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AKG

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[tex]H(\psi _k) = E_k \psi _k[/tex]

Is it true that [itex]\frac{\partial }{\partial t}\psi _k = 0[/itex]? Is it also true that:

[tex]H\left (\exp \left (\frac{-iE_k (t - t_0)}{\hbar }\right )\psi _k\right ) = \exp \left (\frac{-iE_k (t-t_0)}{\hbar }H(\psi _k)[/tex]

Also, the dynamics of the system are described by the wave function that satisfies:

[tex]H(\psi (t)) = i\hbar \frac{\partial }{\partial t}\psi (t)[/tex]

and it's not as though that's a definition for H, right?

Next, is a Hamiltonian Hermitian if and only if it's eigenfunctions span the Hilbert space? Is the previou sentence at least partially true, if not entirely?

Now, some problems:

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Assume that H, A and B are Hermitian operators on a finite-dimensional Hilbert space.

1. Show that if [H,A] = 0, then A and H have a complete set of eigenfunctions in common, i.e. there exists a basis [itex]\{\psi _{\alpha i}\}[/itex] for the Hilbert space such that:

[tex]H\psi _{\alpha i} = E_{\alpha }\psi _{\alpha i},\ A\psi _{\alpha i} = a_i\psi _{\alpha i}[/tex]

Since the operators are Hermitian, I think it suffices to show that for some choice of eigenfunctions, every eigenfuntion of H is an eigenfunction of A. I don't really know how to do this. I've already figured that if f is an eigenfunction for A corresponding to value a, then:

HAf = Haf = aHf

Since HA = AH, it's also true that:

AHf = aHf, so Hf is an eigenfunction of A corresponding to the same eigenvalue as f. Can we pick the f so that Hf = Ef?

2. Show that if A and B are both symmetries of H and [A,B] = 0, then it is possible to construct a basis for the Hilbert space with a triple set of labels [itex]\psi _{\alpha i j}[/itex] such that:

[tex]H\psi _{\alpha i j} = E_{\alpha }\psi _{\alpha i j},\ A\psi _{\alpha i j} = a_i\psi _{\alpha i j},\ B\psi _{\alpha i j} = b_j\psi _{\alpha i j}[/tex]

I'm guessing that the eigenfunctions of H span the Hilbert space (as I asked before, does this follow from the fact that it's Hermitian?), so [H,A] = [H,B] = 0. So from the previous problem, we know that each pair of operators has a basis in common. Although I don't know how to show that they can all share the same basis. I think if I knew how to do question 1, I might know how to do this one.

3. Show that if A and B are both symmetries of H but [A,B] != 0, then it is possible to construct a basis for the Hilbert space with two labelling indices [itex]\{\psi _{\alpha i}[/itex] such that:

[tex]H\psi _{\alpha i} = E_{\alpha }\psi _{\alpha i},\ A\psi _{\alpha i} = a_i\psi _{\alpha i},\ B\psi _{\alpha i} = \sum _j \psi _{\alpha i}M_{ji}(B)[/tex]

where M(B) is a complex matrix.

... No idea for this one.