Editing Operator (physics) (section)

===Linear operators in wave mechanics===

{{Main|Wave function|Bra–ket notation}}

Let {{math|''ψ''}} be the wavefunction for a quantum system, and <math>\hat{A}</math> be any [[linear operator]] for some observable {{math|''A''}} (such as position, momentum, energy, angular momentum etc.). If {{math|''ψ''}} is an eigenfunction of the operator <math>\hat{A}</math>, then

:<math>\hat{A} \psi = a \psi ,</math>

where {{math|''a''}} is the [[Eigenvalues and eigenvectors|eigenvalue]] of the operator, corresponding to the measured value of the observable, i.e. observable {{math|''A''}} has a measured value {{math|''a''}}.

If {{math|''ψ''}}  is an eigenfunction of a given operator <math>\hat{A}</math>, then a definite quantity (the eigenvalue {{math|''a''}}) will be observed if a measurement of the observable {{math|''A''}} is made on the state {{math|''ψ''}}. Conversely, if {{math|''ψ''}} is not an eigenfunction of <math>\hat{A}</math>, then it has no eigenvalue for <math>\hat{A}</math>, and the observable does not have a single definite value in that case. Instead, measurements of the observable {{math|''A''}} will yield each eigenvalue with a certain probability (related to the decomposition of {{math|''ψ''}} relative to the orthonormal eigenbasis of <math>\hat{A}</math>).

In bra–ket notation the above can be written;

:<math>\begin{align}
  \hat{A} \psi &= \hat{A} \psi ( \mathbf{r} ) = \hat{A} \left\langle \mathbf{r} \mid \psi \right\rangle = \left\langle \mathbf{r} \left\vert \hat {A} \right\vert \psi \right\rangle \\
        a \psi &= a \psi ( \mathbf{r} ) = a \left\langle \mathbf{r} \mid \psi \right\rangle = \left\langle \mathbf{r} \mid a \mid \psi \right\rangle \\
\end{align} </math>

that are equal if <math> \left| \psi \right\rangle </math> is an [[eigenvector]], or [[eigenket]] of the observable {{math|''A''}}.

Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the [[del operator]], which is itself a vector (useful in momentum-related quantum operators, in the table below).

An operator in ''n''-dimensional space can be written:

:<math> \mathbf{\hat{A}} = \sum_{j=1}^n \mathbf{e}_j \hat{A}_j </math>

where '''e'''<sub>''j''</sub> are basis vectors corresponding to each component operator ''A<sub>j</sub>''. Each component will yield a corresponding eigenvalue <math>a_j</math>. Acting this on the wave function {{math|''ψ''}}:

:<math> \mathbf{\hat{A}} \psi = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \psi = \sum_{j=1}^n \left( \mathbf{e}_j \hat{A}_j \psi \right) = \sum_{j=1}^n \left( \mathbf{e}_j a_j \psi \right) </math>

in which we have used <math> \hat{A}_j \psi = a_j \psi .</math>

In bra–ket notation:

:<math>\begin{align}
  \mathbf{\hat{A}} \psi = \mathbf{\hat{A}} \psi ( \mathbf{r} ) = \mathbf{\hat{A}} \left\langle \mathbf{r} \mid \psi \right\rangle
    &= \left\langle \mathbf{r} \left\vert \mathbf{\hat{A}} \right\vert \psi \right\rangle \\
  \left ( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right ) \psi = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \psi ( \mathbf{r} ) = \left( \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right) \left\langle \mathbf{r} \mid \psi \right\rangle
    &= \left\langle \mathbf{r} \left\vert \sum_{j=1}^n \mathbf{e}_j \hat{A}_j \right\vert \psi \right\rangle
\end{align}</math>