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The second and the third energy level act like an "almost" degeneracy and therefore it doesn't work. One of the conditions for perturbation theory to work is that the matrix elements of the perturbation can't be bigger than the spacing between the energy levels, which is the case here. A solution would be to introduce a real degeneracy for this "almost" ...


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The same vector x⃗ can be written in terms of the basis vectors ei of the dual space V* as: xiei, why is this true? It's not true. The elements of the dual space are not vectors as we ordinarily conceive of them geometrically, e.g., directed line segments. Rather, they are (geometrically) a set of oriented surfaces with a density proportional to ...


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If $V$ is a (finite dimensional) vector space with no additional structure, then $V^*$ is a different vector space of the same dimension, hence isomorphic to $V$ --- but it would be a great mistake to think of $V$ and $V^*$ as "the same'' or even ``almost the same'', because there is no preferred isomorphism between them. In other words, you can pick a ...



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