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.

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

1) Yes. However, you say that $u$ are orthonormal which contains normalization

2) Yes. I presume, it some sort of a homework. Sometimes it is useful to check/show fact like this in details.

And what $\hat{L}$ have to do with the question?

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  • $\begingroup$ Thanks, Misha. Like I said in my comment to the problem, I really don't see the relevance of the linear operator... HenceI was confused... BTW, what sort of detail could I show? Isn't this all from definition? $\endgroup$ – justcurious Sep 2 '11 at 9:21
  • $\begingroup$ Found the catch to the linear operator bit, there is apparently another bit of the question later on that has to do with it. But I still don't know how to "calulate" the expected energy for each of the states, since it seems to follow from definition...! $\endgroup$ – justcurious Sep 2 '11 at 10:06
  • $\begingroup$ @justcurious It is not definition. State with energy usually means eigenfunction of Hamiltonian. Expectation value of the operator definition is a bit different. To go from one to another you still need one pretty obvious line. \\ You may reformulate the question right here. I can add new details in my answer. It is Ok. $\endgroup$ – Misha Sep 2 '11 at 10:40
  • $\begingroup$ 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... $\endgroup$ – justcurious Sep 2 '11 at 12:45
  • $\begingroup$ Also, what argument would I be integrating with rescet to (since it is not given in the question)? $\endgroup$ – justcurious Sep 2 '11 at 12:50

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