[Article] A Simpeified Method for the Statistical by Linhart G.A.

By Linhart G.A.

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The pairs must be then the basis state vectors of the combined system. 92) where we have simply placed e1 A and e2 B next to each other on the sheet of paper. But we have already seen something similar when we defined an operator in terms of the tensor product ⊗. To avoid possible confusion and to emphasize that we do not really multiply these vectors by each other but merely write them next to each other, let us use the same symbol here: {01}A {10}B → e1 A ⊗ e2 B . 93) The basis of the combined register system can now be described as listed below: Mixing combined register systems e 0 A ⊗ e0 B , e0 A ⊗ e1 B , e 0 A ⊗ e2 B , e0 A ⊗ e3 B , e1A ⊗ e0B , e1A ⊗ e1B , e 1 A ⊗ e2 B , e1 A ⊗ e3 B , e 2 A ⊗ e0 B , e2 A ⊗ e1 B , e 2 A ⊗ e2 B , e2 A ⊗ e3 B , e 3 A ⊗ e0 B , e3 A ⊗ e1 B , e 3 A ⊗ e2 B , e3 A ⊗ e3 B .

77) j On the other hand, we get a different relation for forms: ηi = η, ei = η, Λi j ej Λi = j j j η, ej = Λi j ηj . 78) j We see that form and vector coefficients transform in opposite directions. Vector coefficients (index is up) transform like basis forms (their index is up, too), and form coefficients (index down) transform like basis vectors (their index is down, too). This is good news because it means that expressions such as i ηi v i don’t transform at all. Transformations of ηi and v i cancel each other, so that the resulting scalar η, v is independent of the choice of vector and form bases.

We shall not use it in this text, though, because we shall seldom work with complicated tensor expressions. The placement of indexes on form and vector coefficients is not just a matter Vector and form of esthetics, convenience, and debugging. It reflects transformation properties of transformations these objects, too. Let us suppose that instead of decomposing vector v in basis ei , v = i v i ei , we were to decompose it in another basis, say, ei . The basis vectors ei are not the same as ei , the prime on the index i matters, but they are all linearly independent as basis vectors should be.

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