### Finding Eigenvectors for Coordinate transformation

Posted:

**Tue Jun 21, 2005 11:36 pm**I'm wondering if anyone here might have a solution to a problem I've having. This is a Quantum Mechanics problem I'm doing.

I calculate a 4 by 4 complex Hermitian matrix (H = Hamiltonian) in a basis where it is not diagonal. I diagonalize it numerically (using eispack) and get eigenvalues and eigenvectors, V. With the eigenvector matrix, V, I have verified that the Hamiltonian in the original basis is transformed into a diagonal matrix in the eigenbasis of H. That is,

D = Vadj H V

Now I want to move some other Operators (matrices) from my original basis into the eigenbasis of H. For example, I want to transform matrix X into the eigenbasis of H.

X* = Vadj X V

Here's my problem. There is not a unique V to transform H into D. That is, I could use

Vo = V exp(i B)

where B is a matrix that commutes with D. i.e., BD-DB = 0.

So I still get

Voadj H Vo = exp(-iB) Vadj H V exp(iB) = exp(-iB) D exp(iB) = D

but since XB - BX != 0 (i.e., X and B don't commute), then I get different answers when transforming X into the eigenbasis of H depending on whether my diagonalization routine gives me V or Vo back for the eigenvectors.

Does anyone know what this problem is called? Does anyone know of someplace I can read more, and possibly find a solution?

Thanks,

Philip

I calculate a 4 by 4 complex Hermitian matrix (H = Hamiltonian) in a basis where it is not diagonal. I diagonalize it numerically (using eispack) and get eigenvalues and eigenvectors, V. With the eigenvector matrix, V, I have verified that the Hamiltonian in the original basis is transformed into a diagonal matrix in the eigenbasis of H. That is,

D = Vadj H V

Now I want to move some other Operators (matrices) from my original basis into the eigenbasis of H. For example, I want to transform matrix X into the eigenbasis of H.

X* = Vadj X V

Here's my problem. There is not a unique V to transform H into D. That is, I could use

Vo = V exp(i B)

where B is a matrix that commutes with D. i.e., BD-DB = 0.

So I still get

Voadj H Vo = exp(-iB) Vadj H V exp(iB) = exp(-iB) D exp(iB) = D

but since XB - BX != 0 (i.e., X and B don't commute), then I get different answers when transforming X into the eigenbasis of H depending on whether my diagonalization routine gives me V or Vo back for the eigenvectors.

Does anyone know what this problem is called? Does anyone know of someplace I can read more, and possibly find a solution?

Thanks,

Philip