Let us now derive the uncertainty relation for non-commuting operators and . First, given a state , the Mean Square uncertainty in the physical quantity represented is defined as
Now we will dot into itself to get some information about the uncertainties. The dot product must be greater than or equal to zero.
Plug in that .
This result is the uncertainty for non-commuting operators.
For momentum and position, this agrees with the uncertainty principle we know.
(Note that we could have simplified the proof by just stating that we choose to dot into itself and require that its positive. It would not have been clear that this was the strongest condition we could get.)
Jim Branson 2013-04-22