The variational principle in quantum mechanics, lecture 1

After another long hiatus I’ll be back to more regular blogging: I’m teaching a course entitled the variational principle in quantum mechanics for the summer semester here at the ITP Hannover, and I’ll be posting the notes. I hope to post other things along the way as there are many exciting developments I would like to mention…

Please feel free to comment or make suggestions on the lecture notes, I would sincerely appreciate your feedback!

1. Lecture 1

1.1. Organisation

Lectures will take place on fridays at 12:00 until 14:00 (negotiable). The course comprises 16 lectures in english. Assessment will be via a combination of weekly homework exercises and a final examination.

1.2. Course outline

We will cover the following topics.

  1. Reminder of quantum mechanics and the variational method.
  2. The helium atom.
  3. The many body problem.
  4. Mean field theory and Hartree-Fock theory.
  5. The Gross-Pitaevskii equation.
  6. Density functional theory.
  7. Density functional theory cont.
  8. Matrix product states.
  9. Matrix product states cont.
  10. The density matrix renormalisation group (DMRG).
  11. The DMRG cont.
  12. The time-dependent variational principle (TDVP).
  13. Applications of the TDVP to density functional theory and Hartree-Fock.
  14. Applications of the TDVP to density functional theory and Hartree-Fock cont.
  15. QFT and beyond.
  16. Review.

1.3. Lecture notes

The notes for the lectures will be posted on my blog site: http://tjoresearchnotes.wordpress.com/

1.4. A reminder of quantum mechanics

See handout.

1.5. The variational principle

The variational principle is the basis of a tremendous number of highly successful calculational tools in physics, which is surprising because the basic method is simple enough to cover in an introductory quantum mechanics course. Of course life is never simple, especially for interacting systems of many particles. The trouble is that it is very hard to come up with a variational wavefunction that you can actually calculate with and which bears some resemblence to the actual ground-state. You need to be extremely clever to design reasonable variational wavefunctions, and it often requires deep new insights into the physics of a system to come up with a good one. Some classic examples include the BCS wavefunction in superconductivity, the Gutzwiller wavefunction, and the Laughlin wave function for the fractional quantum Hall effect (ok, the last one is a bit of a cheat, it doesn’t really have any free parameters).

Consider an arbitrary physical system whose hamiltonian {H} is time independent. Let’s initially assume, for simplicity, that the spectrum of {H} is discrete and non degenerate:

\displaystyle  	H|E_n\rangle = E_n|E_n\rangle, \quad n = 0, 1, 2, \ldots, \ \ \ \ \ (1)

with {0 \le E_0 < E_1 < E_2 < \cdots}. Although the hamiltonian is assumed known, this isn’t necessarily the case for {E_n} and {|E_n\rangle}. Generally speaking, the variational method is most useful in the case where we cannot diagonalise {H} exactly, although there are many situations where it can be helpful to exploit the method when exact diagonalisation is, in principle, available because it can be much less expensive computationally.

Ground-state properties

Choose an arbitrary state {|\psi\rangle} from the state space of the system (we don’t assume {|\phi\rangle} is normalised). The expectation value of the energy of the system in this state is given by

\displaystyle  	\langle H \rangle = \frac{\langle \psi|H|\psi\rangle }{\langle \psi |\psi \rangle} \ge E_0, \ \ \ \ \ (2)

with equality occurring if and only if {|\psi\rangle} is proportional to the eigenvector {|E_0\rangle}.

Proof: Exercise. \Box

This simple property is the basis for an approximate determination of the ground-state energy {E_0}: choose a family of trial states {|\psi(\mathbf{x})\rangle} which depend on a number of variational parameters {\mathbf{x} = (x_1, x_2, \ldots, x_n)}. Calculate the expectation value

\displaystyle  	E(\mathbf{x}) \equiv \frac{\langle \psi(\mathbf{x})|H|\psi(\mathbf{x})\rangle}{\langle \psi(\mathbf{x})|\psi(\mathbf{x})\rangle}

with respect to the parameters {\mathbf{x}}. The minimal value {E_* = \inf_{\mathbf{x}} E(\mathbf{x})} so obtained is an approximation to {E_0} which, thanks to the above, always overestimates the ground-state energy: {E_* \ge E_0}. The closer {E_*} gets to {E_0} the more closely the corresponding trial state {|\psi(\mathbf{x}_*)\rangle} should approximate the properties of {|E_0\rangle}.

This is the variational method.

The Ritz theorem

Theorem 1 The expectation value {\langle H \rangle} is stationary in the vicinity of its discrete eigenvalues

Proof: Consider the energy expectation value

\displaystyle  		E(\psi) \equiv \langle H \rangle = \frac{\langle \psi|H|\psi\rangle }{\langle \psi |\psi \rangle}, 	\ \ \ \ \ (3)

as a functional of {|\psi\rangle}. Calculate the increment {\delta E(|\psi\rangle) = E(|\psi\rangle + |\delta \psi\rangle)-E(|\psi\rangle)}; stationarity implies that, to first order in the components of {|\delta \psi\rangle}:

\displaystyle  		\delta E(|\psi\rangle) = 0 	\ \ \ \ \ (4)


\displaystyle  			= \langle \psi| (H- \langle H\rangle )|\delta \psi\rangle + \langle \delta\psi|(H-\langle H\rangle )|\psi\rangle. 	\ \ \ \ \ (5)

Set

\displaystyle  		|\phi\rangle = (H-\langle H\rangle)|\psi\rangle. 	\ \ \ \ \ (6)

Now (5) becomes

\displaystyle  		2\langle \phi|\phi\rangle \delta\lambda = 0 	\ \ \ \ \ (7)

when we write

\displaystyle  		|\delta\psi\rangle = \delta \lambda|\phi\rangle + |\phi^\perp\rangle. 	\ \ \ \ \ (8)

The norm of {|\phi\rangle} is therefore zero, and {|\phi\rangle} must be consequently zero. This means that

\displaystyle  		H|\psi\rangle = \langle H\rangle |\psi\rangle. 	\ \ \ \ \ (9)

Thus the expectation value is stationary if and only if {|\psi\rangle} is an eigenvector of {H}, and the stationary values of {\langle H \rangle} are the eigenvalues. \Box

Thus the variational method may be generalised to the approximate determination of eigenvalues of {H}. If {E(\mathbf{x})} has several extrema, they give the approximate values of some of its energies.

2. Application to a simple example

To illustrate the discussion and to give an idea of how valid the approximations made in the variational method are, we apply the method to the one-dimensional harmonic oscillator

\displaystyle  	H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac12 m\omega^2 x^2. \ \ \ \ \ (10)

Since the hamiltonian is even we know the ground-state must be represented by an even wavefunction.

The trial wavefunctions we explore here are the rational wavefunctions

\displaystyle  	\psi_a(x) = \frac{1}{x^2+a}, \quad a\ge 0. \ \ \ \ \ (11)

The normalisation {\langle \psi_a|\psi_a\rangle} is easily found to be, after a simple integration (exercise)

\displaystyle  	\langle \psi_a|\psi_a\rangle = \frac{\pi}{2a\sqrt{a}}, \ \ \ \ \ (12)

so that (exercise)

\displaystyle  	E(a) = \langle H\rangle = \frac{\hbar^2}{4m} \frac{1}{a} + \frac12 m\omega^2 a. \ \ \ \ \ (13)

The minimum value of {E(a)} is found to be at (exercise)

\displaystyle  	a = a_0 = \frac{1}{\sqrt{2}}\frac{\hbar}{m\omega}, \ \ \ \ \ (14)

where

\displaystyle  	E(a_0) = \frac{1}{\sqrt{2}}\hbar \omega. \ \ \ \ \ (15)

This result is within a relative factor

\displaystyle  	\frac{E(a_0) - \frac12\hbar \omega}{\hbar \omega} = \frac{\sqrt{2}-1}{2} \approx 20\% \ \ \ \ \ (16)

of the exact answer {E_0 = \frac12 \hbar \omega}.

2.1. Caveats

The variational method suffers from two important drawbacks:

  1. There is no a priori guarantee whatsoever that the trial state {|\psi(\mathbf{x}_*)\rangle} corresponding to the minima of the energy expectation value {E(\mathbf{x})} at {\mathbf{x}_*} has any resemblance to the ground state.
  2. For a generally chosen set of trial states {|\psi(\mathbf{x})\rangle} there is typically no way to analytically find the minimum of {E(\mathbf{x})}.

By developing ways to overcome these two crucial drawbacks researchers have built a surprisingly rich and innovative set of methods to apply the variational principle to various physical systems. Roughly speaking, as we’ll see, to overcome the first problem one argues that the variational class is complete, in the sense that a member of the class could, in principle, capture the physics of a given system. To overcome the second problem one relaxes the minimisation via some minimisation heuristic such as Newton’s method. The combination of these two strategies has culminated in what must be the most sophisticated application of the variational method in quantum mechanics: the density matrix renormalisation group.

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4 Responses to The variational principle in quantum mechanics, lecture 1

  1. ektel says:

    great post, love it. I heard variational problem when i was reading Feynman integral

  2. I like what you guys are up too. This type of clever work and exposure!

    Keep up the superb works guys I’ve incorporated you guys to our blogroll.

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