Quantum electrodynamics by imran aziz

Fundamentals of Quantum
Electrodynamics

MOHAMMAD IMRAN AZIZ
Assistant Professor
PHYSICS DEPARTMENT
SHIBLI NATIONAL COLLEGE, AZAMGARH (India).

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Coordinates of Fields
A wave field is specified by its amplitudes at all
points of space and the dependence of these
amplitudes on the time, in much the same way as
a system of particles is specified by the
positional coordinates q and their dependence on
the time. The field evidently has an infinite
number of degrees of freedom, and is analogous
to a system that consists of an infinite number of
particles. It is natural, then, to use the
amplitudes ø(r,t) at all points r as coordinates in
analogy with the particle coordinates q .

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PROPERTIES OF THE FIELDS:
1. In analogy with any physical system, the field is carrier
of energy, momentum, angular momentum and other
observable dynamical quantities.
2. It is to be ideal regarding the definition of the field that
if the values of ø are ordinary or complex numbers then
it is called a classical field, and if the values of ø are
operators in some Hilbert space, then the fields in
question is called a quantum field.
3. The degrees of freedom of the field is represented by
amplitudes ø(r,t) which are infinite in number
corresponding to finite number of points in space.

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Lagrangian Equations
• The Lagrangian L is a function of the time and
a func onal of the possible paths q(t) of the
system. The actual paths are derived from the
variational principle

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By analogy, we expect the field
Lagrangian to be a functional of the
field amplitude ø(r,t). It can usually
be expressed as the integral over all
space of a Lagrangian density L:

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• ǁŚĞƌĞƚŚĞŝŶĮŶŝƚĞƐŝŵĂůǀĂƌŝĂƟŽŶ of is
subject to the restrictions

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This is classical field equation
derived from the Lagrangian
density L( ø, grad ø, ,t).
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Functional Derivatives
In order to pursue further the analogy with particle mechanics, it is
desirable to rewrite Lagrangian equation in terms of L

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which closely resembles the
Lagrangian equations .

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Hamiltonian Equations

and

The Hamiltonian

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We write H as the volume integral of a Hamiltonian density H, and assume that the cells are
small enough so that the difference between a Volume integral and the corresponding cell
summation can be ignored; we then have

It follows from the earlier
discussion of functional
derivatives that

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The classical field equations in Hamiltonian form

This equation also serves to define the Poisson-bracket expression
for two fuctionals of the field variables. In quantum mechanics
Poission-bracket is
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Quantum equations for fields
The
quantum conditions for the canonical
field variables then become

The quantum conditions for the
canonical field variables then
become

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QUANTIZATION OF THE DIRAC EQUATION
• Our procedure consists in treating the one-
particle equation as though it were a classical
field equation. The resulting quantized field
theory represents the motion of a number of
non interacting free electrons.

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Then the Hamiltonian density is

Quantum Equations. As
with the nonrelativistic
Schrodinger equation, it is
convenient to rewrite the
Hamiltonian

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We therefore quantize the field by imposing anticommutation relations on the
components of ø

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Thus the four equa ons are equivalent to the Dirac equa on and follows:

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QUANTIZATION OF ELECTROMAGNETIC FIELDS
Lagrangian Equations:
Maxwell's equations in empty space are
obtained by setting and equal to zero

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Hamiltonian Equations:
The momentum canonically conjugate to
A, is found

The Hamiltonian equations of motion are

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The Hamiltonian is then

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The classical electromagnetic field is converted into a quantum
Quantum equations:
field in the following way

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Commutation Relations for E and H:
The electric and magnetic fields are defined by the equations

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Quantization of Schrodinger equations

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Quantum equations:

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Quantum electrodynamics by imran aziz

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