Can unitary operators on hilbert space behaive like rotations?

In summary, there is a unitary operator on a Hilbert space that can satisfy the equation cU(x)=y, where c is a constant and x and y are fixed non-zero elements in H. This can be achieved by considering a suitable rotation and scaling in R^2, but this approach would not work in higher-dimensional spaces.
  • #1
cyeus
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.
 
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  • #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.
 
1.

What is a unitary operator on a Hilbert space?

A unitary operator on a Hilbert space is a linear transformation that preserves the inner product of vectors in the space. This means that the operator preserves the length and angle between vectors, similar to rotations in Euclidean space.

2.

Can unitary operators on a Hilbert space behave like rotations?

Yes, unitary operators behave like rotations in the sense that they preserve the inner product of vectors. However, this analogy is not perfect as unitary operators can also have other properties, such as reflections and translations, depending on the specific operator.

3.

What are some examples of unitary operators on a Hilbert space?

Some examples of unitary operators on a Hilbert space include the identity operator, orthogonal matrices, and the Fourier transform operator. These operators all have the property of preserving the inner product of vectors.

4.

How are unitary operators related to quantum mechanics?

In quantum mechanics, unitary operators play a crucial role in describing the evolution of quantum systems. They represent the transformations that occur to a quantum state as it evolves in time, and their preservation of the inner product ensures the conservation of probability in quantum systems.

5.

Are unitary operators on a Hilbert space reversible?

Yes, unitary operators are reversible as they have a unique inverse that also preserves the inner product of vectors. This means that a unitary operator can always be undone by applying its inverse, similar to how a rotation can be undone by rotating in the opposite direction.

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