Question

A) Show that the Slater determinant wavefunction for the boron atom in its ground state is antisymmetric. (Please use one of the theorems on determinants). B) State why this is an indistinguishable wavefunction.

Answer 1

In Slater determinant is an expression that wave-function which satisfy anty symmetry and Pauli principle by changing sign upon exchange of two electrons. Since the electrons are indistinguishable we required that the wavefunction is either symmetric or antisymmetric with respect to interchange of electron space and spin indices.

(A) The electonic configuration of ground-state boron is 1s2 2s2

For a multi-electron atomic system, the matrix that one takes the determinant of to create a properly antisymmetrized wavefunction has a row corresponding to each electron, and a column corresponding to each single-electron wavefunction (orbital) of the atom. In writing the single-electron wavefunctions, we have to distinguish between "spin-up" wavefunctions and "spin-down" wavefunctions. I will use an asterisk (*) to indicate a spin-down wavefunction, e.g., 1s*, while spin-up wavefunctions will not have an asterisk.

The matrix, M, for the Slater determinant for boron would then be written as:

1s(1) 1s*(1) 2s(1) 2s*(1)

1s(2) 1s*(2) 2s(2) 2s*(2)

where the numbers in parentheses indicate the electron number

The Slater determinant is then:

(1/sqrt(2!)) * det(M)

(B) Electrons are indistinguishable and the total wavefunction which is written as the product of a spatial part with a spin part must be antisymmetric with respect to particle exchange.