Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.2 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

The theory of quantum mechanics is based on the linear operators and the wave functions belonging to abstract Hilbert space. Schrodinger Eq.: Before going into the discussion of Hilbert space let us review some properties of 3-dimenional Euclidian space. A vector V can be written in Cartesian so-ordinate system as, Vx, Vy and Vz are components of vector V. ex, ey and ez are the unit vectors along three orthogonal directions x, y and z and span the vector space.

The inner product (dot product) of two vector is defined as The length of vector is given by

Hilbert space and wave functions: Linear vector space: A linear vector space consist of two set of elements (i) Set of vectors (ii) set of scalar And two algebraic rules (i) Rule for vector addition (ii) rule for scalar multiplication Known as field

The addition rule has the properties and structure of abelian group: if and are vectors of a space then the sum is also a vector of same space. Commutativity: Associativity: Existence of zero vector: Existence of symmetric or inverse vector:

(II) Multiplication rule: The multiplication of scalars by vectors has following properties (scalars can be real or complex) The product of a scalar with a vector gives another vector If and are two vectors of space then any linear combination is also a vector of the space. Here a and b are two scalars.

Distributive w.r.t addition Associativity For unitary scalar I and zero scalar O,

Examples: The set of n-components of a vector form a vector Space The set of M by N matrices with complex entries form a complex vector space

n by n Hermitian matrices with complex entries form a real vector space. They cannot form a complex vector space because the multiplication of hermitian matrix by a complex number will destroy the hermicity and will yield anti-hermitian matrix and therefore will not be closed under complex scalar multiplication. An example of hermitian matrix is Pauli matrices given by,

Hilbert Space: The Hilbert space H consist of set of Vectors and scalars which satisfy following properties: Hilbert space is a linear space satisfying the properties discussed on last slides (b) Hilbert space H has a well defined scalar product and is strictly positive. The scalar product of two elements is in general complex and thus order of elements matter.

The scalar product of element with other element is equal to the complex conjugate of with Proof: The scalar product of with is linear w.r.t 2nd factor . If ψ = Then we have

The scalar product of with is anti-linear w.r.t 1st factor . If Then, Proof:

The scalar product of vector with itself is a positive real number where equality hold of

The Hilbert space is complete. To understand the meaning of this statement we first need to know the meaning of sequence, convergence of sequence and Cauchy sequence. A sequence say an is some ordered list and it converges if it has some limit L.

A Cauchy sequence is a sequence whose terms become arbitrarily close together as n gets very large. An infinite sequence of vectors in a normed linear space V is called a Cauchy sequence if

In the scalar product the 2nd factor belongs to the Hilbert space whereas the 1st factor belongs to the dual Hilbert space. This is because the scalar product is not commutative and order matters.

Dimension and Basis of vector space: A set of N nonzero vectors is said to be linearly independent if and only if the solution of Eq is It means any vector of the space cannot be expressed as linear combination of other vectors of same space.

If there exist a set scalar which are not all zero such that one of the vector say can be written as linear combination of others i.e.

Dimensions: It is the maximum number of the linear independent vectors of the space. For example if we have N linearly independent vectors then we say it is N dimensional space and any other vector can be expanded as linear combination. Basis: The set of maximum number of possible linearly independent vectors of the space form the basis of that vector space.

The set denoted by is called the basis and is known as base vector. We choose above set of linear independent vectors as orthonormal. It means their scalar product satisfy the condition The basis is called complete if it span the entire space. It means we need not to introduce any other base vector. Any vector of the space can be written as linear combination of base vectors.

The coefficients are known as the components of the vector . The components can be obtained by the scalar product of the vector by base vector Now go back to first slide of present lecture and compare the above discussed properties with the vectors in Euclidean space.

Examples: The orthogonal unit vectors we use in the 3-Dim Euclidian space are linearly independent. Two parallel vectors in a plane are linearly dependent as one can be expressed as multiple of other. The 2 by 2 matrices form a four-dim space. To understand we consider following four linearly independent matrices

Now any two dimension matrix can be written as linear combination of above 4 matrices. If a, b, c and d are real then we have real 4-Dim space and if these are complex then we have complex 4-Dim space. ****From QM by R Shankar:

. Theorem 2: Expansion of a vector in a given basis is unique. See QM by R Shankar.

Square integrable function: Wave functions The scalar product of is given by If above integral diverge scalar product does not exist. A function is called the square integrable if is finite. Scalar product is also known as inner product.

According to Born’s probabilistic interpretation, the total probability of finding the particle somewhere in the space is 1 i.e. Thus wave functions of quantum mechanics are square integrable functions. The space of square integrable functions satisfy the properties of Hilbert space. A wave function which is not square integrable have no physical significance in quantum mechanics. Mathematician calls the Hilbert space as L2.