The variational principle in quantum mechanics, lecture 12

July 7, 2011

1. Lecture 12: the time-dependent variational principle for dissipative dynamics

The time-dependent variational principle of Dirac is a powerful method to simulate the real and imaginary time dynamics of strongly correlated quantum systems. The original formulation has, as far as I can tell, only been formulated in the case of pure states. The generalisation to quantum systems undergoing dissipative dynamics is nontrivial and appears not to have been attempted. Here we describe a natural generalisation.

The pdf version can be found here.

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The variational principle in quantum mechanics, lecture 11

July 1, 2011

1. Lecture 11: the time-dependent variational principle and mean-field theory

In this and the following lectures we are going to apply the time-dependent variation principle (TDVP) to the variational classes we have met previously, namely, product states for quantum spin systems, mean-field states for bosons and fermions, and matrix product states.

The pdf version of these notes can be found here. Read the rest of this entry »


The variational principle in quantum mechanics, lectures 9 and 10

June 23, 2011

In lecture 9 we continued reading the papers on matrix product states that we began in lectures 7 and 8.

1. Lecture 10: the time-dependent variational principle

In this lecture we will discuss the time-dependent variation principle (TDVP), which is a powerful method to simulate the nonequilibrium dynamics of a general quantum system while remaining within a given variational class. The TDVP is due, as far as I’m aware, to Dirac. Additionally, describing the TDVP isn’t especially difficult, so it is rather surprising that this elegant method it hasn’t made into standard textbooks. The general framework of the time-dependent variational principle can be found in [P. Kramer and M. Saraceno, Geometry of the Time-Dependent Variational Principle in Quantum Mechanics (Springer-Verlag, Berlin) (1981)].

These notes can be found in pdf format here.

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The variational principle in quantum mechanics, lecture 7 and 8

June 2, 2011

In lectures 7 and 8 we will discuss the papers:

“Efficient classical simulation of slightly entangled quantum computations”,

found here

and

“Efficient simulation of one-dimensional quantum many-body systems”

found here

by Guifre Vidal.


The variational principle in quantum mechanics, lecture 6

May 18, 2011

1. Lecture 6: density functional theory

In the previous two lectures we discussed several examples of mean-field theory where the variational class is the class of gaussian states or single-particle product states. It turns out that mean-field theory is extremely useful in practice and can give powerful insights into the physics of many strongly interacting systems. This is because it leads to equations which are essentially tractable analytically. Additionally, the solutions often have a direct physical interpretation. Thirdly, the results obtained can often be very good indeed and often match experimentally obtained values to a high precision. Indeed, I’m sure that mean-field theory will continue to be a very useful tool for many years to come, and it should certainly be the first port of call when trying to understand a new system.

In this lecture we turn our attention to another method known as density functional theory (DFT). This can be viewed as, in part, an application of the variational method, although it is not a pure application in the sense of our previous examples.

The pdf version of the lecture notes can be found here. Read the rest of this entry »


The variational principle in quantum mechanics, lecture 5

May 11, 2011

1. Lecture 5: Mean-field theory and the Gross-Pitaevskii equation

In the previous lecture we saw two applications of the variational principle to the class of product states and fermionic gaussian states, respectively. In both cases we obtained an effective equation for the ground-state properties involving only a single effective particle (a single spin in the first case and a single majorana fermion in the second case).

In this lecture we continue our study of mean-field theory in the bosonic setting, in particular to the description of Bose-Einstein condensates. Here we find a similar result: we’ll obtain a nonlinear effective equation for the condensate in terms of a single effective particle degree of freedom. Before we do this we need to review some of the formalism for the description of quantum fields and coherent states.

The pdf version of the notes can be found here. Read the rest of this entry »


The variational principle in quantum mechanics, lecture 4

May 5, 2011

1. Lecture 4: Mean-field theory and Hartree-Fock theory

In this lecture we’ll describe a general strategy to approximately solving the many body problem introduced in the previous lecture. This approach falls broadly under the rubric of mean-field theory and is better known, in various contexts, as Hartree-Fock theory, the self-consistent field method, and the Gutzwiller ansatz.

The idea behind mean-field theory is simple: we take as a variational class one that neglects all quantum correlations between particles and apply the variational method. In the application of the variational method one then sees that the influence of all the other particles on a given one are treated in an averaged way. Our treatment of the Helium atom in lecture 2 could be seen as an application of mean-field theory in an embryonic form.

To explain mean-field theory in this lecture we’ll consider a sequence of simplified examples.

These lecture notes can be found in pdf form here.

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The variational principle in quantum mechanics, lecture 3

April 25, 2011

1. Lecture 3: The many body problem

In this lecture the many body problem is introduced in the context of first and second quantisation. The lecture notes can also be found here in pdf format.

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The variational principle in quantum mechanics, lecture 2

April 13, 2011

1. Lecture 2: The helium atom

In this lecture we are going to apply the variational method to a more complicated example, the helium atom. The simplified example we consider only involves two particles but still exhibits many of the complications and subtleties that arise in the more general {N}-body problem. Lecture notes can be found in pdf format here. Read the rest of this entry »


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