• Support PF! Buy your school textbooks, materials and every day products Here!

Can unitary operators on hilbert space behaive like rotations?

  • Thread starter cyeus
  • Start date
  • #1
1
0

Homework Statement


unitary operators on hilbert space


Homework 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 ?


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.
 

Answers and Replies

  • #2
6,054
390
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.
 

Related Threads for: Can unitary operators on hilbert space behaive like rotations?

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
10
Views
4K
  • Last Post
Replies
1
Views
891
Replies
2
Views
1K
  • Last Post
Replies
4
Views
11K
Replies
5
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
0
Views
1K
Top