Editing Operator (physics) (section)

===Commutation of operators on ''Ψ''===

{{main|Commutator}}

If two observables ''A'' and ''B'' have linear operators <math> \hat{A} </math> and <math> \hat{B} </math>, the commutator is defined by,

:<math> \left[ \hat{A}, \hat{B} \right] = \hat{A} \hat{B} - \hat{B} \hat{A} </math>

The commutator is itself a (composite) operator. Acting the commutator on ''ψ'' gives:

:<math> \left[ \hat{A}, \hat{B} \right] \psi = \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi . </math>

If ''ψ'' is an eigenfunction with eigenvalues ''a'' and ''b'' for observables ''A'' and ''B'' respectively, and if the operators commute:

:<math> \left[ \hat{A}, \hat{B} \right] \psi = 0, </math>

then the observables ''A'' and ''B'' can be measured simultaneously with infinite precision, i.e., uncertainties <math> \Delta A = 0 </math>, <math> \Delta B = 0 </math> simultaneously. ''ψ'' is then said to be the simultaneous eigenfunction of A and B. To illustrate this:

:<math> \begin{align}
  \left[ \hat{A}, \hat{B} \right] \psi &= \hat{A} \hat{B} \psi - \hat{B} \hat{A} \psi \\
    & = a(b \psi) - b(a \psi) \\
    & = 0 . \\
\end{align} </math>

It shows that measurement of A and B does not cause any shift of state, i.e., initial and final states are same (no disturbance due to measurement). Suppose we measure A to get value a. We then measure B to get the value b. We measure A again. We still get the same value a. Clearly the state (''ψ'') of the system is not destroyed and so we are able to measure A and B simultaneously with infinite precision.

If the operators do not commute:

:<math> \left[ \hat{A}, \hat{B} \right] \psi \neq 0, </math>

they cannot be prepared simultaneously to arbitrary precision, and there is an [[uncertainty relation]] between the observables

:<math>\Delta A \Delta B \geq \left|\frac{1}{2}\langle[A, B]\rangle\right|</math> 
even if ''ψ''  is an eigenfunction the above relation holds. Notable pairs are position-and-momentum and energy-and-time uncertainty relations, and the angular momenta (spin, orbital and total) about any two orthogonal axes (such as ''L<sub>x</sub>'' and ''L<sub>y</sub>'', or ''s<sub>y</sub>'' and ''s<sub>z</sub>'', etc.).<ref name=Ballentine1970>{{citation
 | last =Ballentine | first =L. E.
 | title =The Statistical Interpretation of Quantum Mechanics
 | journal =Reviews of Modern Physics 
 | volume =42
 | issue =4
 | pages =358–381
 | year =1970
 | doi =10.1103/RevModPhys.42.358
 | bibcode = 1970RvMP...42..358B
}}</ref>