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Operators for measuring superposition component distinctnes?

  1. Apr 5, 2015 #1

    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.

  2. jcsd
  3. Apr 8, 2015 #2
    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...
    Last edited: Apr 8, 2015
  4. Apr 8, 2015 #3


    Staff: Mentor

    The variance can be derived from any operator, although I don't think its an operator itself:

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

    Its used in the general proof of the uncertainty principle.

  5. Apr 11, 2015 #4
    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).

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