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### Problem-QM-20017

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 $\exp(i\vec{\alpha}\cdot\vec{\sigma}) = \cos{|\vec{\alpha}|} + i\vec{\alpha}\cdot\sigma \sin|\vec{\alpha}|$

kapoor 28/10/18

### Problem-QM-20016

Show that the Pauli matrices anti-commute. $$\sigma_j\sigma_k + \sigma_k\sigma_j =2\delta_{jk}$$ and that the square of each Pauli matrix is identity matrix. $$\sigma_k^2=I, \qquad k=1,2,3$$

kapoor 28/10/18

### Problem-QM-20013

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 ${(s+1)\over s}.$

kapoor 28/10/18

### QM-20 :: Spin and Identical Particles

Summary (or abstract) not available for this node.
kapoor 28/10/18

### Problem-QM-20015

Prove that the Pauli matrices satisfy the commutation relations.
$$[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$$

kapoor 28/10/18

### Problem-QM-20014

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 $\vec{S}^2$?

kapoor 28/10/18

### Problem-QM-20012

1. Using the known results for H- atom, write the wave functions for electron in Hydrogen atom for the following quantum numbers
(i) $n=2 , \ l=1,\ m_l=1,0,-1$
(ii) $n=2, \ l=2, m_l=2,1,0,-1,-2$.
2. Write the wave function for an electron in a state with $j={1\over 2}, \ m_j= {1\over 2}$
kapoor 28/10/18

### Problem-QM-20011

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.

kapoor 28/10/18

### Problem-QM-20010

For an electron with $\ell=1$ what are the allowed values of the total angular momentum ? Write the total angular momentum states $\vert{jm}\rangle$ in terms of  $\vert{\ell=1\,m_l,~s=\frac{1}{2}\, m_s}\rangle$ states for all possible values of $j,m$.

kapoor 28/10/18

### Problem-QM-20009

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 $S=1$.

kapoor 28/10/18