1 / 7

The Fermionic Fock Space

The Fermionic Fock Space. Vladimir Aleksandrovich Fock (1898–1974) was a Soviet physicist, who did foundational work on quantum mechanics.

manuelmjohnson
Download Presentation

The Fermionic Fock Space

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Fermionic Fock Space Vladimir Aleksandrovich Fock (1898–1974) was a Soviet physicist, who did foundational work on quantum mechanics. His primary scientific contribution lies in the development of quantum physics, although he also contributed significantly to the fields of mechanics, theoretical optics, theory of gravitation, physics of continuous medium. In 1926 he generalized the Klein-Gordon equation. He gave his name to Fock space [Z. Phys.,75 (1932) 622], the Fock representation and Fock state, and developed the Hartree-Fock method in 1930. Fock was a professor at Saint Petersburg State University

  2. Many-fermion operators and permutational symmetry Let us start from the Hamiltonian operator: Kinetic energy term Interaction with external field Two-body (pairwise) interaction one-particle operators (sums of contributions from individual particles) single-particle degrees of freedom describes self-interaction. If such a term exists, it can always be included in V1 Three-body interaction:

  3. Exchange operators eigenvalues of are For identical particles, measurements performed on quantum states and have to yield identical results - exchanges particles i and j is hermitian and unitary: (identical particles cannot be distinguished) A principle, supported by experiment This principle implies that all mmany-body wave functions are eigenstates of is a basis of one-dimensional representation of the permutation group There are only two one-dimensional representations of the permutation group: for all i,j - fully symmetric representation for all i,j - fully antisymmetric representation

  4. Consequently, systems of identical particles form two separate classes: bosons (integer spins) fermions (half-integer spins) For spin-statistics theorem, see W. Pauli, Phys. Rev. 58, 716-722(1940)

  5. Many-body space One-particle (single-particle, s.p.) basis Dimension of the s.p. space (can be infinite) The corresponding Hilbert space is called - one particle Hilbert space. On the level of it is irrelevant whether one is dealing with fermions or bosons. However, this is not the case for a two-particle space

  6. There are such states! Two-particle space tensor product, has a product basis Is this wave function a good candidate for a two-particle state? NO! … but is a basis in The above state can also be written as: labels all permutations How to generalize this result for the A-particle case?

  7. There are such states! A-particle space A-times p - permutation of A elements: a Slater determinant spanned by one-particle states! There is only one state for A=M (why?) In addition to 1-, 2-, M-particle space, there also exists a vacuum space, containing only one state, a vacuum. One cannot associate a wave function with the vacuum space (a function of no arguments?) Fock space A Hilbert space being a sum of all many-body spaces: One can now talk about wave functions which do not have a fixed number of particles! Dimension:

More Related