Question

(a) Construct the completely anti symmetric wave function $\mathit{\psi }\mathbf{(}{\mathbf{x}}_{\mathbf{A}}\mathbf{,}{\mathbf{x}}_{\mathbf{B}}\mathbf{,}{\mathbf{x}}_{\mathbf{C}}\mathbf{)}$for three identical fermions, one in the state ${\mathit{\psi }}_{\mathbf{5}}$, one in the state ${\mathbf{\psi }}_{\mathbf{7}}$_,and one in the state ${\mathbf{\psi }}_{\mathbf{17}}$

(b)Construct the completely symmetric wave function $\mathit{\psi }\mathbf{(}{\mathbf{x}}_{\mathbf{A}}\mathbf{,}{\mathbf{x}}_{\mathbf{B}}\mathbf{,}{\mathbf{x}}_{\mathbf{C}}\mathbf{)}$for three identical bosons (i) if all are in state ${\mathbf{\psi }}_{\mathbf{11}}$(ii) if two are in state ${\mathbf{\psi }}_{\mathbf{19}}$and another one is role="math" localid="1658224351718" ${\mathbf{\psi }}_{\mathbf{1}}$c) one in the state ${\mathbf{\psi }}_5$, one in the state ${\mathbf{\psi }}_7$_,and one in the state${\mathbf{\psi }}_{\mathbf{17}}$

Short answer

(a)The total wave function for fermions is

$\Psi (x_A,x_B,x_c)=\frac{A^3}{\sqrt{6}}\left[sin\right(k_5{x}_A\left)sin\right(k_7{x}_B\left)sin\right(k_{1}7x_C)-sin(k_5{x}_A\left)sin\right(k_{17}{x}_B\left)sin\right(k_7{x}_C\left)\right]$

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

(b)The total wave function for bosons are

(i) $\Psi (x_A,x_B,x_C)=A^3\sqrt{6}sin\left(k_{11}{x}_A\right)sin\left(k_{11}{x}_B\right)sin\left(k_{11}{x}_C\right)$

(ii) $$\begin{gathered} \Psi (x_A,x_B,x_C)=\sqrt{\frac{2}{3}}{A}^3\left[sin\right(k_1{x}_A\left)sin\right(k_1{x}_B\left)sin\right(k_{1}9x_C)+sin(k_1{x}_A\left)sin\right(k_1{x}_C\left)sin\right(k_{1}9x_B\left)\right]+ \\ \sqrt{\frac{2}{3}}{A}^3\left[sin\right(k_1{x}_C\left)sin\right(k_1{x}_B\left)sin\right(k_{1}9x_A\left)\right] \end{gathered}$$

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

Step-by-step solution

Define Fermions and bosons

• 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.

Determining the anti-symmetric wave function

a)

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

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

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

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

The total wave function for fermions is

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

Determining the symmetric wave function

(b)

For bosons, we use the following formula,

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

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

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

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

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

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

$k_{19}=\frac{19\pi }{a}{k}_1=\frac{\pi }{a},A=\sqrt{\frac{2}{a}}$

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

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

Therefore the total wave function for bosons are

(i)$\Psi (x_A,x_B,x_C)=A^3\sqrt{6}sin\left(k_{11}{x}_A\right)sin\left(k_{11}{x}_B\right)sin\left(k_{11}{x}_C\right)$

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

$\sqrt{\frac{2}{3}}{A}^3\left[sin\right(k_1{x}_C\left)sin\right(k_1{x}_B\left)sin\right(k_{19}{x}_A\left)\right]$

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