Operators for measuring superposition component distinctnes?

[Agrippa]

Hello,

I'm wondering, is it possible to define an operator that gives information about the "distinctness" of superposition components?

As a simple example, imagine that we have two particles. Let |3> designate the state in which they are 3 meters apart, let |5> designate the state in which they are 5 meters apart, and |7> for seven metres apart. Now imagine the three possible states:

(s1) a|3> + b|3> = |3>
(s2) a|3> + b|5>
(s3) a|3> + b|7>

Where a and b are amplitudes such that |a|²+|b|²=1.
Is it possible to define an operator that gives a null result for s1 (no distinctness) while giving nonzero values for s2 and s3 and ranking them so that s3 gets a higher value (more distinct?)?

If there are real quantum mechanical applications for such operators I would be very interested to learn of them. If there are no known applications even though they are nonetheless in principle definable, then I would still be very interested.

Thanks!

 

[TangledMind]

It's not an expression. It's a physical entity.

Superposition = information based on preparation.
Thus, you have presented us three different preparations, three different physical systems.
Surely there must be a way to tell them apart.

If those are systems where "internal space/distance" of meter scale plays a role, then they must have... charge? (Or mass). This, with distance, translates to some potential energy...

 

[bhobba]

If there are real quantum mechanical applications for such operators I would be very interested to learn of them. If there are no known applications even though they are nonetheless in principle definable, then I would still be very interested.

The variance can be derived from any operator, although I don't think its an operator itself:
http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

If its zero then that state will always give that outcome if observed.

Its used in the general proof of the uncertainty principle.

Thanks
Bill

 

[Agrippa]

The variance can be derived from any operator, although I don't think its an operator itself:
http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics
If its zero then that state will always give that outcome if observed.

QM variance refers to variation in possible measurement outcomes for specific observable given specific state.
So if state is |Ψ> and |Ψ> is an eigenstate of observable O then variance [<Ψ|O²|Ψ> - (<Ψ|O|Ψ>)²] = zero; meaning no variance.
If instead state is noneigenstate of O and O is two-valued (e.g. spin observable) then variance = one; meaning (I think) one variation i.e. two possible outcomes.

That's not quite what I'm after.
I'm not looking for a measure of the variation of possible measurement outcomes given state and observable.
I'm looking for a measure of the variation of superposition components given a state and a basis (which defines those components).

I think what will do the trick is a linear map in the Hilbert space that maps the vectors mentioned above (s1, s2, s3, etc.) to positive real-valued multiples of the basis vectors (of which |3>, |5>, |7> are examples). Those vectors are eigenvectors whose eigenvalues measure the relevant variance. For example, s1 will get mapped to a basis vector with eigenvalue 0, s2 will get mapped to a basis vector with eigenvalue 2 etc. (I think only one basis vector V need be used here, since the ray along which it lies (its 1D subspace) will contain all the desired values. )

This will be a non-hermitian operator: eigenvectors of a Hermitian operator (that don't share the same eigenvalue) are all mutually orthogonal. This does not hold for my operator. But such a non-Hermitian operator is still a well-defined operator (I think).

Its used in the general proof of the uncertainty principle.

The link was helpful but I couldn't find discussion of use of variance to derive uncertainty principle.