Module 1 : Quantum Mechanics
Chapter 7 : Miscellaneous topics
 
  in Eq.(6.62) where $\vert l,m>$ are eigenstates of $L^2$ and $L_z$,  
  \begin{displaymath} L^2\vert l,m>=\hbar^2 l(l+1)\vert l,m> \quad L_z\vert l,m>=\hbar m\vert l,m>. \end{displaymath} (7.11)
  The maximum and minimum values of $m$ have to lead to zeros in Eq.(7.10) which implies  
  $\displaystyle m=m_{\rm {max}}:\quad (l-m)(l+m+1)=0\Rightarrow m_{\rm {max}}=l$  
  $\displaystyle m=m_{\rm {min}}:\quad (l+m)(l-m+1)=0\Rightarrow m_{\rm {min}}=-l$  
  $\displaystyle \qquad \Rightarrow m_{\rm {max}}-m_{\rm {min}}=2l.$ (7.12)
  Now, since $m_{\rm {max}}$ is reached from $m_{\rm {min}}$ in integer number of steps as in Eq.(7.10), one has  
  \begin{displaymath} 2l=\rm {integer}. \end{displaymath} (7.13)
  This implies that the angular momentum algebra allows $l$ to be a half integer. Of course,
spatial description in terms of spherical harmonics puts in the condition that $m$ is an integer.
Therefore, there is no spatial description of $l$= half integer states which are described as intrinsic
spin $1/2$ or higher half integer intrinsic spin states. 		 
  For spin 1/2 particle, the basic states may be described as  
  \begin{displaymath} \vert 1/2,1/2>=\vert\alpha>,\quad \vert 1/2,-1/2>=\vert\beta> \end{displaymath} (7.14)
  which are eigenstates of spin angular momentum,  
  $\displaystyle S^2 \vert 1/2,\pm 1/2>=(1/2)(3/2)\hbar^2 \vert 1/2,\pm 1/2>$  
  $\displaystyle S_z \vert 1/2,\pm 1/2>=\pm (\hbar/2)\vert 1/2,\pm 1/2>.$ (7.15)