with the
program "harmonic_wavefunction.py",

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that implements an exact recursion relation
for the Hermite polynomials

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The wave functions are zero in the limit x
going to - infinity and x going to + infinity.

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In addition, they are normalized, which means
that

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their integral from -infinity to infinity
of psi_n squared is equal to one.

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Finally, the wavefunctions are orthogonal.
You don't have to believe me that the wavefunctions

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computed in "harmonic_wavefunctions.py" actually
solve the Schrödinger equation.

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You can check this for yourself.

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To do so let's rewrite the Schrödinger equation
as H psi / psi is equal to E,

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and let's write a little program "harmonic_wavefunctions_check.py"
with a discrete approximations

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for the second derivative. Sure enough, for
the groundstate wavefunction psi_0 we find

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H psi_0/psi_0 is equal to 1/2 for all x.

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And for the first excited state, we find H
psi_1 / psi_1 equals to 3/2 everywhere.

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Now let us move right away into Quantum Statistical
Mechanics.

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Quantum means that for a particle in the state
n, the probability to be at the position x is

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given by the absolute value of psi_n(x)^2.

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But we are also doing statistical mechanics,
and the probability to be in the state n is

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given by exp(-E_n/ kT) or exp(-beta E_n) where
beta = 1/k_b T.

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Now let's put the two pieces together, and
we find that the probability to be in state

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n and at position x is proportional to exp(-beta
E_n) * | psi_n(x)|^2.

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Before plunging into this subject, please
take a moment to download, run and modify

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the two programs we discussed in this section.
On the coursera website, you will find the

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program harmonic_wavefunction.py that implements
the recursion of Hermite polynomials. There

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is also the nice program harmonic_wavefunctions_check.py
that checks that the Schrödinger equation

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is solved, that the wavefunctions are normalized,
and that they are orthogonal.

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As we discussed a few moments ago, the probability
to be in state n and at position x is proportional

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to e^-(beta * En) psi_n(x) psi_n*(x).

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In this equation, the asterisk refers to the
complex conjugate.

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In this lecture, and this your homework this week,
the wavefunctions are real-valued,

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so psi∗ = psi, but in this week's tutorial,
we have to take into account complex wavefunctions,

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so we better use the correct formulas from
the beginning. Notice that in this equation,

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we have two different types of probabilities.
We have the thermal probability of the Boltzmann

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distribution, and the quantum-mechanical probability
of the wavefunctions: two completely separate

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worlds meet in this equation.

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However, the energy levels and wave functions
cannot *normally* be computed , and this expression leads

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nowhere, even for simple problems!
To make progress, we discard the information

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about the energy levels
and consider what is called the (diagonal)

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density matrix: the probability to be at x
which is proportional to the density matrix

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rho(x, x, beta) equals to the Σ_n e^(-beta E_n)
psi_n(x) psi_n*(x).

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We also consider a more general object, the
non-diagonal density matrix, which is equal

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to rho(x, x', beta) = Σ_n psi_n(x) psi_n*(x').

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This is the central object of Quantum Statistical
Mechanics. For example, the partition function

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Z(beta) is given by the Trace of the density
matrix.

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As discussed in previous weeks, the partition
function Z is the sum of the probabilities π_n;

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but here the n are no longer positions
in space, but the energy levels.

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We next discuss the three fundamental properties
of the density matrix:

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First of all, each density matrix possesses
the convolution property.

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This means that the integral over x'
of rho(x, x', beta_1)*rho(x', x'', beta_2) can be

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written as an integral over x' over a double
sum over n and m. This can be exchanged into

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a double sum over n and m over the integral
in x'. The orthogonality property that we

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just discussed allows us to write this as
a Σ_n psi_n(x) e^(-(beta_1 + beta_2) E_n) psi_n*(x''):

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in other words, the density matrix rho(x, x'',
beta_1 + beta_2).

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In this exact equation let us set beta_1 equal
to beta_2. We find that the integral over

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x' of rho(x, x', beta)*rho(x', x'', beta) is equal
to the density matrix rho(x, x'', 2beta). Now

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realize that beta = 1/Temperature. So in this
equation we compute the density matrix at

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2beta, that means at low temperature, through
a product over density matrices at high temperature.

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We can use this equation if we know the density
matrix at high temperature, to compute it

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at twice lower temperature. Then we can use
it again to compute it at 4 times lower temperature,

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8 times lower temperature, and so on, and
so on... until we reach the full quantum regime.

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The second property is the free density matrix.
We will derive this equation in the beginning

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of this week's tutorial, and make sure that
you understand the role of the non-diagonal

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elements in this density matrix, and we will
illustrate this in nice pictures of the entire

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density matrix at high temperature; and lower,
and lower temperatures. As beta becomes larger,

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the variance of the gaussian becomes larger,
and the system becomes more and more quantum.

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Finally, the third property of the density
matrix concerns the high-temperature limit

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for a Hamiltonian H = H_free plus a potential
V, the density matrix at small beta (high temperature)

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is given by rho(x, x', beta) = e^(-beta/2 V(x))
* rho_free(x, x', beta) e^(-beta/2 V(x')).

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So you see, at high temperature, the correction
of the density matrix to the free density

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matrix is given by a simple Boltzmann factor
e^(-beta V(x)) split into half between x and x'.

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But notice that through this expression, we
have an explicit formula for the density matrix

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rho(x, x', beta) without solving the Schrödinger
equation, for any potential. So this is the

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density matrix at high temperature (small beta).

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So now, let us involve the convolution property,
and from this density matrix at temperature beta,

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let's compute it at inverse temperature
2beta, 4beta, 8beta, 16beta, and so on... We can reach

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the full quantum regime.

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Please take a moment to download and to run
this program as written, for the harmonic oscillator.

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You can then modify it for other
potentials by just changing an exponential factor,

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not by solving a new Schrödinger
equation.

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We will pursue this great story further in
this week's homework session.

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In matrix squaring, the subject of the last
section, we convoluted two density matrices

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at temperature T to obtain a new density
matrix at temperature T/2.

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By iterating this process, we could
go to lower and lower temperatures

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starting from the high-temperature
quasi-classical limit.

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Normally, however, we cannot do this
matrix squaring analytically. For a large

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number of particles, we soon ran out of space
to store a reasonable discretized approximation

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of rho(x, x', beta) on the computer, so we cannot
do the matrix squaring numerically here.

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We now see how the Feynman path integral overcomes
this problem, how it leads to the use of Monte-Carlo

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methods and to the idea of path sampling.
Instead of evaluating the convolution integrals

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one after the other, as we did in matrix squaring,
let us write them out all together.

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So we write the density matrix rho(x, x', beta)
= integral dx'' rho(x, x'', beta/2) rho(x'', x', beta/2).

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Each of the density matrices at beta/2 can be
written as an integral over two density matrices

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at temperature beta/4. This gives an integral
over dx'', dx''', dx'''' of rho on temperature

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beta/4, beta/4, beta/4 and beta/4. Now, each of the
density matrices at beta/4 can again be written

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as a product over two density matrices at
beta/8, and thus this would lead us to multiple

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integrals over dx''''' dx'''''' dx'''''''
and dx''''''''. The idea we are pursuing is

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great, but we are having a notational nightmare..

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Let us write {x0, x1, x2, x3 ...} instead
of the cumbersome {x, x', x'', x'''...}.

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This gives the density matrix... [formula on screen]

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For the partition function, which is the trace
of the density matrix as we discussed before,

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we find that... [formula on screen]
x0, x1... xN in these integrals is called

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a "path", and we can imagine the variable xk
to be at position k beta/N of an imaginary time

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variable tau that goes from 0 to beta in little
steps of Delta tau which is equal to beta/N. Density

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matrices and partition functions can thus
be expressed as multiple integrals over paths

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variables, so called paths integrals.

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In Markov-chain Monte-Carlo, we can move from
one path configuration to the next by choosing

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one position x_k and making a little displacement
delta x that can be positive or negative. We compute

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the weight after the move and before the move
and accept this move with the Metropolis acceptance

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probability. Note that we can also move x0
which is between x1 and x(N-1) so that the

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path can move as a whole.

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Configurations of a Markov chain simulation
for the Harmonic oscillator are shown here.

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The histogram of the x-position in this simulation
is given by the probability of the particle

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to be at position x, or in other words, the
density matrix rho(x, x, beta), the diagonal

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density matrix.

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In Python this gives the program naive_harmonic_path.py,
that I ask you to download and to run from

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the Coursera website. You will modify this
program in this week's homework where you

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will do your own Markov-chain Monte Carlo
simulation of a Quantum system, or a Path-Integral

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Monte-Carlo simulation.

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In conclusion, we have plunged in this session
of Statistical Mechanics: Algorithms and Computations

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into the world of quantum
physics and quantum statistical mechanics.

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What I have shown you, the case of the harmonic
oscillator, can be greatly generalized, as

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we will see in the coming weeks.

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The solution of the Schrödinger equation
needs a new technique for each potential,

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and Feynman Path Integral is more general.
It is for this reason that it is so famous.

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It naturally leads to the idea of Monte Carlo
simulations, and Path integral Monte Carlo algorithms.

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As we are becoming to be great
experts in Monte Carlo simulation, of course

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this approach is just for us...

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So as I said, the story will continue to unfold
in homeworks, lectures, and tutorials, and

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I hope to keep up your interest in this fascinating
subject.

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Finally, let me thank you for your attention,
and see you again, in further sessions

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of this lecture course.