• If an arbitrary quantum state vector is presented that is not normalized, normalizing the quantum state vector before performing any other computations is absolutely necessary to finding the correct probability calculations.

• In general, a quantum state vector is normalized if $\left\vert\langle \psi\vert\psi \rangle\right\vert^{2}=1$. So, if we have an arbitrary wave state

$$\vert\psi \rangle=ae^{i \alpha}\vert+ \rangle + be^{i \beta}\vert- \rangle \; \; , $$

the quantum state vector is normalized if

$$\left\vert ae^{i \alpha}\right\vert^{2} + \left\vert be^{i \beta}\right\vert^{2}=1 \; \; ,$$

$$a^{2}+b^{2}=1 \; \; .$$

• Let the students know that if this is not the case for a given quantum state vector, insert a normalization constant $\frac{1}{N}$ into the quantum state vector, that is,

$$\vert\psi \rangle=\frac{1}{N}\left(ae^{i \alpha}\vert+ \rangle + be^{i \beta}\vert- \rangle\right) \; \; $$

and solve for N such that $$\frac{a^{2}+b^{2}}{N^{2}}=1 \; \; .$$