#### Question

A convenient and compact way of expressing multiparticle states of antisymmetric character for many fermions is the Slater determinant: $$\begin{matrix} \Psi_{n 1}(x 1) ms_1 & \Psi_{\mathrm{n} 2}\left(x_{1}\right) \mathrm{ms}_2 & \Psi_{n 3}(x 1) ms_3 & \Psi_{\mathrm{nN}}(x_1) \mathrm{ms}_N\\ \Psi_{\mathrm{n}1}(x_2) \mathrm{ms}_1 & \Psi_{\mathrm{n}2}(x_2) \mathrm{ms}_2 & \Psi_{n 3}( x_2) m s_3 & \Psi_{\mathrm{nN}}(x_2) \mathrm{ms}_N\\ \Psi_{\mathrm{n}1}(x_3) \mathrm{ms}_1 & \Psi_{\mathrm{n}2}(x_3) \mathrm{ms}_2 & \Psi_{n 3}( x_3) m s_3 & \Psi_{\mathrm{nN}}(x_3) \mathrm{ms}_N\\ \cdot & \cdot & \cdot & \cdot\\ \Psi_{\mathrm{n}1}(x_N) \mathrm{ms}_1 & \Psi_{\mathrm{n}2}(x_N) \mathrm{ms}_2} & \text{\Psi_{n 3}( x_N) m s_3 & \Psi_{\mathrm{nN}}(x_N) \mathrm{ms}_N\\ \end{matrix}$$ It is based on the fact that for N fermions there must be N different individual-particle states, or sets of quantum numbers. The ith state has spatial quantum numbers (which might be ni; $\ell i$ and $m \ell i)$ represented simply by ni*and spin quantum number msi. Were it occupied by the $\psi_{n1}(x_j)ms_i$ A column corresponds to a given state and a row to a given particle. For instance, the first column corresponds to individual-particle state $\psi_{n1}(x_j)ms_1$ where j progresses (through the rows) from particle I to particle N. The first row corresponds to particle 1, 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?

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The given Slater Determinant for N fermions there must be N different individual-particle state, or sets of quantum numbers:

$\begin{vmatrix} \psi_{n_{1}}(x_{1})m_{s1} & \psi_{n_{2}}(x_{1})m_{s2} & \psi_{n_{3}}(x_{1})m_{s3}& \dots & \psi_{n_{N}}(x_{1})m_{sN} \\ \psi_{n_{1}}(x_{2})m_{s1} & \psi_{n_{2}}(x_{2})m_{s2} & \psi_{n_{3}}(x_{2})m_{s3}& \dots & \psi_{n_{N}}(x_{2})m_{sN}\\ \psi_{n_{1}}(x_{3})m_{s1} & \psi_{n_{2}}(x_{3})m_{s2} & \psi_{n_{3}}(x_{3})m_{s3}& \dots & \psi_{n_{N}}(x_{3})m_{sN}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \psi_{n_{1}}(x_{N})m_{s1} & \psi_{n_{2}}(x_{N})m_{s2} & \psi_{n_{3}}(x_{N})m_{s3}& \dots & \psi_{n_{N}}(x_{N})m_{sN}\\ \end{vmatrix}$

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