Simple Bra-Ket Manipulations
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Here is an operator corresponding to some observable (like for energy), and its eigenvalues are λn. The corresponding eigenvectors are . is a generic state ket. an is the projection of onto . One of the keys to what follows is that the eigenvectors form an orthonormal basis. That just means that any vector can be built from a unique combination of , just like any vector in 3-D space can be built from a unique combination of x, y, and z, or any complex number can be built from 1 and i.
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Basics
Notation Definitions
Note that the ket symbols don't "do" anything except surround a normal column vector. However, the bra symbols do perform an operation: they represent taking the adjoint (conjugate transpose) of a column vector.