# Two-state system problem

Given a 2-state system with (complete set) orthonormal eigenstates $u_1, u_2$ with eigenvalues $E_1, E_2$ respectively, where $E_2>E_1$, and there exists a linear operator $\hat{L}$ with eigenvalues $\pm1$,

1. would the normalized eigenfunctions (in terms of the given eigenstates) just be $$u_1\over \sqrt{\int u_1^*u_1}$$ ? Since I am not given an argument/coordinate system for the eigenstates, perhaps I should first project them into one? But I don't have any info on the nature of the quantum system...

2. A second question asks for the expectation values of the energy in the respective states. But isn't that just $E_1$ and $E_2$ respectively??

Grateful for any enlightenment.

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And what does that linear operator have to do with anything? – justcurious Sep 2 '11 at 8:46

1) Yes. However, you say that $u$ are orthonormal which contains normalization
And what $\hat{L}$ have to do with the question?
Thanks again, Misha. :) So $\langle E \rangle = \langle H \rangle$? Then we have to do $\int u_1^*Hu_1$ where $H$ is the Hamiltonian? Unfortunately, the question only has the information I have given in the question above... – justcurious Sep 2 '11 at 12:45