Editing Operator (physics) (section)

===Examples of applying quantum operators===

The procedure for extracting information from a wave function is as follows. Consider the momentum ''p'' of a particle as an example. The momentum operator in position basis in one dimension is:

:<math>\hat{p} = -i\hbar\frac{\partial }{\partial x}</math>

Letting this act on ''ψ'' we obtain:

:<math>\hat{p} \psi = -i\hbar\frac{\partial }{\partial x} \psi ,</math>

if ''ψ'' is an eigenfunction of <math>\hat{p}</math>, then the momentum eigenvalue ''p'' is the value of the particle's momentum, found by:

:<math> -i\hbar\frac{\partial }{\partial x} \psi = p \psi.</math>

For three dimensions the momentum operator uses the [[nabla symbol|nabla]] operator to become:

:<math>\mathbf{\hat{p}} = -i\hbar\nabla .</math>

In Cartesian coordinates (using the standard Cartesian basis vectors '''e'''<sub>x</sub>, '''e'''<sub>y</sub>, '''e'''<sub>z</sub>) this can be written;

:<math>\mathbf{e}_\mathrm{x}\hat{p}_x + \mathbf{e}_\mathrm{y}\hat{p}_y + \mathbf{e}_\mathrm{z}\hat{p}_z = -i\hbar\left ( \mathbf{e}_\mathrm{x} \frac{\partial }{\partial x} + \mathbf{e}_\mathrm{y} \frac{\partial }{\partial y} + \mathbf{e}_\mathrm{z} \frac{\partial }{\partial z} \right ),</math>

that is:

:<math> \hat{p}_x = -i\hbar \frac{\partial}{\partial x}, \quad \hat{p}_y = -i\hbar \frac{\partial}{\partial y} , \quad \hat{p}_z = -i\hbar \frac{\partial}{\partial z} \,\!</math>

The process of finding eigenvalues is the same. Since this is a vector and operator equation, if ''ψ'' is an eigenfunction, then each component of the momentum operator will have an eigenvalue corresponding to that component of momentum. Acting <math> \mathbf{\hat{p}} </math> on ''ψ'' obtains:

:<math> \begin{align}
\hat{p}_x \psi & = -i\hbar \frac{\partial}{\partial x} \psi = p_x \psi \\
\hat{p}_y \psi & = -i\hbar \frac{\partial}{\partial y} \psi = p_y \psi \\
\hat{p}_z \psi & = -i\hbar \frac{\partial}{\partial z} \psi = p_z \psi \\
\end{align} \,\!</math>