Fundamentals of Quantum
                           Electrodynamics

                                     MOHAMMAD IMRAN AZ...
Coordinates of Fields
                             A wave field is specified by its amplitudes at all
                    ...
PROPERTIES OF THE FIELDS:
                     1. In analogy with any physical system, the field is carrier
              ...
Lagrangian Equations
      • The Lagrangian L is a function of the time and
        a func onal of the possible paths q(t)...
By analogy, we expect the field
          Lagrangian to be a functional of the
          field amplitude ø(r,t). It can us...
• ǁŚĞƌĞƚŚĞŝŶĮŶŝƚĞƐŝŵĂůǀĂƌŝĂƟŽŶ of is
        subject to the restrictions




                                             ...
This is classical field equation
                      derived from the Lagrangian
                      density L( ø, gra...
Functional Derivatives
              In order to pursue further the analogy with particle mechanics, it is
              d...
which closely resembles the
                                                                                              ...
Hamiltonian Equations




            and


     The Hamiltonian

                                                pdfMachi...
We write H as the volume integral of a Hamiltonian density H, and assume that the cells are
    small enough so that the d...
The classical field equations in Hamiltonian form




   This equation also serves to define the Poisson-bracket expressio...
Quantum equations for fields
                                           The
                          quantum conditions f...
QUANTIZATION OF THE DIRAC EQUATION
      • Our procedure consists in treating the one-
        particle equation as though...
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                              ...
Then the Hamiltonian density is




        Quantum Equations. As
        with the nonrelativistic
        Schrodinger equ...
We therefore quantize the field by imposing anticommutation relations on the
            components of ø




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




                                              ...
QUANTIZATION OF ELECTROMAGNETIC FIELDS
                       Lagrangian Equations:
                      Maxwell's equati...
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                              ...
Hamiltonian Equations:
                       The momentum canonically conjugate to
                       A, is found



...
The Hamiltonian is then




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                         A pdf writ...
The classical electromagnetic field is converted into a quantum
                                  Quantum equations:
     ...
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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                     ...
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                              ...
Quantum equations:




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Quantum electrodynamics by imran aziz

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Quantum electrodynamics by imran aziz

  1. 1. Fundamentals of Quantum Electrodynamics MOHAMMAD IMRAN AZIZ Assistant Professor PHYSICS DEPARTMENT SHIBLI NATIONAL COLLEGE, AZAMGARH (India). pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  2. 2. 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 . pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  3. 3. 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. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  4. 4. 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 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  5. 5. 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: pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  6. 6. • ǁŚĞƌĞƚŚĞŝŶĮŶŝƚĞƐŝŵĂůǀĂƌŝĂƟŽŶ of is subject to the restrictions pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  7. 7. This is classical field equation derived from the Lagrangian density L( ø, grad ø, ,t). pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  8. 8. Functional Derivatives In order to pursue further the analogy with particle mechanics, it is desirable to rewrite Lagrangian equation in terms of L pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  9. 9. which closely resembles the Lagrangian equations . pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  10. 10. Hamiltonian Equations and The Hamiltonian pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  11. 11. 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 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  12. 12. 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 pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  13. 13. Quantum equations for fields The quantum conditions for the canonical field variables then become The quantum conditions for the canonical field variables then become pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  14. 14. 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. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  15. 15. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  16. 16. Then the Hamiltonian density is Quantum Equations. As with the nonrelativistic Schrodinger equation, it is convenient to rewrite the Hamiltonian pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  17. 17. We therefore quantize the field by imposing anticommutation relations on the components of ø pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  18. 18. Thus the four equa ons are equivalent to the Dirac equa on and follows: pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  19. 19. QUANTIZATION OF ELECTROMAGNETIC FIELDS Lagrangian Equations: Maxwell's equations in empty space are obtained by setting and equal to zero pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  20. 20. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  21. 21. Hamiltonian Equations: The momentum canonically conjugate to A, is found The Hamiltonian equations of motion are pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  22. 22. The Hamiltonian is then pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  23. 23. The classical electromagnetic field is converted into a quantum Quantum equations: field in the following way pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  24. 24. Commutation Relations for E and H: The electric and magnetic fields are defined by the equations pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  25. 25. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  26. 26. Quantization of Schrodinger equations pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  27. 27. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  28. 28. Quantum equations: pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!
  29. 29. pdfMachine A pdf writer that produces quality PDF files with ease! aziz_muhd33@yahoo.co.in Produce quality PDF files in seconds and preserve the integrity of your original documents. Compatible across nearly all Windows platforms, simply open the document you want to convert, click “print”, select the “Broadgun pdfMachine printer” and that’s it! Get yours now!

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