15. Formalism of Quantum Mechanics: Vector Space

In quantum mechanics, the state of a system is described by an element of abstract vector space, the so-called state space. In Dirac notation, an element of this space is called a ket and is denoted by the symbol $\left| {} \right\rangle$

A vector space consists of a set of vectors | α >, | β >, | γ > ... together with a set of scalars (a, b, c …). In fact, we will be working with vectors that live in spaces of infinite dimension and the scalars will be ordinary complex numbers. Two important operations are: vector addition and scalar multiplication.

Vector addition:

The ‘sum’ of any two vectors is another vector: | α > + | β > = | γ >

Commutative: | α > + | β > = | β > + | α >         

Associative: | α > + (| β > + | γ >) = (| α > + | β >) + | γ >                    

There is a zero vector | 0 > with the property: | α > + | 0 > = | α >     

The associated inverse vector | -α > has the property: | α > + | -α >  = | 0 >

Scalar multiplication:
The ‘product’ of a scalar with a vector is another vector: a | α > = | γ >

Distributive:     a (| α > + | β >) = a | α > + a | β >

                           (a + b) | α > = a | α > + b | β >

Associative: a (b | α >) = (ab) | α >
Multiplication with scalars 0 and 1 has the properties: 0 | α > = 0; 1 | α > = | α > and |-α> = (-1)|α> 

Two important definitions in linear algebra are: linear combinations and linear dependence, these are closely related to systems of linear equations. A vector | v >  is a linear combination of vectors | α >, | β >, | γ > ... if there exist scalars (k₁, k₂, k₃ …) such that:

| v > = k₁ | α > + k₂ | β > + k₃ | γ > + ... + k_n | ω >                                         (15.1)

that is, if the vector equation:

| v > = x₁ | α > + x₂ | β > + x₃ | γ > + ...+ x_n | ω >                                        (15.2)

Has a solution where x_i are the unknown scalars.

Vectors | α >, | β >, | γ > ... are linearly dependent  if there exist scalars (k₁, k₂, k₃ …), not all zero, such that:

k₁ | α > + k₂ | β > + k₃ | γ > + ... + k_n | ω > = 0                                      (15.3)

that is, if the vector equation:

x₁ | α > + x₂ | β > + x₃ | γ > + ... + x_n | ω > = 0                                   (15.4)

has a nonzero solution where x_i are unknown scalars. Otherwise, the vectors are said to be linearly independent. For instance, in three dimensions the unit vector $\hat{k}$ is linearly independent of $\hat{i}$ and $\hat{j}$, but any vector in the xy-plane is linearly dependent on $\hat{i}$ and $\hat{j}$. A set of vectors is linearly independent if each one is linearly independent of all the rest. A collection of vectors span the space if every vector can be written as a linear combination of the members of this set. A set of linearly independent vectors that spans the space is called a basis. The number of vectors in any basis is the dimension of the space. Any given vector with n-tuple of its components | α > ↔ (a₁, a₂, a₃ ... a_n), can be represented with respect to a prescribed basis | e₁ >, | e₂ >, | e₃ > ... | e_n >:

| α > = a₁ | e₁ > + a₂ | e₂ > + a₃ | e₃ > + ... + a_n | e_n >                                 (15.5)

It is often more convenient to work with the components than with the ‘abstract’ vectors. For instance, addition of two vectors can be done by adding the corresponding components:

| α > + | β > ↔ (a₁ + b₁, a₂ + b₂, a₃ + b₃ ...  a_n + b_n)                                 (15.6)

Multiplication by a scalar can be simply done by multiplying each component:

c | α > ↔ (ca₁, ca₂, ca₃ ... ca_n)                                               (15.7)

Component of zero vector is represented by a string of zeroes:

| 0 > ↔ (0, 0, 0 ... 0)                                                     (15.8)

And the components of the inverse vector have their signs reversed:

| -α > ↔ (-a₁, -a₂, -a₃ ... -a_n)                                                (15.9)

There are two kinds of vector products in three dimensional space: the inner/dot product and cross product.

Vector space that is formed by inner product is called an inner product space. The dot product of two vectors | α > and | β > which is written as: < α | β >,  is a complex number with the following properties:

< β | α > = < α | β >^*                                                   (15.10)

< α | α >  ≥  0, and < α | α > = 0 ↔ | α > = | 0 >                                 (15.11)

< α | (b | β > + c | γ >) = b < α | β > + c < α | γ >                                (15.12)

The inner product of any vector is a non-negative number therefore, its square root is real. We call this the norm or the “length” of the vector:

|| α || ≡ (< α | α >)^1/2                                                   (15.13)

A unit vector with norm is 1, is said to be normalized and two vectors whose inner product is equal to zero are called orthogonal. An orthonormal set is a collection of mutually orthogonal normalized vectors, which can be defined as follows,

< α_i | α_j > ≡ δ_ij                                                    (15.14)

It is always possible and convenient to choose and orthonormal basis, so that the dot product can be written in terms of their components:

< α | β > = $a_{1}^{*}{{b}_{1}}+a_{2}^{*}{{b}_{2}}+...+a_{n}^{*}{{b}_{n}}$                          (15.15)

And the squared norm of the vector becomes:

< α | β > = | a₁ |² + | a₂ |² + ... + | a_n |²                                   (15.16)

With the components themselves expressed in term of basis < e_i | :

a_i = < e_i | α >                                                      (15.17)

The question then arises as to what is the angle between two vectors? In ordinary vector analysis the angle between two vectors is given by:

$\cos \theta =\frac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}$                                                    (15.18)

And by means of Schwarz inequality:

|< α | β >|²  ≤  < α | α > < β | β >                                                 (15.19)

The angle between | α > and | β > can be generalized by the formula:

$\cos \theta =\sqrt{\frac{<\alpha |\beta ><\beta |\alpha >}{<\alpha |\alpha ><\beta |\beta >}}$                                                 (15.20)