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?