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

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

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

starting from the high-temperature quasi-classical limit.

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

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

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

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

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

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

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

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

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

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

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

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

great, but we are having a notational nightmare..

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

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

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

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

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

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

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

variables, so called paths integrals.

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

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

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

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

path can move as a whole.

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

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

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

density matrix.

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

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

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

Monte-Carlo simulation.

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

into the world of quantum physics and quantum statistical mechanics.

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

we will see in the coming weeks.