I'm teaching myself QM using Zettili's book. I'v been chugging along seeing how linear algebra works in infinite-dimensional Hilbert space. Now the author has introduced the "completeness relation," that for an infinite set of orthonormal kets |Xn> forming a basis, Sigma(n=1 to infinity) of |Xn><Xn|= I (the unit operator).(adsbygoogle = window.adsbygoogle || []).push({});

Now this notation for the unit operator keeps coming up in proofs, e.g. in showing how to change bases, etc. I can see that in a matrix representation with |Xn> a column vector of coordinates and |Xn> a row vector of conjugates of coordinates, |Xn><Xn| gives a matrix, i.e. is a linear operator.

I don't have any sort of physical intuition though, as to what this sigma expression means. I know the bra-ket is a scalar product and can be thought of as the projection of the ket onto the bra. So the sigma-ket-bra is an infinite sum of linear operators (matricies). Is there anything more that can be said about it? Could somebody make this sigma-ket-bra expression a little less opaque? Thanks.

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# What exactly is the Sigma-ket-bra?

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