Bra-Ket Notation

Bra-ket notation was used by Dirac. Grassman had used something similar a hundred years earlier for Inner Products.

  • This is an inner product: \langle\begin{matrix}u,v\rangle\end{matrix}
  • This is a bra-ket pair: \langle\begin{matrix}u|v\rangle\end{matrix}

\langle\begin{matrix}u|v\rangle\end{matrix} = \langle\begin{matrix}v|u\rangle\end{matrix}*

\forall v, \langle\begin{matrix}v|v\rangle\end{matrix} \geq 0

\langle\begin{matrix}v|v\rangle\end{matrix}=0 \leftrightarrow v=0

We have linearity in the second argument:

\langle\begin{matrix}u|c_1v_1 + c_2v_2\rangle\end{matrix} = c_1\langle\begin{matrix}u|v_1\rangle\end{matrix} + c_2\langle\begin{matrix}u|v_2\rangle\end{matrix}

(notice that this is the identity that includes the two tests for Linearity)

We have antilinearity in the first argument:

\langle\begin{matrix}c_1u_1 + c_2u_2|v\rangle\end{matrix} = c_1^*\langle\begin{matrix}u_1|v\rangle\end{matrix} + c_2^*\langle\begin{matrix}u_2|v\rangle\end{matrix}

\langle\begin{matrix}u|v\rangle\end{matrix}=0 \leftrightarrow vectors u and v are orthogonal

The norm: |v|^2=\langle\begin{matrix}v|v\rangle\end{matrix}

  • — —

We make a comment that, although his intentions were good, the decision to create a new notation for the subject made things harder for a good student. Said student is going to want to go back and confirm that what is done using the new notation matches up to the math and chemistry that we had before the new stuff came along. The only student to benefit is the one who doesn’t check things out and plays a game to memorize and understand just enough to get by when taking the test–a student who lacks sincerity.