Probability amplitudes

Read Pgs. 91-108 in Text.

$$\vert \psi ({\bf r}) \vert^{2} dx dy dz$$ (1.1)

is the probability to find a particle in volume dx dy dz. We require that $\int d^{3} {\bf r} \vert\psi({\bf r})\vert^2$ =1. If $\psi_{1}({\bf r})$, and $\psi_{2}({\bf r})$ are possible physical amplitudes, then so is

$$\psi({\bf r}) =c_{1} \psi_{1}({\bf r})+ c_{2}\psi_{2}({\bf r})$$ (1.2)

where c_i are complex numbers. We define a scalar, or inner, product

$$(\phi,\psi) \equiv \int d^{3}{ \bf r} \, \phi^{*}({\bf r}) \psi({\bf r}).$$ (1.3)

Note

$$(\phi,\psi)=(\psi,\phi)^{*}$$ (1.4)

also

$$(\phi, c_{1} \psi_{1}+ c_{2} \psi_{2})= c_{1} (\phi,\psi_{1})+ c_{2}(\phi,\psi_{2})$$ (1.5)

$$c_{1} (\phi_{1},\psi)+ c_{2}(\phi_{2}, \psi)= c^{*}_{1} (\phi_{1},\psi)+ c^{*}_{2}(\phi_{2},\psi)$$ (1.6)

The former is a linear relation, whereas the latter is anti-linear.

If a scalar product vanishes, the pair of functions are said to be orthogonal. For physical states we require,

$$(\psi,\psi) >0$$ (1.7)

the quantity $\sqrt{(\psi,\psi)}$ is called the norm of $\psi$. The wave functions also obey the Schwarz inequality

$$\vert(\psi_{1},\psi_{2})\vert <= \sqrt{(\psi_{1},\psi_{1})} \sqrt{(\psi_{2},\psi_{2})}$$ (1.8)