If i have an arbitrary ket then i know it can always be expressed as a linear combination of the basis kets.I now have an operator A which has 2 eigenvalues +1 and -1.(adsbygoogle = window.adsbygoogle || []).push({});

The corresponding eigenvectors are | v >_{+}= k | b > + m | a > and | v >_{-}= n | c > where | a > , | b > and | c > are linear combinations of the basis vectors.

The arbitrary ket is expressed as | ψ > = a | a > + b | b > + c | c > where | a |^{2}gives the probability of a measurement giving the eigenvalue corresponding to | a >. A question asks what is the probability of measuring the eigenvalue +1 . It gives the answer as | b |^{2}+ | a |^{2}.

Finally to my question ; how or why does the expansion theorem apply to this situation as the eigenvector | v >_{+}only exists as a combination of | a > and | b >

Hoping you can understand my question. Thanks

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# I Eigenvalues, eigenvectors and the expansion theorem

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