Bra-ket Notation - Notation Used By Mathematicians

Notation Used By Mathematicians

The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product space).

Let be a Hilbert space and is a vector in . What physicists would denote as is the vector itself. That is

.

Let be the dual space of . This is the space of linear functionals on . The isomorphism is defined by where for all we have

,

where and are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises when identifying and with and respectively. This is because of literal symbolic substitutions. Let and let . This gives

One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we are dealing with linear operators and composition acts like a ring multiplication.

Moreover, mathematicians usually write the dual entity not at the first place, as the physicists do, but at the second one, and they don't use the *-symbol, but an overline (which the physicists reserve to averages) to denote conjugate-complex numbers, i.e. for scalar products mathematicians usually write

whereas physicists would write for the same quantity