Fock Space - Wave Function Interpretation

Wave Function Interpretation

Often the one particle space is given as, the space of square integrable functions on a space with measure (strictly speaking, the equivalence classes of square integrable functions where functions are equivalent if they differ on a set of measure zero). The typical example is the free particle with the space of square integrable functions on three dimensional space. The Fock spaces then have a natural interpretation as symmetric or anti-symmetric square integrable functions as follows. Let and, etc. Consider the space of tuples of points which is the disjoint union

.

It has a natural measure such that and the restriction of to is . The even Fock space can then be identified with the space of symmetric functions in whereas odd Fock space can be identified with the space of anti-symmetric functions. The identification follows directly from the isometric identification

.

Given wave functions, the Slater determinant

\Psi(x_1, \ldots x_n) = \frac{1}{\sqrt{n!}}\left|\begin{matrix} \psi_1(x_1) & \ldots & \psi_n(x_1) \\ \vdots & & \vdots \\ \psi_1(x_n) & \dots & \psi_n(x_n) \\ \end{matrix} \right|

is an antisymmetric function on . It can thus be naturally interpreted as an element of -particle section of the odd Fock space. The normalisation is chosen such that if the functions are orthonormal. There is a similar "Slater permanent" with the determinant replaced with the permanent which gives elements of -sector of the even Fock space.

Read more about this topic:  Fock Space

Famous quotes containing the words wave and/or function:

    “Through the dark cold and the empty desolation,
    The wave cry, the wind cry, the vast waters
    Of the petrel and the porpoise. In my end is my beginning.”
    —T.S. (Thomas Stearns)

    “The function of comedy is to dispel ... unconsciousness by turning the searchlight of the keenest moral and intellectual analysis right on to it.”
    —George Bernard Shaw (1856–1950)