Dirac Notation of Vector Spaces
Overview
The Dirac notation expresses a vector as
(see inner product below for motivation)
{% \vec{v} = |v> %}
When there is a
basis set of vectors,
{% v_1,v_2,...,v_n %} ,
it is common to write {% v_i %} as {% |i> %}.
Then, any vector can be expressed as a sum over the basis vectors.
{% |v> \; = \sum_i \; |i> %}
Inner Products
The motivation for the Dirac notation comes from the
inner product.
The inner product between two vectors {% v_1 %} and {% v_2 %} is
{% <v_1 | v_2 > %}
In the Dirac notation, a vector {% v_2 %} becomes {% |v_2 > %} (often called a ket), and there exists a function
associated with {% v_1 %},
labeled {% < v_1 | %} (often called a bra) which when applied to the ket, returns the value of the inner product
between {% v_1 %} and {% v_2 %} (known as the bracket).
{% < v_1 | ( |v_2 > ) = <v_1 | v_2 > %}
