Matrix representations of wave functions and operators, Commuting observables,

Commutator algebra – Physical significance in Quantum Mechanics, Unitary

transformations

Unit_II_Lect_3

• A linear vector space consists of two sets of elements !

and two algebraic rules: (i) Addition (ii) Multiplication (i) Vector addition : • It has following property:

Linear Algebra

Commutative

Associative

Zero/Null vector

Inverse vector

(ii) Multiplication:

Linear Algebra

• It has following property:

DistributiveScalar addition

Associative

• A linear combination of the is given as:

• A vector is said to be linearly independent of the vectors of the set if it cannot be written as a linear combination.

• By extension, a set of vectors is linearly independent if each one is linearly independent of all the rest.

• A collection of vectors said to span the space if every vector can be written as a linear combination of the members of the set.

• A set of linearly independent vectors that spans the space is called basis.

• The number of vectors in any basis is called the dimension of space.

For n finite dimension space with the basis

The vector may be written as;

Its components may be written as;

Ket ‘a’=

Ket ‘a’=

where e1, e2 and e3 are the orthonormal basis.

A Hilbert space (Hil) consists of a set of and a set of , which satisfy the following four properties:

Hilbert Space

•Hil is a linear space. •Hil has a defined inner product that is strictly positive..

•Hil is separable. •Hil is complete.

Inner Product: The inner product of two vectors is a complex number which is given as; having following properties:

The physical state of a system is represented in quantum mechanics by elements of a Hilbert space; these elements are called state vectors. The state vectors in different bases by means of function expansions. This is analogous to specifying an ordinary (Euclidean) vector by its components in various coordinate systems. The meaning of a vector is, of course, independent of the coordinate system chosen to represent its components. Similarly, the state of a microscopic system has a meaning independent of the basis in which it is expanded.

Dirac Notation

Kets: elements of a vector space

Dirac Notation

Bras: elements of a dual space

Bra-ket: Dirac notation for the inner/scalar product

Properties of kets, bras, and bra-ketsDirac Notation

• Every ket has a corresponding bra

• Properties of the inner/scalar product

• The norm is real and positive

• Schwarz inequality

• Triangle inequality

• Orthogonal states

• Orthonormal states

• Forbidden quantities

• Forbidden quantities

Operators with Bra and Ket vectors

Now an arbitrary vector in term of Ket vector:

Then

Examples Operators!

Unity operator: !

The gradient operator: !

The linear momentum operator: !

The Laplacian operator: !

The parity operator:

Products of operators

Products of operators

•Expectation of operator

•Products of operators

• Products of the type (i.e., when an operator stands on the right of a ket or on the left of a bra) are forbidden.

• Hermitian Adjoint

•Properties of Adjoint

• Hermitian operator (Observables are represented by Hermitian operators)

•Skew-Hermitian operator

•Commutator Algebra

•Physical significance in QM

Unitary Transformations

Properties of unitary transformations

Properties of unitary transformations

Conclusion:

For Hilbert space

For a set of function {fn}, a function said to be normalised if its inner product with itself is 1; and two functions are orthogonal if their inner product is 0. The {fn} is orthonormal if:

Finally, a set of function is complete if any other function in Hilbert space can be expressed as a linear combination of them.

The measurement of Q on such a state is certain to yield the eigen value q. The collection of all eigen values of an operator is called spectrum. Sometimes two or more linearly independent eigenfunctions share the same eigen value; in that case the spectrum is said to be degenerate.

This is similar to determinate state of total energy are eigenfunctions of the Hamiltonian.

Determinate states are eigen functions of operator Q.

Eigenfunctions of a hermitian operator.

It is physically determinate states of observables. This is classify into two types: (i) Discrete and (ii) Continuous

The discrete spectrum have constituted physically separate realisable states and normalised eigenfunctions of a hermitian operator have two properties (i) The eigen values are real. (ii) Eigen functions belonging to distinct eigenvalues are

orthogonal.

Other hand the continuous spectrum are not normalizable and proof of the two properties for discrete are failed.

Matrix Representation of Kets, Bras, and Operators

Consider a discrete, complete, and orthonormal basis which is made of an kets set

The orthonormality condition of the base kets is expressed by

The completeness, or closure, relation for this basis is given by

The unit operator acts on any ket, it leaves the ket unchanged.

Matrix Representation of Kets and Bras

Consider a vector within the context of the basis set

where =

where

Matrix Representation of Operators

Operator A can be represented in the form of matrix.

where !This can be represented by