## Uncertainty Principle for Non-Commuting Operators

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

where we define (just to keep our expressions small)

Since and are just constants, notice that

OK, so much for the definitions.

Now we will dot into itself to get some information about the uncertainties. The dot product must be greater than or equal to zero.

This expression contains the uncertainties, so lets identify them.

Choose a to minimize the expression, to get the strongest inequality.

Plug in that .

This result is the uncertainty for non-commuting operators.

If the commutator is a constant, as in the case of , the expectation values can be removed.

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