# State after energy measurement in NRQM

Let $H$ be the Hamiltonian of a system with a set of degenerate energies, so that $H |1⟩ = E_1 |1⟩$, $H |2⟩ = E_1 |2⟩$ and $H |3⟩ = E_2 |3⟩$. Given a linear combination of these three states, I want to know what is the state after you measure the energy of the system and get $E_1$.

For example, if you have a normalized state $|\psi⟩ = \frac{1}{\sqrt{14}}(2|1⟩ + |2⟩ - 3|3⟩)$, and $E_1$ is measured, is the state immediately after $$|\psi⟩ = \frac{1}{\sqrt{2}}(|1⟩ + |2⟩),$$

where the probabilities of $|1⟩$ and $|2⟩$ are the same?

Or does the state preserve the relative probabilities between $|1⟩$ and $|2⟩$ and it's just the original renormalized state without the $|3⟩$ component? i. e. $$|\psi⟩ = \frac{1}{\sqrt{5}}(2|1⟩ + |2⟩)?$$

I am interested in the general case where the probabilities of the degenerate eigenstates are different in the initial state.

The appropriate projection operator for this type of measurement is $\Pi_{E_1}=\vert 1\rangle \langle 1\vert + \vert 2\rangle \langle 2\vert$, i.e. simple unweighted sum over all the projectors corresponding to energy $E_1$. Of course you will need to renormalize your ket after the projection.