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Normalized kets

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The state space of a certain physical system is 3-D.
Let { |u1 , | u2 , | u3 } be an orthonormal basis of this space. The kets | phi0 ? and | phi1 ? are defined by:

| phio = (1/sqrt2 ) | u1 + (i/2) | u2 + (1/2) | u3

| phi1 = (1/sqrt3 ) | u1 + (i/sqrt3) | u3

1.Are these kets normalized?
2.Calculate the matrices rhoo and rho1 representing, in the { |u1 ?, | u2 ?, | u3 ?} basis, the projection operators onto the state | phio ? and onto the state | phi1 ?. Verify that these matrices are Hermitian.

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Solution Summary

This solution provides calculations with kets and matrices.

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ψ_0 is normalized; ψ_1 is not

Since ψ_0 is normalized, the projection of any vector (x, y, z) (in the u_i-basis) is

(x, y, z)*(1/sqrt 2, i/2, 1/2) (1/sqrt 2, i/2, 1/2) = {x/sqrt 2, i.y/2 + z/2} (1/sqrt 2, i/2, 1/2) where * is the dot product, and ...

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