- #1

happyparticle

- 400

- 20

- Homework Statement
- U unitary operator that commute with H.

##| \psi_n \rangle## an eigenstate of H with eigenvalue ##E_n##

##| \phi_n \rangle = U | \psi_n \rangle##

Thus,

##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i \rangle##

- Relevant Equations
- ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i \rangle##

Hi,

I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it.

From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle##

Also, a linear combination of all states is also an solution.

If U commutes with H then they have the same eigenstates (and same eigenvalues ?)

Thus, ##U|\psi_n \rangle = E_n | \psi_n \rangle##

I have hard time to put all those things together or seeing what that really means.

Thank you

I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it.

From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle##

Also, a linear combination of all states is also an solution.

If U commutes with H then they have the same eigenstates (and same eigenvalues ?)

Thus, ##U|\psi_n \rangle = E_n | \psi_n \rangle##

I have hard time to put all those things together or seeing what that really means.

Thank you