1.8: Using Bras to pick Kets
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 ⟨im|.
For example, if we have an expansion of a mixed state Φ in eigenstates in: ˆH|Φ⟩=∑nˆH|in⟩⟨in|Φ⟩, we can remove the sum by ⟨im|:
⟨im|ˆH|Φ⟩=⟨im|∑nˆH|in⟩⟨in|Φ⟩=Em⟨im|Φ⟩
This works because ⟨im|ˆH|in=Emδnm; it is analogous to taking components of a vector.