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Mike2
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So the equations of QM give eigenfunctions and eigenvalues. The eigenfunctions form a complete set with which any state is a combination of such. When measuring, the superposition of states collapse to one of the eigenfunctions. And the probability that some state with be measured in a particular eigenfunction is formed like an inner product of two states, etc.
All this to ask the question: the eigenfunction is a function that maps one manifold to another. And each eigenfunction is a different manifold from the others. There is an inner product between these manifolds to form the probability of going from one to the other. I wonder if the inner product tells us that all the separate eigenfunctions rest within a larger manifold. This would be a manifold of manifolds. Is this a valid way of looking at things? If so, then is there some more general equation that specifies this manifold of eigenfunctions, perhaps some symmetry principle?
Thanks.
All this to ask the question: the eigenfunction is a function that maps one manifold to another. And each eigenfunction is a different manifold from the others. There is an inner product between these manifolds to form the probability of going from one to the other. I wonder if the inner product tells us that all the separate eigenfunctions rest within a larger manifold. This would be a manifold of manifolds. Is this a valid way of looking at things? If so, then is there some more general equation that specifies this manifold of eigenfunctions, perhaps some symmetry principle?
Thanks.