Spin States (3-325)

2. Use the following definitions

$S_x =(\hbar/2)\{|+><-|+|-><+|\}$   and $S_y =(\hbar/2)\{-i|+><-|+i|-><+|\}$

to show that $S_z=S_x \pm iS_y$

Notation $|m><n|=|m,n|$

$S_x=ℏ/2(|+,-| + |-,+|)$   and $S_y=-iℏ/2(|+,-| -|-,+|)$
Therefore

$S_+=ℏ*|+,-| =S_x+iS_y$     and $S_-=ℏ*|-,+|=S_x-iS_y$

4. Use the operator forms of $S_x$ and $S_z$ to evaluate $[S_x,S_z]$

$S_x=ℏ/2(|+,-| + |-,+|)$    and  $S_z=ℏ/2(|+,+| -|-,-|)$
$[S_x,S_z ]=S_x S_z-S_z S_x=$

$=ℏ^2/4*(|+,-|+|-,+|)(|+,+|-|-,-|)-$

$-ℏ^2/4(|+,+|-|-,-|)(|+,-|+|-,+|)=$

$=ℏ^2/4*\{(|+,-||+,+|)-(|+,-||–|)+(|-,+||+,+|)- (|-,+||-,-|)\}-$

$-ℏ^2/4*\{(|+,+||+,-|)+ (|+,+||-,+|)-(|-,-||+,-|)-(|-,-||-,+|)\}=$

$=ℏ^2/4*(-|+,-|+|-,+|)-$

$-ℏ^2/4*(|+,-|-|-,+|)=-ℏ^2/2*(|+,-|-|-,+|)=-iℏ*S_y$

5. For the $S_z+$ state of a spin 1/2 system , find the dispersion of $S_x$, $S_y$ and $S_z$

Prove that the generalized uncertainty relation is satisfied by the product of the dispersions of  $S_x$ and $S_y$.

a)

The $S_z+$  state is the $|+>=\begin{pmatrix}1\\0\end{pmatrix}$   state.

In short notation  $<(ΔS_i )^2> = <S_i^2>-<S_i >^2$

$S_x=ℏ/2*\begin{pmatrix}0&1\\1&0\end{pmatrix}$;   $S_y=ℏ/2*\begin{pmatrix}0&-i\\i&0\end{pmatrix}$;   $S_z=ℏ/2*\begin{pmatrix}1&0\\0&-1\end{pmatrix}$
$<S_x> = <+|S_x |+> =ℏ/2*(1, 0)\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=0$;

$<S_y>=<+|S_y |+> =⋯=0$;   $<S_z>=⋯=ℏ/2$

$<S_x^2>=ℏ^2/4*(1, 0)\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=ℏ^2/4*(0, 1)\begin{pmatrix}0&1\\1&0\end{pmatrix}\begin{pmatrix}1\\0\end{pmatrix}=$

$=ℏ^2/4*(1, 0)\begin{pmatrix}1\\0\end{pmatrix}=ℏ^2/4$

and

$<S_y^2>=⋯.=ℏ^2/4$     and   $<S_z^2> =⋯=ℏ^2/4$
Therefore

$<(ΔS_x )^2> =ℏ^2/4-0^2=ℏ^2/4$;   $<(ΔS_y )^2> =ℏ^2/4-0=ℏ^2/4$;   $<(ΔS_z )^2> =ℏ^2/4-ℏ^2/4=0$
b)

$(<(ΔS_x )^2><(ΔS_y )^2>) ≥(1/2) |<[S_x,S_y ]>| =$

$=1/2*|i*ϵ_{xyz}*ℏ<S_z>|=1/2*ℏ*ℏ/2=ℏ^2/4$    TRUE