Dirac notation enables us to do calculations
in a general Hilbert space
without needing to worry about the space’s representation.
It is the lingua franca of quantum mechanics.
In Dirac notation there are
kets from the Hilbert space
and bras from its dual space .
Crucially, the bras and kets are from different Hilbert spaces
and therefore cannot be added,
but every bra has a corresponding ket and vice versa.
Bras and kets can be combined in two ways: the inner product
, which returns a scalar, and the outer product
, which returns a linear operator
that maps kets to other kets .
Recall that by definition the Hilbert inner product must satisfy:
So far, nothing has been said about the actual representation of bras or kets.
If we represent kets as -dimensional columns vectors,
the corresponding bras are given by the kets’ adjoints,
i.e. their transpose conjugates:
The inner product is then just the familiar dot product :
Meanwhile, the outer product creates an matrix,
which can be thought of as applying an operation to any vector it multiplies:
If the kets are instead represented by continuous functions of ,
then the bras are functionals
that take an arbitrary function as an argument and return a scalar:
Consequently, the inner product is simply the following familiar integral:
However, the outer product is then rather abstract:
a continuous analogue of a matrix:
This maybe makes more sense if we surround it
by a bra and a ket and rearrange:
- R. Shankar,
Principles of quantum mechanics, 2nd edition,