quantum electrodynamics by imran aziz
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Fundamentals of Quantum Electrodynamics
MOHAMMAD IMRAN AZIZAssistant Professor
PHYSICS DEPARTMENTSHIBLI NATIONAL COLLEGE, AZAMGARH (India).
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Coordinates of Fields
A wave field is specified by its amplitudes at allpoints of space and the dependence of theseamplitudes on the time, in much the same way asa system of particles is specified by thepositional coordinates q and their dependence onthe time. The field evidently has an infinitenumber of degrees of freedom, and is analogousto a system that consists of an infinite number ofparticles. It is natural, then, to use theamplitudes ø(r,t) at all points r as coordinates inanalogy 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 fieldLagrangian to be a functional of thefield amplitude ø(r,t). It can usuallybe expressed as the integral over allspace of a Lagrangian density L:
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� ǁŚĞƌĞ�ƚŚĞ�ŝŶĮŶŝƚĞƐŝŵĂů�ǀĂƌŝĂƟŽŶ� 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 DerivativesIn order to pursue further the analogy with particle mechanics, it isdesirable to rewrite Lagrangian equation in terms of L
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which closely resembles the Lagrangian equations .
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Hamiltonian Equations
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and
The Hamiltonian
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We write H as the volume integral of a Hamiltonian density H, and assume that the cells aresmall enough so that the difference between a Volume integral and the corresponding cellsummation can be ignored; we then have
It follows from the earlierdiscussion of functionalderivatives 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 fieldsThe
quantum conditions for the canonical field variables then become
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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. Aswith the nonrelativisticSchrodinger equation, it isconvenient to rewrite theHamiltonian
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We therefore quantize the field by imposing anticommutation relations on thecomponents 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
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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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Quantum equations:The classical electromagnetic field is converted into a quantum 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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