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

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