91Ó°ÊÓ

• Fermions are typically associated with matter, whereas bosons are commonly associated with force carrier particles.
• However, in today's particle physics, the distinction between the two concepts is blurred.
• Under extreme conditions, weakly interacting fermions can also exhibit bosonic behavior.

a)

If one fermion is in state${\mathrm{ψ}}_5$, second in state ${\mathrm{ψ}}_7$and third in state ${\mathrm{ψ}}_{17}$, write down total antisymmetric wave function$\mathrm{ψ}\left({\mathrm{x}}_{\mathrm{A}},{\mathrm{x}}_{\mathrm{B}},{\mathrm{x}}_{\mathrm{C}}\right)$. If we assume that fermions are in a square well of with , we can write down a wave function of one particle:

$$\begin{gathered} \mathrm{Ψ}\left(x\right)=Asin\left(k_{n}x\right) \\ {k}_n=\frac{n\mathrm{π}}{A}=\sqrt{\frac{2}{a}} \\ ⇒{\mathrm{Ψ}}_5=Asin\left(k_{5}x\right),{\mathrm{Ψ}}_7=Asin\left(k_{7}x\right),{\mathrm{Ψ}}_{1}7=Asin\left(k_{1}7x\right) \\ {k}_5=\frac{5\mathrm{π}}{a},k_7=\frac{7\mathrm{π}}{a},k_{17}=\frac{17\mathrm{π}}{a} \end{gathered}$$

$$\begin{gathered} \mathrm{ψ}\left(x_A,x_B,x_C\right)=\frac{1}{\sqrt{6}}\left[{\mathrm{ψ}}_A\left(x_A\right){\mathrm{ψ}}_b\left(x_B\right){\mathrm{ψ}}_c\left(x_C\right)-{\mathrm{ψ}}_a\left(x_A\right){\mathrm{ψ}}_c\left(x_B\right){\mathrm{ψ}}_b\left(x_C\right)-{\mathrm{ψ}}_c\left(x_A\right){\mathrm{ψ}}_b\left(x_B\right){\mathrm{ψ}}_a\left(x_C\right)\right]+ \\ =\frac{1}{\sqrt{6}}\left[-{\mathrm{ψ}}_b\left(x_A\right){\mathrm{ψ}}_b\left(x_B\right){\mathrm{ψ}}_c\left(x_C\right)+{\mathrm{ψ}}_c\left(x_A\right){\mathrm{ψ}}_a\left(x_B\right){\mathrm{ψ}}_b\left(x_C\right)+{\mathrm{ψ}}_b\left(x_A\right){\mathrm{ψ}}_c\left(x_B\right){\mathrm{ψ}}_a\left(x_C\right)\right] \end{gathered}$$

Where ${\mathrm{Ψ}}_a={\mathrm{Ψ}}_5,{\mathrm{Ψ}}_b={\mathrm{Ψ}}_7,{\mathrm{Ψ}}_c={\mathrm{Ψ}}_{17}$

The total wave function for fermions is

$$\begin{gathered} \mathrm{ψ}\left(x_A,x_B,x_C\right)=\frac{A}{\sqrt{6}}\left[\sin \left(k_A{x}_A\right)\sin \left(k_7{x}_B\right)\sin \left(k_{17}{x}_C\right)-\sin \left(k_5{x}_A\right)\sin \left(k_{17}{x}_B\right)\sin \left(k_7{x}_C\right)\right] \\ +\frac{A^3}{\sqrt{6}}\left[\sin \left(k_{17}{x}_A\right)\sin \left(k_7{x}_B\right)\sin \left(k_5{x}_C\right)-\sin \left(k_7{x}_A\right)\sin \left(k_5{x}_B\right)\sin \left(k_{17}{x}_C\right)\right] \\ +\frac{A^3}{\sqrt{6}}\left[\sin \left(k_{17}{x}_A\right)\sin \left(k_5{x}_B\right)\sin \left(k_7{x}_C\right)-\sin \left(k_7{x}_A\right)\sin \left(k_{17}{x}_B\right)\sin \left(k_5{x}_C\right)\right] \end{gathered}$$

(b)

For bosons, we use the following formula,

$$\begin{gathered} \mathrm{ψ}\left(x_A,x_B,x_C\right)=\frac{1}{\sqrt{6}}\left[{\mathrm{ψ}}_a\left(x_A\right){\mathrm{ψ}}_b\left(x_B\right){\mathrm{ψ}}_c\left(x_C\right)+{\mathrm{ψ}}_a\left(x_A\right){\mathrm{ψ}}_c\left(x_B\right){\mathrm{ψ}}_b\left(x_C\right)+{\mathrm{ψ}}_c\left(x_A\right){\mathrm{ψ}}_b\left(x_B\right){\mathrm{ψ}}_a\left(x_C\right)\right]+ \\ =\frac{1}{\sqrt{6}}\left[-{\mathrm{ψ}}_b\left(x_A\right){\mathrm{ψ}}_a\left(x_B\right){\mathrm{ψ}}_c\left(x_C\right)+{\mathrm{ψ}}_c\left(x_A\right){\mathrm{ψ}}_a\left(x_B\right){\mathrm{ψ}}_b\left(x_C\right)+{\mathrm{ψ}}_b\left(x_A\right){\mathrm{ψ}}_c\left(x_B\right){\mathrm{ψ}}_a\left(x_C\right)\right] \end{gathered}$$

(i) If they all are the same state role="math" localid="1658227599599" ${\mathrm{ψ}}_{\mathrm{a}}={\mathrm{ψ}}_{\mathrm{b}}={\mathrm{ψ}}_{\mathrm{c}}={\mathrm{ψ}}_{11}$, then we have

$$\begin{gathered} \mathrm{Ψ}(x_A,x_B,x_C)=A^3\sqrt{6}sin\left(k_{11}{x}_A\right)sin\left(k_{11}{x}_B\right)sin\left(k_{11}{x}_C\right) \\ {k}_{11}=\frac{11\mathrm{π}}{a}A=\sqrt{\frac{2}{a}} \end{gathered}$$

(ii) If two of them are same state ${\mathrm{ψ}}_{\mathrm{a}}={\mathrm{ψ}}_{\mathrm{b}}={\mathrm{ψ}}_1$and the third one is a different state

${\mathrm{ψ}}_{\mathrm{c}}={\mathrm{ψ}}_{19}$

$$\begin{gathered} \mathrm{ψ}\left(x_A,x_B,x_C\right)=\sqrt{\frac{2}{3}}{A}^3\left[\sin \left(k_1{x}_A\right)\sin \left(k_1{x}_B\right)\sin \left(k_{19}{x}_C\right)+\sin \left(k_1{x}_A\right)\sin \left(k_1{x}_C\right)\sin \left(k_{19}{x}_B\right)\right]+ \\ \sqrt{\frac{2}{3}}{A^3}^3\left[\sin \left(k_1{x}_C\right)\sin \left(k_1{x}_B\right)\sin \left(k_{19}{x}_A\right)\right] \end{gathered}$$

(iii)If all of them are in different states ${\mathrm{Ψ}}_a={\mathrm{Ψ}}_5,{\mathrm{Ψ}}_b={\mathrm{Ψ}}_7,{\mathrm{Ψ}}_c={\mathrm{Ψ}}_{17}$then total wave

$$\begin{gathered} \mathrm{ψ}\left(x_A,x_B,x_C\right)=\frac{A^3}{\sqrt{6}}\left[\sin \left(k_5{x}_A\right)\sin \left(k_7{x}_B\right)\sin \left(k_{17}{x}_C\right)-\sin \left(k_5{x}_A\right)\sin \left(k_{17}{x}_B\right)\sin \left(k_7{x}_C\right)\right] \\ =\frac{A^3}{\sqrt{6}}\left[\sin \left(k_{17}{x}_A\right)\sin \left(k_7{x}_B\right)\sin \left(k_5{x}_C\right)-\sin \left(k_7{x}_A\right)\sin \left(k_5{x}_B\right)\sin \left(k_{17}{x}_C\right)\right] \\ =\frac{A^3}{\sqrt{6}}\left[\sin \left(k_{17}{x}_A\right)\sin \left(k_5{x}_B\right)\sin \left(k_7{x}_C\right)-\sin \left(k_7{x}_A\right)\sin \left(k_{17}{x}_B\right)\sin \left(k_5{x}_C\right)\right] \end{gathered}$$

Therefore the total wave function for bosons are

(ii) $\mathrm{Ψ}(x_A,x_B,x_C)=\sqrt{\frac{2}{3}}{A}^3\left[sin\right(k_1{x}_A\left)sin\right(k_1{x}_B\left)sin\right(k_{19}{x}_C)+sin(k_1{x}_A\left)sin\right(k_1{x}_C\left)sin\right(k_{19}{x}_B\left)\right]+$

(iii) $$\begin{gathered} \mathrm{Ψ}(x_A,x_B,x_c)=\frac{A^3}{\sqrt{6}}\left[sin\right(k_5{x}_A\left)sin\right(k_7{x}_B\left)sin\right(k_{17}{x}_C)-sin(k_5{x}_A\left)sin\right(k_{1}7x_B\left)sin\right(k_7{x}_C\left)\right] \\ +\frac{A^3}{\sqrt{6}}\left[sin\right(k_{17}{x}_A\left)sin\right(k_7{x}_B\left)sin\right(k_5{x}_C)-sin(k_7{x}_A\left)sin\right(k_5{x}_B\left)sin\right(k_{17}{x}_C\left)\right] \\ +\frac{A^3}{\sqrt{6}}\left[sin\right(k_{17}{x}_A\left)sin\right(k_5{x}_B\left)sin\right(k_7{x}_C)-sin(k_7{x}_A\left)sin\right(k_{17}{x}_B\left)sin\right(k_5{x}_C\left)\right] \end{gathered}$$