1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Can unitary operators on hilbert space behaive like rotations?
  1. Hilbert Space (Replies: 4)