Vector Space and Dirac Notation
A vector space V is a collection of elements called vectors. It is a field F of elements called scalars and two operations, vector addition and scalar multiplication. A field is a set of scalars with the property that if a and b belong to F then a + b, a – b, ab, and a/b are also in F.
Addition: The addition operation takes any two vectors u and v in V and produces a third vector in V, written as u + v in V. The addition operation obeys the following conditions:
1. u + v is a vector in V (closure)
2. u + v = v + u (commutativity)
3. (u + v) + w = u + (v + w) (associativity)
4. There is a zero vector 0 in V such that for every u in V, (u + 0) = u (identity)
5. For every u in V, there is a vector in V denoted by -u such that u +(-u) = 0 (inverse)
Multiplication: The scalar multiplication that takes a scalar c in F, a vector v in V, and produces a new vector written as cv in V that satisfies the following conditions:
1. cv is in V (closure)
2. c(u + v) = cu + cv (distributivity)
3. (c + d)u = cu + du (distributivity)
4. c(du) = (cd)u (associativity)
5. 1(u) = u (identity)
In quantum mechanics, a different notation is called Dirac notation. It is used to represent quantum states.
In this notation, the inner product of two vectors u and v are denoted by
<u | v>. The left part
<u | is called bra and the right part
| v> is called ket. Thus, in the Dirac notation also known as the bra-ket notation, an inner product is denoted by a
< > (bracket).