Quantum Mechanics - PPT 1

Matrix representations of wave functions
and operators, Commuting observables,

Commutator algebra – Physical significance
in Quantum Mechanics, Unitary
transformations
Unit_II_Lect_3
Linear Algebra
• 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:
Commutative
Associative
Zero/Null vector
Inverse vector
Linear Algebra
(ii) Multiplication:
• It has following property:

Distributive
Scalar 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.

Hilbert Space
A Hilbert space (Hil) consists of a set of
and a set of , which satisfy the
following four properties:
•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:
Dirac Notation
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
Bras: elements of a dual space
Bra-ket: Dirac notation for the inner/scalar product

Dirac Notation
Properties of kets, bras, and bra-kets
• 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.
Determinate states are eigen functions of operator Q.
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.
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

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Matrix representations of wave functions

and operators, Commuting observables,

and two algebraic rules: (i) Addition (ii) Multiplication

• It has following property:

• A linear combination of the is given

For n finite dimension space with the basis

The vector may be written as;

Its components may be written as;

where e1, e2 and e3 are the orthonormal basis.

•Hil is a linear space.

Inner Product: The inner product of two vectors

Bras: elements of a dual space

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

• Properties of the inner/scalar product

Operators with Bra and Ket vectors

• Products of the type (i.e., when an operator

For Hilbert space

For a set of function {fn}, a function said to be normalised if

Finally, a set of function is complete if any other function in

The measurement of Q on such a state is certain to yield the

This is similar to determinate state of total energy are

It is physically determinate states of observables. This is

The discrete spectrum have constituted physically

Other hand the continuous spectrum are not normalizable

Consider a discrete, complete, and orthonormal basis

The orthonormality condition of the base kets is expressed

The completeness, or closure, relation for this basis is

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

Consider a vector within the context of the basis set

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