Quadrature Operators & Uncertainty Principle

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In summary, the conversation discusses the uncertainty principle and how it applies to operators of the form aX+bP. The person asking the question initially believed that this operator violated the uncertainty principle, but then realizes their misconception. They also discuss quadrature operators and their consistency with the uncertainty principle. Finally, the other person clarifies that being an eigenstate of a linear combination of X and P does not necessarily mean being an eigenstate of X and P separately.
  • #1
McLaren Rulez
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Hi,

This may seem like a silly question but if we have an operator of the form [itex]aX+bP[/itex] 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 [itex]a=cos\theta[/itex] and [itex]b=sin\theta[/itex]. So how is this consistent with the uncertainty principle?

Thank you.
 
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  • #2
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...
 
  • #3
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.
 

FAQ: Quadrature Operators & Uncertainty Principle

What are quadrature operators?

Quadrature operators are mathematical operators used in quantum mechanics to describe the position and momentum of a particle. They are represented by the position and momentum operators, which are Hermitian conjugates of each other.

How do quadrature operators relate to the uncertainty principle?

The uncertainty principle states that it is impossible to know both the exact position and momentum of a particle simultaneously. Quadrature operators play a crucial role in this principle, as their commutation relation leads to the Heisenberg uncertainty principle.

What is the significance of the commutation relation of quadrature operators?

The commutation relation of quadrature operators is fundamental to understanding the uncertainty principle and the behavior of quantum systems. It describes the non-commutativity of position and momentum, which leads to the uncertainty in their simultaneous measurement.

How are quadrature operators used in experiments?

Quadrature operators are used in experiments to make precise measurements of position and momentum, as well as other physical quantities. They are also used in the creation and manipulation of quantum states, such as in quantum computing and quantum information processing.

Are there other types of quadrature operators besides position and momentum?

Yes, there are other types of quadrature operators that describe different physical quantities in quantum systems. These include angular momentum, energy, and spin operators. They all have similar commutation relations and play important roles in understanding the behavior of quantum systems.

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