Question

Slater Determinant: A convenient and compact way of expressing multi-particle states of anti-symmetric character for many fermions is the Slater determinant:

$\left|\begin{array}{ccccc}{\psi }_{n_1}\left(x_1\right)m_{31}& {\psi }_{n_2}\left(x_1\right)m_{32}& {\psi }_{n_3}\left(x_1\right)m_{33}& ···& {\psi }_{n_N}\left(x_1\right)m_{sN}\\ {\psi }_{n_1}\left(x_2\right)m_{11}& {\psi }_{n_2}\left(x_2\right)m_{32}& {\psi }_{n_3}\left(x_2\right)m_{33}& ···& \psi {\psi }_{n_1}\left(x_2\right)m_{sN}\\ {\psi }_{n_3}\left(x_3\right)m_{31}& {\psi }_{n_2}\left(x_3\right)m_{12}& {\psi }_{n_3}\left(x_3\right)m_{33}& & {\psi }_{n_N}\left(x_3\right)m_{sN}\\ ···& ···& ···& ···& ···\\ {\psi }_{n_1}\left(x_N\right)m_{11}& {\psi }_{n_2}\left(x_N\right)m_{32}& {\psi }_{n_3}\left(x_N\right)m_{33}& ···& {\psi }_{n_N}\left(x_N\right)m_{sN}\end{array}\right|$

It is based on the fact that for N fermions there must be Ndifferent individual-particle states, or sets of quantum numbers. The ith state has spatial quantum numbers (which might be ${\mathit{n}}_{\mathbf{i}}\mathbf{,}{\mathit{\ell }}_{\mathbf{i}}$, and ${\mathit{m}}_{\mathbf{f}\mathbf{i}}$) represented simply by${\mathit{n}}_{\mathbf{i}}$ and spin quantum number ${\mathit{m}}_{\mathbf{s}\mathbf{i}}$. Were it occupied by the ith particle, the slate would be${\mathit{\psi }}_{\mathbf{n}\mathbf{i}}\left(x_j\right){\mathit{m}}_{\mathbf{s}\mathbf{i}}$ a column corresponds to a given state and a row to a given particle. For instance, the first column corresponds to individual particle state ${\mathit{\psi }}_{\mathbf{n}}\left(x_j\right){\mathit{m}}_{\mathbf{s}\mathbf{1}}$. Where jprogresses (through the rows) from particle 1 to particle N. The first row corresponds to particle I. which successively occupies all individual-particle states (progressing through the columns). (a) What property of determinants ensures that the multiparticle state is 0 if any two individual particle states are identical? (b) What property of determinants ensures that switching the labels on any two particles switches the sign of the multiparticle state?

Short answer

(a) The property is when two columns of a determinant are identical then the determinant is considered to be zero.

(b) The property is that if two rows or two columns of a determinant are interchanged then the sign of the determinant will also be changed.

Step-by-step solution

Given data

Multi particle state is 0.

Concept of Determinant

A determinant is defined only for square matrix.

$$\begin{gathered} A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right] \\ \det A=\left(a_{41}\right)\det \left[\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right]+\left(a_{33}\right)\det \left[\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right] \\ =a_{11}\left[a_{22}\left(a_{33}\right)-a_{32}\left(a_{23}\right)\right]-a_{42}\left[a_{23}\left(a_{33}\right)-a_{31}\left(a_{23}\right)\right]+a_3\left[a_{21}{a}_{32}-a_{31}{a}_{22}\right] \end{gathered}$$

Step 3(a): Find what property of determinants ensures that the multiparticle state is 0

• A matrix is an array of number.
• A scalar quantity that can be calculated from a square matrix of order n by n is called determinant.
• A determinant is defined only for square matrix.
• In case two columns of a determinant are identical then the determinant is considered to be zero.
• It is a property of N by N matrix.
• It assures that when two of the one-electron states have same orbital wave function and also same spin, and then wave function is identically zero.

Step 4(b): Find what property of determinants ensures that switching the labels on any two particles switches the sign of the multiparticle state

• A matrix is an array of number.
• A scalar quantity that can be calculated from a square matrix of order n by n is called determinant.
• A determinant is defined only for square matrix.
• When two rows or two columns of a determinant are interchanged then the sign of the determinant will also be changed.
• It is a property of N by N matrix.
• It assures that when given pair of coordinates are interchanged.
• The wave function represented by the slater determinant will have changed sign.