Quantum mechanics/Mathematical background
- 1 Vector Spaces
- 2 Inner Product Spaces
- 3 Dirac notation
- 4 Subspaces
- 5 Linear Operator
- 6 Matrix Elements of Linear Operators
- 7 Active and Passive Transformations
- 8 Eigenvalue
- 9 Functions of Operators
- 10 Generalization to Infinite Dimensions
- 11 Reference and Further reading
A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below.
- The first operation, called vector addition or simply addition + : V × V → V, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. (Note that the resultant vector is also an element of the set V ).
- The second operation, called scalar multiplication · : F × V → V， takes any scalar a and any vector v and gives another vector av. (Similarly, the vector av is an element of the set V ).
Elements of V are commonly called vectors. Elements of F are commonly called scalars.
In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively.
To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms. In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.
|Associativity of addition||u + (v + w) = (u + v) + w|
|Commutativity of addition||u + v = v + u|
|Identity element of addition||There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.|
|Inverse elements of addition||For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0.|
|Compatibility of scalar multiplication with field multiplication||a(bv) = (ab)v This axiom and the next refer to two different operations: scalar multiplication: bv; and field multiplication: ab. They do not assert the associativity of either operation. More formally, scalar multiplication is a group action of the multiplicative group of the field F on the vector space V.|
|Identity element of scalar multiplication||1v = v, where 1 denotes the multiplicative identity in F.|
|Distributivity of scalar multiplication with respect to vector addition||a(u + v) = au + av|
|Distributivity of scalar multiplication with respect to field addition||(a + b)v = av + bv|
These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands:
- (xv, yv) + (xw, yw) = (xw, yw) + (xv, yv).
Likewise, in the geometric example of vectors as arrows, v + w = w + v since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space.
Subtraction of two vectors and division by a (non-zero) scalar can be defined as
- v − w = v + (−w),
- v/a = (1/a)v.
When the scalar field F is the real numbers R, the vector space is called a real vector space. When the scalar field is the complex numbers C, the vector space is called a complex vector space. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed field F. The notion is then known as an F-vector spaces or a vector space over F. A field is, essentially, a set of numbers possessing addition, subtraction, multiplication and division operations. For example, rational numbers form a field.
In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general vector spaces, no notion of nearness, angles or distances.
Inner Product Spaces
Definition and equivalent expression of dot product
Definitions of inner product space, orthogonal, normalized vector, orthonormal basis.
Matrix Elements of Linear Operators
Active and Passive Transformations
Functions of Operators
Generalization to Infinite Dimensions
Reference and Further reading
- Ramamurti Shankar, Principle of Quantum Mechanics, 2nd ed. (1994), Chap. 1. Mathematical Introduction
- David J. Griffith, Introduction to Quantum Mechanics, 2nd ed. (2005), Chap. 3. Formalism
- Quantum Mechanics Made Simple:Lecture Notes (2012), Chap. 5. Mathematical Preliminaries