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Homework Help: Can unitary operators on hilbert space behaive like rotations?

  1. Sep 25, 2012 #1
    1. The problem statement, all variables and given/known data
    unitary operators on hilbert space

    2. Relevant equations
    is there a unitary operator on a (finite or infinite) Hilbert space so that cU(x)=y, for some
    constant (real or complex), where x and y are fixed non-zero elements in H ?

    3. The attempt at a solution
    I know the answer in R^2, it is enough to consider U a suitable rotation so that U(x)
    be a point on the straight line Ry={ry; r ε R}, and then there is a scaler r in R so that
    rU(x)=y. I guess this is true for R^n too.
  2. jcsd
  3. Sep 25, 2012 #2
    You would need to define "rotation" in a Hilbert space. In 2D Euclidean space it is a one-parameter linear transformation. Scaling is also a one-parameter linear transformation. Together that is two parameters, which is just enough to describe any element in 2D. This, of course, would not work in a higher-dimensional space, not even Euclidean 3D.
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