More than 2 states

We have been labelling our basis kets as $\ket +$ and $\ket -$. Another approach is to label the kets according to the value of whatever our "analyzer" has measured. Since the S-G apparatus measures $S_z$ to be either $+\hbar/2$ or $-\hbar/2$, we could also label our states this way, as: $$\ket +\equiv \ket{+\hbar/2}\ \ \text{ and }\ \ \ket - \equiv \ket{-\hbar/2}.$$ Some other measurement apparatus may find 3,4, or more different, quantized values (or "observables") for whatever it measures.

The set of all possible observables from some apparatus is $a_1$, $a_2$, $a_3$...$a_n$. Then the corresponding kets form a set of basis vectors $\ket{a_1}$, $\ket{a_2}$, $\ket{a_3}$...$\ket{a_n}$. where the kronecker delta is defined to be 1 if $i=j$ and 0 if $i\neq j$. The kets are all orthonormal: $$\innerp{a_i}{a_i}=1,$$ and $$\innerp{a_i}{a_j}=\delta_{ij},$$ where the kronecker delta is defined to be 1 if $i=j$ and 0 if $i\neq j$.

The basis vectors 'span the space' (completeness) of all possible state vectors: $$\ket \psi=\sum_{i=1}^n\big(\innerp{a_i}{\psi}\big)\,\ket{a_i}.$$

Finally...

Immerse yourselves in the mysticism of nature that is Experiment 4 (page 9)!