# 1.8: Using Bras to pick Kets


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.

This page titled 1.8: Using Bras to pick Kets is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.