Undergrad Probabilities for degenerate eigenvalues?

[LarryS]

In non-relativistic QM, given a wave function that has a degenerate eigenvalue for some observable, say energy. There is a whole subspace of eigenvectors associated with that single degenerate eigenvalue. How is the measurement probability for that degenerate eigenvalue computed from the eigenvectors in the subspace?

Thanks in advance.

 

[hilbert2]

It's the sum of the squared absolute values of the amplitudes of the components belonging to that subspace. After the measurement, if you got that degenerate eigenvalue as a result, the state has collapsed to its projection on that eigen-subspace.

In more precise terms, if the system is in state $\left|\right.\psi\left.\right>$ and the degenerate eigenstates with eigenvalue $\lambda$ are denoted by $\left|\right.\phi_i \left.\right>$, the probability of getting result $\lambda$ in a measurement is

$P(\lambda ) = \sum\limits_i |\left<\phi_i \left|\right. \psi\right>|^2$,

and the state after getting that result is

$\sum\limits_i \left<\phi_i \left|\right. \psi\right>\left|\right.\phi_i \left.\right>$.

 

[LarryS]

It's the sum of the squared absolute values of the amplitudes of the components belonging to that subspace. After the measurement, if you got that degenerate eigenvalue as a result, the state has collapsed to its projection on that eigen-subspace.

In more precise terms, if the system is in state $\left|\right.\psi\left.\right>$ and the degenerate eigenstates with eigenvalue $\lambda$ are denoted by $\left|\right.\phi_i \left.\right>$, the probability of getting result $\lambda$ in a measurement is

$P(\lambda ) = \sum\limits_i |\left<\phi_i \left|\right. \psi\right>|^2$,

and the state after getting that result is

$\sum\limits_i \left<\phi_i \left|\right. \psi\right>\left|\right.\phi_i \left.\right>$.

Ok, thanks.

I find it interesting that the individual amplitudes for the eigenstates in the subspace are squared before they are summed. Normally, in QM, if you are given all the different ways something can happen (an eigenvalue can be measured), you add the individual amplitudes first and then square the sum to get the final probability.

 

[hilbert2]

Normally, in QM, if you are given all the different ways something can happen (an eigenvalue can be measured), you add the individual amplitudes first and then square the sum to get the final probability.

If you do that in this case, you can have zero probability for some result despite some of the corresponding amplitudes being nonzero, and then the sum of all probabilities would not be 1.