5.5.4.1 So­lu­tion com­plexsb-a

Ques­tion:

As an ex­am­ple of the or­tho­nor­mal­ity of the two-par­ti­cle spin states, ver­ify that $\langle{\uparrow}{\uparrow}\vert{\downarrow}{\uparrow}\rangle$ is zero, so that ${\uparrow}{\uparrow}$ and ${\downarrow}{\uparrow}$ are in­deed or­thog­o­nal. Do so by ex­plic­itly writ­ing out the sums over $S_{z1}$ and $S_{z2}$.

An­swer:

The in­ner prod­uct is by de­f­i­n­i­tion

$$\langle{\uparrow}{\uparrow}\vert{\downarrow}{\uparrow}\rangl... ...z1}){\uparrow}(S_{z2})\;{\downarrow}(S_{z1}){\uparrow}(S_{z2})$$

or writ­ing out the sec­ond sum ex­plic­itly

$$\langle{\uparrow}{\uparrow}\vert{\downarrow}{\uparrow}\rangl... ...rrow}(S_{z1}){\uparrow}(-{\textstyle\frac{1}{2}}\hbar) \right]$$

or writ­ing out the first sum also ex­plic­itly

$$\begin{eqnarray*}\langle{\uparrow}{\uparrow}\vert{\downarrow}{\uparrow}\rangle &... ...style\frac{1}{2}}\hbar){\uparrow}(-{\textstyle\frac{1}{2}}\hbar) \end{eqnarray*}$$

and not­ing that ${\uparrow}(-{\textstyle\frac{1}{2}}\hbar)$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0 and ${\downarrow}(+{\textstyle\frac{1}{2}}\hbar)$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0, you see that all terms are zero.