Meaning of notation <a|b>

Question

I came across this notation in a question. Where a and b are column vectors. What does it mean? To me, it looks like an inner product but it is separated by a pipeline instead of a comma.

Answer 1

This looks like the bra-ket notation from Quantum Mechanics.

The connection between $\langle a |$ and $|a\rangle$ is that one is the conjugate transpose of the other.

For example, if $\displaystyle{ |a\rangle = \left(\begin{array}{c} 2-\mathrm i \\ 4+3\mathrm i\end{array}\right)}$ then $\langle a| = (2+\mathrm i,4-3\mathrm i)$.

That way, the Hermitian product of $\langle a|$ and $\langle b |$ is given by $\langle a||b\rangle$ which is written as $\langle a|b\rangle$.

The dagger notation for the conjugate transpose is common: $\langle a|^{\dagger} = |a\rangle$.

Answer 2

That's the bra-ket notation used in quantum mechanics.

Answer 3

It is an inner product, in the "bra-ket" notation common in quantum mechanics. $|b\rangle$ denotes a vector ("ket"), and $\langle a|$ is a functional ("bra").