Editing Bra–ket notation (section)

== Notation used by mathematicians ==

The object physicists are considering when using bra–ket notation is a Hilbert space (a [[Complete metric space|complete]] inner product space).

Let <math>(\mathcal H, \langle\cdot,\cdot\rangle)</math> be a Hilbert space and {{math|''h'' ∈ {{mathcal|H}}}} a vector in {{math|{{mathcal|H}}}}. What physicists would denote by {{math|{{ket|''h''}}}} is the vector itself. That is,
<math display="block"> |h\rangle\in \mathcal{H} .</math>

Let {{math|{{mathcal|H}}*}} be the dual space of {{math|{{mathcal|H}}}}. This is the space of linear functionals on {{math|{{mathcal|H}}}}. The [[embedding]] <math>\Phi:\mathcal H \hookrightarrow \mathcal H^*</math> is defined by <math>\Phi(h) = \varphi_h</math>, where for every {{math|''h'' ∈ {{mathcal|H}}}} the linear functional <math>\varphi_h:\mathcal H\to\mathbb C</math> satisfies for every {{math|''g'' ∈ {{mathcal|H}}}} the functional equation <math>\varphi_h(g) = \langle h, g\rangle = \langle h\mid g\rangle</math>.
Notational confusion arises when identifying {{math|''φ<sub>h</sub>''}} and {{math|''g''}} with {{math|{{bra|''h''}}}} and {{math|{{ket|''g''}}}} respectively. This is because of literal symbolic substitutions. Let <math>\varphi_h = H = \langle h\mid</math> and let {{math|1=''g'' = G = {{ket|''g''}}}}. This gives
<math display="block"> \varphi_h(g) = H(g) = H(G)=\langle h|(G) = \langle h|\bigl(|g\rangle\bigr) \,. </math>

One ignores the parentheses and removes the double bars.

Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they usually use not an [[asterisk]] but an overline (which the physicists reserve for averages and the [[Dirac equation#Conservation of probability current|Dirac spinor adjoint]]) to denote [[complex conjugate]] numbers; i.e., for scalar products mathematicians usually write
<math display="block">\langle\phi ,\psi\rangle=\int \phi (x)\overline{\psi(x)}\, dx \,,</math>
whereas physicists would write for the same quantity
<math display="block"> \langle\psi |\phi \rangle = \int dx \, \psi^*(x) \phi(x)~.</math>