Formal Definitions
Dual Vector Space
Let V be a finite dimensional vector space over a field K and let dim V = n.
In most cases in physics, the field K is either the set of complex numbers C
or the set of real numbers R. In particular, quantum mechanics is formulated
using a vector space over K = C.
The dual vector space V ∗ of V is defined as the set of all linear maps
from V to K,
V ∗ ≡ { f | f : V → K linear }.
(1)
The vector space structure is provided by the addition and scalar multiplication which are straightforwardly defined using the field structure of K:
Let f, g ∈ V ∗ , v ∈ V , and λ ∈ K, then
(f + g)(v) ≡ f (v) + g(v),
(λf )(v) ≡ λ f (v).
(2)
For finite dimensional V , its dual space has the same dimension, dim V =
dim V ∗. Therefore, the two spaces are isomorphic, i.e. there exists a bijective (one-to-one) map between them. Thus we can uniquely identify an
element of V with an element of V ∗ . This is precisely what the ”dual correspondency” does in quantum mechanics.
Given a basis of V we would like to construct a basis of V ∗ , that is a
basis for all linear maps from V to K. Let {ai }, i = 1, . . . , n, be a basis
of V . Then any element v ∈ V can be written as
v=
n
X
vi ai .
(3)
i=1
In order to unambiguously describe an element in V ∗ , is therefore sufficient
to specify its action on the basis {ai } of V . This follows from (3) and
linearity. Thus we can construct a basis {bj } of V ∗ by specifying its action
on {ai }. The most convenient construction is to take
bj (ai ) = δji .
(4)
From this it also becomes clear what the most straightforward identification
of elements in V with elements in V ∗ looks like. We construct the bijection B
1
by taking
B:V
→ V∗
B(ai )
=
bi .
(5)
The elements of V ∗ are sometimes also referred to as covectors or one-forms.
Inner Product
From now on we take K = C in order to be specific and because this is
the case most relevant for quantum mechanics. The inner product or scalar
product is a map
h | i:V ×V →C
bilinear,
(6)
which for v, w, u ∈ V and λ ∈ C satisfies
hw|vi = hv|wi∗
h(λw + u)|vi = λ∗ hw|vi + hu|vi
hv|vi ≥ 0, and hv|vi = 0 only for v = 0.
(7)
Here ∗ denotes complex conjugation. Due to the isomorphy of V ∗ and V in
the finite dimensional case, one can also interpret this as a map from V ∗ × V
to C, as it is done in quantum mechanics.
Linear Operators and Outer Product
A linear operator O is a map
O:V →V
linear.
(8)
Mathematicians also call such a map an endomorphism. The Hermitian
adjoint operator O† is defined by
hv|Owi = hvO† |wi
(9)
for any v, w ∈ V . Given a vector w ∈ V and a dual vector u ∈ V ∗ we can
construct a linear operator. The outer product P (u, w) is a linear map
→ V,
P (u, w) : V
P (u, w)v
=
w(v)u.
(10)
Note that, by definition, P (u, w) is a linear operator and it simply maps a
vector v on the vector u multiplied by the scalar w(v). Often this product
is referred to as tensor product and denoted as
P (u, w) = u ⊗ w.
2
(11)
Application to Quantum Mechanics
In the following table you find the entities which were formally defined above
and the corresponding objects you have encountered in the quantum mechanics course:
Mathematical object
Correspondent in quantum mechanics
vector space V
ket space spanned by
eigenkets of Hermitian operator
dual vector space V ∗
bra space spanned by
eigenbras of Hermitian operator
basis of vector space {ai }
eigenkets |ai i of Hermitian operator A
dual basis {bi }
eigenbras hai | of Hermitian operator A
inner product h | i
scalar product ha|bi of bra and ket
linear operator O on V
linear operator O |bi acting on kets
Hermitian adjoint O†
linear operator ha| O† acting on bras
outer product P (u, w)
linear operator |bi ha| acting on kets
3