Quadrature Operators & Uncertainty Principle

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The discussion centers on the relationship between quadrature operators, specifically the form aX + bP, and the uncertainty principle in quantum mechanics. The initial question raises concerns about whether this operator allows for simultaneous measurement of position and momentum, potentially violating the uncertainty principle. Clarifications reveal that the operator A, when a and b are non-zero, is distinct from the individual position (X) and momentum (P) operators, thus not contradicting the uncertainty principle. The misconception about eigenstates of linear combinations of X and P is addressed, emphasizing that being an eigenstate of such a combination does not imply being an eigenstate of X and P individually. Overall, the discussion highlights the nuanced understanding of quantum operators and their implications for measurement.
McLaren Rulez
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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.
 
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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...
 
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.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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