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1) We are looking at a Hamiltonian H. I make a rotation in Hilbert space by the transformation

[tex]

{\cal H} = \mathbf a^\dagger\mathsf H \mathbf a =

\mathbf a^\dagger \mathsf U\mathsf U^\dagger\mathsf H \mathsf U\mathsf U^\dagger\mathbf a = \mathbf b^\dagger \mathsf D \mathbf b

[/tex]

where D is diagonal.

Now, is there a difference between a rotation of this kind and the transformation I perform when I go to momentum-representation?

2) When I want to write my second quantization operators in momentum-space, I write them as

[tex]

a(\mathbf k) = \sum_\nu <\mathbf k | \psi_\nu>a_\nu.

[/tex]

In my book they write this as

[tex]

a(\mathbf r) = \sum_{\mathbf k}{e^{i\mathbf k\cdot r}} a(\mathbf k).

[/tex]

How do I show this rigorously?

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# Representations and change of basis

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