1.13: Bra and Ket Notation

Also known as Dirac notation, Bra and Ket notation is a convenient shorthand for the integrals above.

The wavefunction is represented by a Ket:

\[ \psi(x) \rightarrow |\psi\rangle \nonumber \]

The complex conjugate is represented by a Bra:

\[ \psi^{*}(x) \rightarrow \langle \psi| \nonumber \]

Together, the bracket \( \langle \psi| \psi \rangle\) (hence Bra and Ket) symbolizes an integration over all space:

\[ \int^{+\infty}_{-\infty}\psi^{*}(x)\psi(x) \rightarrow \langle \psi| \psi \rangle \nonumber \]

Thus, in short form the expectation value of x is

\[ \langle x\rangle = \frac{\langle \psi|x|\psi \rangle}{\langle \psi|\psi\rangle} \nonumber \]

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