Editing Probability amplitude (section)

=== Examples ===
An example of the discrete case is a quantum system that can be in [[two-state quantum system|two possible states]], e.g. the [[light polarization|polarization]] of a [[photon]]. When the polarization is measured, it could be the horizontal state <math>|H\rangle</math> or the vertical state <math>|V\rangle</math>. Until its polarization is measured the photon can be in a [[Quantum superposition|superposition]] of both these states, so its state <math>|\psi\rangle</math> could be written as

:<math>|\psi\rangle = \alpha |H\rangle + \beta|V\rangle</math>,

with <math>\alpha</math> and <math>\beta</math> the probability amplitudes for the states <math>|H\rangle</math> and <math>|V\rangle</math> respectively. When the photon's polarization is measured, the resulting state is either horizontal or vertical. But in a random experiment, the probability of being horizontally polarized is <math>|\alpha|^2</math>, and the probability of being vertically polarized is <math>|\beta|^2</math>.

Hence, a photon in a state <math display="inline">|\psi\rangle = \sqrt{\frac{1}{3}} |H\rangle - i \sqrt{\frac{2}{3}}|V\rangle</math> would have a probability of <math display="inline">\frac{1}{3}</math> to come out horizontally polarized, and a probability of <math display="inline">\frac{2}{3}</math> to come out vertically polarized when an [[statistical ensemble (mathematical physics)|ensemble]] of measurements are made. The order of such results, is, however, completely random.

Another example is quantum spin. If a spin-measuring apparatus is pointing along the z-axis and is therefore able to measure the z-component of the spin (<math display="inline">\sigma_z</math>), the following must be true for the measurement of spin "up" and "down":
:<math>\sigma_z |u\rangle = (+1)|u\rangle </math>
:<math>\sigma_z |d\rangle = (-1)|d\rangle</math>
If one assumes that system is prepared, so that +1 is registered in <math display="inline">\sigma_x</math> and then the apparatus is rotated to measure <math display="inline">\sigma_z</math>, the following holds:
:<math>\begin{align}
\langle r|u \rangle &= \left(\frac{1}{\sqrt{2}}\langle u| + \frac{1}{\sqrt{2}}\langle d|\right) \cdot |u\rangle \\
 &= \left(\frac{1}{\sqrt{2}} \begin{pmatrix}1\\0\end{pmatrix} + \frac{1}{\sqrt{2}} \begin{pmatrix}0\\1\end{pmatrix}\right) \cdot \begin{pmatrix}1\\0\end{pmatrix}  \\
&= \frac{1}{\sqrt{2}}
\end{align}</math>
The probability amplitude of measuring spin up is given by <math display="inline">\langle r|u\rangle</math>, since the system had the initial state <math display="inline"> | r \rangle</math>. The probability of measuring <math display="inline">|u\rangle</math> is given by
:<math>P(|u\rangle) = \langle r|u\rangle\langle u|r\rangle = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}</math>
Which agrees with experiment.