# [QM] Conservation probability to degenerate eigenvalues

1. Oct 31, 2014

### Saverio

Hello =) I have a question regarding the conservation of probability in quantum mechanics.

We know that the probability of a measurement of a given observable, is preserved in time if the observable commutes with the Hamiltonian.
But this is also true if the value of the measurement rapresented by the eigenvalues is degenerate?

For example, if I have the observable $\Omega$ which commutes with the Hamiltonian, and suppose that the system state is a linear combination of eigenstates of H and hence of $\Omega$, and I want to know the probability that a measurement give me a value of $\omega_1$ at any time

$P (\omega_1, t) = | <\omega_1 | Psi, t> | ^ 2 = | <\omega_1 | e ^ {- i / h H} | \Psi, 0> | ^ 2 = | e ^ {- i / h E_1} c_ {1, t = 0} |^ 2 =$

$= e^ {i / h E_1} \tilde{c} _ {1, t = 0} e^ {- i / h E_1} c_ {1, t = 0} = \tilde {c} _ {1, t = 0 c_ {1}, t = 0} = P (\omega_1, t = 0)$

but in the degenerate case that is the same thing?

2. Nov 5, 2014

### Staff: Admin

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 5, 2014

### dextercioby

Is the angular momentum eigenvalue of the H-atom conserved in time when the H-atom's bound state evolves according to SE ?

4. Nov 5, 2014

### HomogenousCow

A general linear combination of eigenstates is itself not an eigenstate.

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