quantum algorithms:
basics
Examples
In quantum physics and especially in condensed matter theory, Slater determinants are certain wavefunctions expressing the joint quantum state of multiple electrons (or possiby other fermions) as skew-symmetrized products of given single-particle wavefunctions.
Concretely, given a linear basis
for the Hilbert space of single-electron wavefunctions, being square-integrable functions on the ambient Euclidean -space
a Slater determinant for particles is a function on the -fold product space of the following determinant-form
where is an -tuple of indices, with the corresponding -tuple of 1-electron wavefunctions.
(Here denotes the symmetric group of permutations of ordered elements, and denotes the signature of a given permutation .)
In fact, for actual electrons the wavefunctions are also functions of their spin, which means, in the non-relativistic case, that the depend also on an argument in , in addition to their dependence on , and the corresponding Slater determinant states are obtained by skew-symmetrizing over all of these degrees of freedom.
The point of this construction is that it enforces the skew-symmetry under permutation of position of electrons, which is their characteristic property as fermions. As the multi-index set ranges, the corresponding Slater determinants span the Hilbert space of -electron quantum states.
In the practice of computing ground states etc. in solid state physics, one tries to use as few multi-indices as possible:
In the extreme case, the Hartree-Fock method tries to approximate a multi-electron system by the clever choice of a single Slater determinant. More accurate approximation methods use linear combinations of more and more Slater determinants, as the multi-index set ranges. If, in principle, the full space of Slater determinants is used, one speaks of the configuration interaction method.
The construction was maybe first made explicit as eq. (15) in
It is named after:
Review:
Attila Szabo, Neil S. Ostlund, Sec. 2.2.3 of: Modern Quantum Chemistry – Introduction to Advanced Electronic Structure Theory, Macmillan (1982), McGraw-Hill (1989), Dover (1996) pdf
C. Lanczos, R. C. Clark, G. H. Derrick (eds.), p. 196 in: Mathematical Methods in Solid State and Superfluid Theory, Springer (1986) doi:10.1007/978-1-4899-6435-9
Pablo Echenique, J. L. Alonso, around (33) in: A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry, Molecular Physics 105 (2007) 3057-3098 doi:10.1080/00268970701757875
See also:
Last revised on May 13, 2022 at 08:12:55. See the history of this page for a list of all contributions to it.