1.8: Using Bras to pick Kets

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One of the most useful algebraic tricks in quantum mechanics is to multiply a sum of terms by a complex conjugate wavefunction, and integrate the product over all space. Orthogonality often means that this procedure can be used to ‘pick’ a single term from the sum. In Dirac notation this procedure simply becomes applying \(\langle i_m|\).

For example, if we have an expansion of a mixed state \(\Phi\) in eigenstates in: \(\hat{H} |\Phi \rangle = \sum _n \hat{H} | i_n \rangle \langle i_n | \Phi \rangle\), we can remove the sum by \(\langle i_m|\):

\[\langle i_m | \hat{H} | \Phi \rangle = \langle i_m | \sum_n \hat{H} | i_n \rangle \langle i_n | \Phi \rangle = E_m \langle i_m | \Phi \rangle \nonumber\]

This works because \(\langle i_m | \hat{H} | i_n = E_m \delta_{nm}\); it is analogous to taking components of a vector.