Orthonormal Basis Vectors
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Given two sets of complete orthonormal basis vectors: {|u1>, |u2>, |u3>,...} and {|v1>, |v2>, |v3>,...}. Use Dirac notation to prove:
1. That for any linear operator Â, the trace is independent of choice of basis i.e.
Tr(Â) = Σ<ui| Â|ui> = Σ<vi| Â|vi>.
2. Given that Tr(Â) = Σ<ui| Â|ui>. Prove that Tr(ÂĈ) = Tr(ĈÂ) for any two linear operators  and Ĉ.
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Solution Summary
Orthonormal basis vectors are proving using Dirac notation.
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1. The trace of an operator A is the sum of the diagonal elements in its matrix representation, relative to some (complete orthonormal) basis. That is, if
Tr (A) = Σ;_{i=1}^n < u_i | A | u_i > = < u_1 | A | u_1 > + .... + < u_n | A | u_n >
where {|u_i>} is a (complete orthonormal) basis for the (Hilbert) space on which T is defined.
To show that this definition is ...
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