Quadrature Operators & Uncertainty Principle

[McLaren Rulez]

Hi,

This may seem like a silly question but if we have an operator of the form $aX+bP$ where a and b are some numbers and X and P are the position and momentum operators, doesn't this violate the uncertainty principle. Isn't it sort of measuring position and momentum simulataneously?

I recently came across quadrature operators where $a=cos\theta$ and $b=sin\theta$. So how is this consistent with the uncertainty principle?

Thank you.

 

[dextercioby]

I don't understand, what has the new operator have to do with the uncertainty principle ? If b≠0 for a≠0 then the operator, call it A, is different than both X and P which enter the uncertainty principle...

 

[McLaren Rulez]

Sorry, I think I may have had a bit of a misconception there.

I thought that being an eigenstate of a linear combination of X and P meant that the state was an eigenstate of X and P separately as well. Now its obvious that this was wrong. Sorry about that. Thank you for replying dextercioby.