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Find the spin wave function for a spin ${1\over2}$ particle. It is  given that spin projection along the $x$- axis has a definite value 
(a)$\displaystyle {\hbar\over 2}$                         (b) $\displaystyle -{\hbar\over2}$

A particle is in the spin state given by $$   \left( \begin{array}{r} \displaystyle{1\over\sqrt{5}}\\[4mm]\displaystyle {-2\over\sqrt{5}}  \end{array} \right) $$

Find the normalized  spin wave functions of a spin $1/2$ particle for the following cases.

PROBLEM-QM-20-LEVEL1

Consider two  particles, $A,B$ each having spin 1. Construct all possible states $\vert{SM}\rangle$ with definite values of $S^2$ and $S_z$. Use the notation $\vert{A  m_1}\rangle\vert{B    m_2}\rangle$ to represent the states of the two particles having definite $z-$ projections $m_1,m_2$ of spin. Verify that the states are symmetric under exchange of $m_1$ and $m_2$ when total spin is $S=2,0$ and antisymmetric for $S=1$.

Find the spin wave functions for a spin $1/2$ particle with spin projections $\pm 1/2$ along $(1,1,1)$. Verify that the two wave functions are orthogonal.

Construct spin matrices for spin $3/2$. Using your answer find the matrix for $\vec{S}^2$. Do you need to construct all the three spin matrices to get an answer for $\vec{S}^2$?

Show that for a system of two identical particles having spin $s$, the ratio of the number of states, symmetric under exchange of spins, to the number of the antisymmetric states is given by ${(s+1)\over s}.$

If $\vec{\alpha}$ is a vector $(\alpha_1,\alpha_2,\alpha_3)$, show that  $$ (\vec{\alpha}\cdot\vec{\sigma})^2 = |\vec{\alpha}|^2$$  where \(|\alpha|=\sqrt{\alpha_1^2+\alpha_2^2+\alpha_3^2}\)  Use this result and show that \[\exp(i\vec{\alpha}\cdot\vec{\sigma}) =
\cos{|\vec{\alpha}|} + i\vec{\alpha}\cdot\sigma \sin|\vec{\alpha}|\]

Show that  a  \( 2\times 2 \) complex matrix, which anticommutes with all the three Pauli matrices, must be null matrix.