Paul's Exclusion Principle

The equations involve variables like i, j, n, m, and Greek letters (possibly representing quantum numbers or wavefunctions). There are no diagrams or other visual elements. The handwriting is neat and legible, with clear spacing between characters and lines. There are mathematical equations, formulas, and explanations of statistical mechanics concepts. The diagrams are simple and represent basic concepts. The overall layout is somewhat organized, with text and diagrams placed in different sections. Ques. 16 [\psi(\vec{r})=\frac{1}{\sqrt{\pi a_{0}^{3}}}e^{-(\gamma/a_{0})}] Rage-66 $ = 12=27 [\hat{\phi}\rangle=\langle z^{2}\rangle-\langle\tau^{2}\rangle] z= rose : (\langle\gamma^{2}\rangle=\frac{n^{2}}{2}(5\eta^{2}+1-3il(1+1)(\frac{\eta_{0}}{2})^{2}) [x=0\ g*\end{matrix}] or [\langle r^{2}\rangle=3{4^{\circ}}^{2}] For s-orbital. (groundstate) [\langle r^{2}\rangle=\frac{1}{\pi q_{0}^{3}}\int_{0}^{\infty}e^{-\frac{2\gamma}{a_{0}}}\gamma^{-4}d\gamma] 4! (2/9)5 [\langle x^{2}\rangle=\langle y^{2}\rangle=\langle z^{1}\rangle] • (\langle r^{2}\rangle=\langle x^{2}\rangle+\langle y^{2}\rangle+iz^{2}) [3a_{0}^{2}=3\langle z^{2}\rangle] 2 = 90 \[\langle\hat{\phi}\rangle={a_{o}}^{2}-3{a_{o}}^{2}=-2{a_{o}}^{2}\] Ques.16 ophond) [\psi=\sqrt{\frac{8}{21}}\psi_{20}-\sqrt{\frac{3}{7}}\Psi_{310}+\sqrt{\frac{4}{21}}\Psi_{321}] 21 [y=\frac{5}{7}] 21 [=\frac{(18+24)\hbar^{2}}{21}=\frac{\prime\prime}{21}=\alpha\hbar^{2}] for ground stalk [n=1,4=0,me0] The ground stade energy of the system pothwell \[E=\frac{\pi^{2}\hbar^{2}}{\alpha^{2}m\iota^{2}}+\frac{4~\pi^{2}\hbar^{2}}{2m~L^{2}}\] \[\psi_{Tefel}=\] Antisymmetric (for 2e-system) \[X_{s}=\] \[\psi_{fpac}=\] symme bric Anti symm. Porsibilities \[\psi_{1}(x_{1})\psi_{2}(n_{2})\] \[\psi_{1}(\eta_{2})\psi_{2}(\eta_{1})\] Y should be antisummetaty after superpossing \[\psi_{\beta pace}=\frac{1}{\sqrt{2}}[\psi_{1}(\gamma_{1})\psi_{2}(\gamma_{2})-\psi_{1}(\gamma_{2})\psi_{2}(\gamma_{1})]\] (eption-d) The text includes variables like n, m, a, and Greek letters (likely representing quantum numbers or wavefunctions). There are no diagrams or other visual elements. The handwriting is somewhat neat, with generally clear spacing between characters and lines. The energy of first excited stater [=\frac{\pi^{2}\hbar^{2}}{2mL^{2}}+2\times\frac{4\pi^{2}\hbar^{2}}{2mL^{2}}] [=9(\frac{\pi^{2}\hbar^{2}}{\alpha^{2}mL^{2}})] for g.s and first excited state energy in Unit of 2m² is 6. and 9. (option-b) alles.12 2-D rigid box page-78 [\epsilon_{n}=(n_{x}^{2}+n_{y}^{2})\frac{\pi^{2}\hbar^{2}}{8m_{l}^{2}}] [n_{x},n_{y}=1,2,3-] Ey \[\epsilon_{1}=\frac{2\pi^{2}\hbar^{2}}{2~m\dot{L}^{2}}\] \[\epsilon_{2}=\frac{5\pi^{2}\hbar^{2}}{2mL^{2}},m_{x}=1,n_{y}=\] \[n_{xx}=2,n_{y}=1\] \[\epsilon_{3}\] 62 The ground state energy [E=\alpha\epsilon_{1}+4\epsilon_{2}] [=\alpha\times\frac{\alpha\pi^{2}\hbar^{2}}{2m^{2}}+4\times\frac{5\pi^{2}b^{2}}{2mL^{2}}] [=-\alpha^{\prime}4(\frac{a^{4}\hbar^{2}}{2m~L^{2}})=\frac{\underline{2}4}{Ans}.]