{"id":148,"date":"2018-11-14T06:25:05","date_gmt":"2018-11-14T06:25:05","guid":{"rendered":"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/?post_type=chapter&p=148"},"modified":"2019-04-30T09:18:42","modified_gmt":"2019-04-30T09:18:42","slug":"perturbation-theory","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/chapter\/perturbation-theory\/","title":{"rendered":"Perturbation Theory"},"content":{"raw":"
\"epgp<\/a>\r\n<\/span><\/div>\r\n
<\/div>\r\n
<\/div>\r\n
\u00a0 \u00a0 1. Learning outcomes<\/strong><\/div>\r\n
    \r\n \t
  1. Learn what is electromagnetic interaction and its strength.<\/li>\r\n \t
  2. Learn why perturbation theory can be applied for most of the processes \u2013\u00a0involving charged particles and radiation field.<\/li>\r\n \t
  3. Learn how the quantum theory of radiation is applied to\u00a0 absorption and\u00a0emission of light by electrons in atom\u00a0 using perturbation theory.<\/li>\r\n \t
  4. Learn what is time dependent perturbation theory and the transition probability\u00a0per unit time.<\/li>\r\n<\/ol>\r\n
    \u00a0 \u00a0 2. Introduction<\/strong><\/div>\r\n
    <\/div>\r\n
    Whenever light radiation is incident on <\/span>matter it shows a lot of interesting processes. It can be scattered, absorbed or it can eject electrons of the atom and ionize the atom. It is seen that the effective size of the atom is around 10 8 cm. <\/span>Also lots of atoms emit light radiation where the wavelength of the light emitted is of the order of 10 5 cm. (this region of <\/span>spectrum is known as visible spectrum). The strength of the interaction of light radiation with mater is determined by the charge of the participating particle and is well known. It is small as shown by the smallness of the \u201ccoupling\u00a0<\/span>the electromagnetic processes it is the electrons of the atoms of the matter that take part in such processes. Therefore we shall study the interaction of electrons with the electromagnetic field. The notable processes are the absorption and emission of light, the <\/span>black body radiation, the photoelectric effect, the famous Lamb shift, and some scattering processes.<\/span><\/div>\r\n
    <\/div>\r\n
    The electron is a negatively charged particle and obeys Fermi-Dirac statistics. In the visible range of light the energy of the electrons is such that they can be treated as non-relativistic particles and so we shall describe electrons of the atoms by means of Schrodinger equation.<\/span><\/div>\r\n
    <\/div>\r\n
    We shall here use the famous <\/span>time dependent perturbation theory and obtain the expressions for transition probability per unit time, which helps to calculate the life time\u00a0<\/span>for decay of atoms and also differential cross-section for\u00a0<\/span>scattering processes.. By treating the interaction term as a small perturbation we shall study induced\/stimulated emission, absorption and spontaneous emission of radiation by the electrons. In the next two modules a detailed study of spontaneous emission of radiation in the Dipole, Quadrupole and Magnetic Dipole transitions is made.<\/span><\/div>\r\n
    <\/div>\r\n
    \u00a0 \u00a0 3. Interaction of Radiation with Matter<\/strong><\/div>\r\n
    <\/div>\r\n
    Radiation consists of electromagnetic waves which react with the charged particles of the matter. In this case we shall study the interactions of electrons of the atoms with the electromagnetic waves. The electromagnetic waves are described by the vector potential <\/span>A<\/strong> . We work in the Coulomb or radiation gauge. We shall treat electrons as non-relativistic particles and assume that they satisfy Schrodinger equation.<\/span><\/div>\r\n
    \r\n\r\n \r\n\r\n3.1 Schrodinger Equation for matter interacting with electromagnetic waves<\/strong>\r\n\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n\u00a0 \u00a0 \u00a0 where e<\/em> is the charge of the electron. This gives us the following Schrodinger equation\r\n\r\n\"\"\r\n

    We shall take as quantized field vector and scalar potential is taken in the electron Coulomb interaction term variables as it is not a quantized variable in Coulomb gauge. If N electrons of an atom participate in the interaction the total Hamiltonian H satisfies the equation<\/p>\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n \r\n

    The first part of HO is the particle kinetic interaction of the electrons and the last term energy plus the instantaneous Coulomb is E.M. field energy.<\/p>\r\n\r\n<\/div>\r\n\"\"\r\n

    <\/div>\r\n
    \r\n\r\n\u00a0 \u00a0 3.2 Quantized Electromagnetic Field<\/strong>\r\n\r\n \r\n

    We have already studied Quantization of the electromagnetic field using radiation or Coulomb gauge. As studied earlier the vector potential operator? (? , ?) in Heinsenberg picture is written as<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    <\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n\"\"\r\n\r\n \r\n\r\nEq. 3.14 and 3.15 are the two most important results of quantized radiation theory.<\/strong>\r\n\r\n \r\n\r\n4. Time dependent Perturbation Theory<\/strong>\r\n\r\n \r\n

    As the total Hamiltonian H = H0+HI is time dependent, the energy of the system is not conserved and so there are no stationary states. In this case we shall apply the time dependent perturbation theory. We shall assume that the unperturbed Hamiltonian 0 is time independent and forms the major part of the Hamiltonian, and has stationary, orthogonal and complete set of states. Let 0 be the wave function of the Hamiltonian 0. The wave function carries the time dependence. This is like Schrodinger picture. We assume that the interaction Hamiltonian is small (a perturbation) compared to 0 and is time dependent and produces a change or mixing of the eigen wave functions of 0. This is Heisenberg picture for . This mixed picture for the whole Hamiltonian is called the Interaction picture. We shall seek the solution of the Schrodinger\u2019s equation<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\n<\/div>\r\n
    \r\n\r\n\"\"\r\n\r\n4.1 Initial Condition and Expansion in powers of<\/strong>\r\n\r\n \r\n

    Suppose now that perturbation H1<\/sub> begins to act at time t=0, and at t = 0, when the perturbation has not acted yet, let\u00a0 only one unperturbed state\u00a0??<\/sub>0<\/sup> be there.<\/p>\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n

    and so on. Eq.(4.11) can be interpreted as transition in two stages. First from state i<\/em> to state m<\/em> and next from state m<\/em> to state n<\/em> . The state m<\/em> is called the intermediate state.<\/p>\r\n \r\n\r\n\"\"\r\n\r\n<\/div>\r\n

    \r\n\r\n\"\"\r\n\r\nNow in the final state we have a number of photon states lying in the range and + . So the transition probability to go to state n has to be summed over all the states available for transition. As the energy interval is small we assume that the matrix element is constant and multiply the transition probability eq. (4.17) by the number of states in the small interval of energy dE, that is by the density of states d = dE,\r\n\r\n\"\"\r\n\r\n<\/div>\r\nThus the transition probability for emission of photons in the states ? and ? + ?? is given by\r\n\r\n\"\"\r\n\r\nusefulness in the next module for absorption or decay of atom and for scattering processes..\r\n
      \r\n \t
    1. Summary<\/strong><\/li>\r\n<\/ol>\r\n

      \u00a0 \u00a0 In this module we have studied the emission and absorption of photons by charged particles like electrons in the atoms. It is seen that by treating electrons non-relativistically and describing photons by quantized radiation field we obtain the transition probability per unit time by using the techniques of time dependent perturbation theory.<\/p>\r\n \r\n

      The application of the expressions of transition probabilities under for various processes like absorption , emission will be taken up in detail in the next few modules.<\/p>\r\n\r\n\r\n\r\n
      you can view video on Perturbation Theory<\/strong><\/td>\r\n\"\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\nLearn More<\/strong>\r\n\r\n \r\n\r\nThe following books are some among many.\r\n
        \r\n \t
      1. Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education, Singapore 1998)<\/li>\r\n<\/ol>\r\n
          \r\n \t
        1. Quantum Mechanics Vol. 3 by L.D. Landau and E.M. Lifshitz (Pergamon Press, Oxford, Reprinted 1981)<\/li>\r\n<\/ol>\r\n
            \r\n \t
          1. Quantum Electrodynamics, Vol. 4 by V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii (Pergamon Press, Oxford, 1982)<\/li>\r\n<\/ol>\r\n\u00a0 \u00a0 Web Links<\/strong>\r\n\r\n \r\n\r\nwww.tcm.phy.cam.ac.uk\/~bds10\/aqp\/lec18pdf<\/a>\r\n\r\n \r\n

            This is more about the application of time dependent perturbation theory to simple but important systems like two level systems.<\/p>\r\n \r\n\r\nInteresting Facts<\/strong>\r\n\r\n \r\n

            Non relativistic perturbation theory in the form discussed here was started by P.A,M. Dirac in 1927 and was used by particle physicists for about 25 years. The relativistic theory with Feynman diagrams then took over . However the condensed matter theorists still use it in many cases.<\/p>\r\n \r\n

            The Golden Rule is taught in Quantum mechanics courses. Here we have given a short revision to enable us to apply it for the electromagnetic field. See the web link for other applications. The name Golden rule was given by Enrico Fermi in his course on quantum mechanics give at University of Chicago, USA.<\/p>\r\n ","rendered":"

            \"epgp<\/a>
            \n<\/span><\/div>\n
            <\/div>\n
            <\/div>\n
            \u00a0 \u00a0 1. Learning outcomes<\/strong><\/div>\n
              \n
            1. Learn what is electromagnetic interaction and its strength.<\/li>\n
            2. Learn why perturbation theory can be applied for most of the processes \u2013\u00a0involving charged particles and radiation field.<\/li>\n
            3. Learn how the quantum theory of radiation is applied to\u00a0 absorption and\u00a0emission of light by electrons in atom\u00a0 using perturbation theory.<\/li>\n
            4. Learn what is time dependent perturbation theory and the transition probability\u00a0per unit time.<\/li>\n<\/ol>\n
              \u00a0 \u00a0 2. Introduction<\/strong><\/div>\n
              <\/div>\n
              Whenever light radiation is incident on <\/span>matter it shows a lot of interesting processes. It can be scattered, absorbed or it can eject electrons of the atom and ionize the atom. It is seen that the effective size of the atom is around 10 8 cm. <\/span>Also lots of atoms emit light radiation where the wavelength of the light emitted is of the order of 10 5 cm. (this region of <\/span>spectrum is known as visible spectrum). The strength of the interaction of light radiation with mater is determined by the charge of the participating particle and is well known. It is small as shown by the smallness of the \u201ccoupling\u00a0<\/span>the electromagnetic processes it is the electrons of the atoms of the matter that take part in such processes. Therefore we shall study the interaction of electrons with the electromagnetic field. The notable processes are the absorption and emission of light, the <\/span>black body radiation, the photoelectric effect, the famous Lamb shift, and some scattering processes.<\/span><\/div>\n
              <\/div>\n
              The electron is a negatively charged particle and obeys Fermi-Dirac statistics. In the visible range of light the energy of the electrons is such that they can be treated as non-relativistic particles and so we shall describe electrons of the atoms by means of Schrodinger equation.<\/span><\/div>\n
              <\/div>\n
              We shall here use the famous <\/span>time dependent perturbation theory and obtain the expressions for transition probability per unit time, which helps to calculate the life time\u00a0<\/span>for decay of atoms and also differential cross-section for\u00a0<\/span>scattering processes.. By treating the interaction term as a small perturbation we shall study induced\/stimulated emission, absorption and spontaneous emission of radiation by the electrons. In the next two modules a detailed study of spontaneous emission of radiation in the Dipole, Quadrupole and Magnetic Dipole transitions is made.<\/span><\/div>\n
              <\/div>\n
              \u00a0 \u00a0 3. Interaction of Radiation with Matter<\/strong><\/div>\n
              <\/div>\n
              Radiation consists of electromagnetic waves which react with the charged particles of the matter. In this case we shall study the interactions of electrons of the atoms with the electromagnetic waves. The electromagnetic waves are described by the vector potential <\/span>A<\/strong> . We work in the Coulomb or radiation gauge. We shall treat electrons as non-relativistic particles and assume that they satisfy Schrodinger equation.<\/span><\/div>\n
              \n

               <\/p>\n

              3.1 Schrodinger Equation for matter interacting with electromagnetic waves<\/strong><\/p>\n

               <\/p>\n

              \"\"<\/p>\n<\/div>\n

              \n

              \u00a0 \u00a0 \u00a0 where e<\/em> is the charge of the electron. This gives us the following Schrodinger equation<\/p>\n

              \"\"<\/p>\n

              We shall take as quantized field vector and scalar potential is taken in the electron Coulomb interaction term variables as it is not a quantized variable in Coulomb gauge. If N electrons of an atom participate in the interaction the total Hamiltonian H satisfies the equation<\/p>\n<\/div>\n

              \n

              \"\"<\/p>\n

               <\/p>\n

              The first part of HO is the particle kinetic interaction of the electrons and the last term energy plus the instantaneous Coulomb is E.M. field energy.<\/p>\n<\/div>\n

              \"\"<\/p>\n

              <\/div>\n
              \n

              \u00a0 \u00a0 3.2 Quantized Electromagnetic Field<\/strong><\/p>\n

               <\/p>\n

              We have already studied Quantization of the electromagnetic field using radiation or Coulomb gauge. As studied earlier the vector potential operator? (? , ?) in Heinsenberg picture is written as<\/p>\n

              \"\"<\/p>\n<\/div>\n

              <\/div>\n
              \n

              \"\"<\/p>\n

              \"\"<\/p>\n

              \"\"<\/p>\n

               <\/p>\n

              Eq. 3.14 and 3.15 are the two most important results of quantized radiation theory.<\/strong><\/p>\n

               <\/p>\n

              4. Time dependent Perturbation Theory<\/strong><\/p>\n

               <\/p>\n

              As the total Hamiltonian H = H0+HI is time dependent, the energy of the system is not conserved and so there are no stationary states. In this case we shall apply the time dependent perturbation theory. We shall assume that the unperturbed Hamiltonian 0 is time independent and forms the major part of the Hamiltonian, and has stationary, orthogonal and complete set of states. Let 0 be the wave function of the Hamiltonian 0. The wave function carries the time dependence. This is like Schrodinger picture. We assume that the interaction Hamiltonian is small (a perturbation) compared to 0 and is time dependent and produces a change or mixing of the eigen wave functions of 0. This is Heisenberg picture for . This mixed picture for the whole Hamiltonian is called the Interaction picture. We shall seek the solution of the Schrodinger\u2019s equation<\/p>\n

              \"\"<\/p>\n<\/div>\n

              \n

              \"\"<\/p>\n<\/div>\n

              \n

              \"\"<\/p>\n

              4.1 Initial Condition and Expansion in powers of<\/strong><\/p>\n

               <\/p>\n

              Suppose now that perturbation H1<\/sub> begins to act at time t=0, and at t = 0, when the perturbation has not acted yet, let\u00a0 only one unperturbed state\u00a0??<\/sub>0<\/sup> be there.<\/p>\n

              \"\"<\/p>\n<\/div>\n

              \n

              \"\"<\/p>\n

              and so on. Eq.(4.11) can be interpreted as transition in two stages. First from state i<\/em> to state m<\/em> and next from state m<\/em> to state n<\/em> . The state m<\/em> is called the intermediate state.<\/p>\n

               <\/p>\n

              \"\"<\/p>\n<\/div>\n

              \n

              \"\"<\/p>\n

              Now in the final state we have a number of photon states lying in the range and + . So the transition probability to go to state n has to be summed over all the states available for transition. As the energy interval is small we assume that the matrix element is constant and multiply the transition probability eq. (4.17) by the number of states in the small interval of energy dE, that is by the density of states d = dE,<\/p>\n

              \"\"<\/p>\n<\/div>\n

              Thus the transition probability for emission of photons in the states ? and ? + ?? is given by<\/p>\n

              \"\"<\/p>\n

              usefulness in the next module for absorption or decay of atom and for scattering processes..<\/p>\n

                \n
              1. Summary<\/strong><\/li>\n<\/ol>\n

                \u00a0 \u00a0 In this module we have studied the emission and absorption of photons by charged particles like electrons in the atoms. It is seen that by treating electrons non-relativistically and describing photons by quantized radiation field we obtain the transition probability per unit time by using the techniques of time dependent perturbation theory.<\/p>\n

                 <\/p>\n

                The application of the expressions of transition probabilities under for various processes like absorption , emission will be taken up in detail in the next few modules.<\/p>\n\n\n\n
                you can view video on Perturbation Theory<\/strong><\/td>\n\"\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                Learn More<\/strong><\/p>\n

                 <\/p>\n

                The following books are some among many.<\/p>\n

                  \n
                1. Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education, Singapore 1998)<\/li>\n<\/ol>\n
                    \n
                  1. Quantum Mechanics Vol. 3 by L.D. Landau and E.M. Lifshitz (Pergamon Press, Oxford, Reprinted 1981)<\/li>\n<\/ol>\n
                      \n
                    1. Quantum Electrodynamics, Vol. 4 by V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii (Pergamon Press, Oxford, 1982)<\/li>\n<\/ol>\n

                      \u00a0 \u00a0 Web Links<\/strong><\/p>\n

                       <\/p>\n

                      www.tcm.phy.cam.ac.uk\/~bds10\/aqp\/lec18pdf<\/a><\/p>\n

                       <\/p>\n

                      This is more about the application of time dependent perturbation theory to simple but important systems like two level systems.<\/p>\n

                       <\/p>\n

                      Interesting Facts<\/strong><\/p>\n

                       <\/p>\n

                      Non relativistic perturbation theory in the form discussed here was started by P.A,M. Dirac in 1927 and was used by particle physicists for about 25 years. The relativistic theory with Feynman diagrams then took over . However the condensed matter theorists still use it in many cases.<\/p>\n

                       <\/p>\n

                      The Golden Rule is taught in Quantum mechanics courses. Here we have given a short revision to enable us to apply it for the electromagnetic field. See the web link for other applications. The name Golden rule was given by Enrico Fermi in his course on quantum mechanics give at University of Chicago, USA.<\/p>\n

                       <\/p>\n","protected":false},"author":3,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["prof-n-panchapakesan","prof-j-d-anand"],"pb_section_license":""},"chapter-type":[],"contributor":[58,59],"license":[],"class_list":["post-148","chapter","type-chapter","status-publish","hentry","contributor-prof-n-panchapakesan","contributor-prof-j-d-anand"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/148","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/users\/3"}],"version-history":[{"count":9,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/148\/revisions"}],"predecessor-version":[{"id":261,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/148\/revisions\/261"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/148\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/media?parent=148"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapter-type?post=148"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/contributor?post=148"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/license?post=148"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}