91影视

• "Particles come in two types: the particles that make up matter, known as 'fermions,' and the particles that transport forces, known as 'bosons,' according to Carroll.
• Fermions take up space, whereas bosons can be stacked on top of one another.
• A Slater determinant is a formula in quantum mechanics that describes the wave function of a multi-fermionic system.

• It satisfies anti-symmetry criteria, and thus the Pauli principle, by changing sign when two electrons are exchanged (or other fermions)

$$\begin{gathered} \mathrm{When}\mathrm{N}\mathrm{fermions}\mathrm{are}\mathrm{present},\mathrm{the}\mathrm{whole}\mathrm{wave}\mathrm{function}\mathrm{can}\mathrm{be}\mathrm{represented}\mathrm{as}: \\ {\mathrm{x}}_2,...,{\mathrm{x}}_{\mathrm{N}})=\frac{1}{\sqrt{\mathrm{N}!}}\left|\begin{array}{cccc}{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right)& {\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_1\right)& ...& {\mathrm{蠄}}_{\mathrm{N}}\left({\mathrm{x}}_1\right)\\ {\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right)& {\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_1\right)& ...& {\mathrm{蠄}}_{\mathrm{N}}\left({\mathrm{x}}_2\right)\\ ...& & & \\ {\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_{\mathrm{N}}\right)& {\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_{\mathrm{N}}\right)& ...& {\mathrm{蠄}}_{\mathrm{N}}\left({\mathrm{x}}_{\mathrm{N}}\right)\end{array}\right| \\ \mathrm{The}\mathrm{Slater}\mathrm{determinant}\mathrm{is}\mathrm{the}\mathrm{name}\mathrm{for}\mathrm{this}\mathrm{formula}. \end{gathered}$$

(a)

The total wave function for identifiable particles is simply the combination of three wave functions:

$\mathrm{蠄}\left({\mathrm{x}}_{1,}{\mathrm{x}}_{2,}{\mathrm{x}}_3\right)={\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)$

(b)

When we permute any two particles, the entire wave function for identical bosons must be symmetric:

$$\begin{gathered} \mathrm{蠄}\left({\mathrm{x}}_{1,}{\mathrm{x}}_{2,}{\mathrm{x}}_{3,}\right)-\frac{1}{\sqrt{6}}\left[{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)\right] \\ +\frac{1}{\sqrt{6}}\left[{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)\right] \end{gathered}$$

(c)

When we permute any two fermions, the whole wave function must be antisymmetric for identical fermions:

$$\begin{gathered} \mathrm{蠄}\left({\mathrm{x}}_{1,}{\mathrm{x}}_{2,}{\mathrm{x}}_{3,}\right)-\frac{1}{\sqrt{6}}\left[{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)\right] \\ +\frac{1}{\sqrt{6}}\left[{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)+{\mathrm{蠄}}_{\mathrm{a}}\left({\mathrm{x}}_1\right){\mathrm{蠄}}_{\mathrm{b}}\left({\mathrm{x}}_2\right){\mathrm{蠄}}_{\mathrm{c}}\left({\mathrm{x}}_3\right)\right] \end{gathered}$$