{"id":211,"date":"2018-11-15T05:02:33","date_gmt":"2018-11-15T05:02:33","guid":{"rendered":"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=211"},"modified":"2019-04-30T09:51:59","modified_gmt":"2019-04-30T09:51:59","slug":"higher-order-interactions_landmarks-on-the-way-to-quantum-field-theory","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/chapter\/higher-order-interactions_landmarks-on-the-way-to-quantum-field-theory\/","title":{"rendered":"Higher Order Interactions_Landmarks on the way to Quantum Field Theory"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/KYt2lAHiwjU\" target=\"_blank\" rel=\"noopener\"><img src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a>\r\n<\/span><\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Learning outcomes<\/strong>\r\n<ul>\r\n \t<li>Estimate the value of higher order interactions like spin, quadrupole and magnetic dipole.<\/li>\r\n \t<li>Learn to estimate the power by which each order of perturbation decreases.<\/li>\r\n \t<li>Learn the derivation of Planck\u2019s Law from QED with hermiticity playing a crucial role.<\/li>\r\n \t<li>Learn about two landmarks before quantum theory of radiation :\u00a0 Planck\u2019s<\/li>\r\n \t<li>Law derivation by (1) Einstein in 1917 which first identified Spontaneous Emission and (2) by Satyendra Nath Bose in 1924 which led to Bose-Einstein statistics.<\/li>\r\n \t<li>Understand , that the principle of detailed balance used by Einstein is a consequence of hermiticity of the Hamiltonian used in Quantum mechanics.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 2. Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The interaction due to the magnetic moment of the electron, not included in our interaction Hamiltonian so far is next studied and shown to be of higher order and hence smaller by thousand times in the amplitude . The momentm <strong>k<\/strong> of photon in <strong>k.x<\/strong> is of the order of (~1<em>\/<\/em> <em>)<\/em> and <strong>x<\/strong> is limited in the integral over the radius of the atom <em>a<\/em> . The amplitude is smaller by <strong>|kx|<\/strong> (~<em>a\/<\/em> <em>~<\/em> 10\u22123). and the transition probability is smaller by a\u00a0factor of , square of that (10\u22126). where <em>a<\/em> is the radius of the atom and is the wave length of light emitted. We shall study this in section 3.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Some times the first term 1 , in the expansion of plane wave , leading to dipole emission or absorption is forbidden by selection rules. The next term in the expansion the leads to the quadrupole and the magnetic dipole terms. These are of higher order and hence smaller as seen above . In studying them it is convenient to expand the plane wave part \u00a0\u00a0\u00a0.\u00a0 \u00a0in terms of spherical Bessel function and Legender polynomials. We do this in section 4.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We can apply our results obtained to calculate Planck\u2019s law for black body radiation, the law which started off the quantum revolution and brought Planck\u2019s constant h into physics. We make use of the hermiticity of the interaction Hamiltonian in doing this. A charged particle like electron emits radiations as well as absorbs electromagnetic radiations. So in deriving Planck\u2019s Law inside a back body, one can consider a system of atoms in thermodynamic equilibrium. In emitting and absorption of photons the electrons of the atoms can be treated non-relativistically. The atom in a state A emits a photon of energy \u03c9 and goes to state B. The atom in state A absorbs a photon of energy\u00a0<span style=\"text-align: initial;font-size: 1em\">and goes to state B.\u00a0 By treating <\/span>these system<span style=\"text-align: initial;font-size: 1em\"> of atoms\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">in thermodynamic equilibrium, we derive the Planck\u2019s Law of radiation in section 5.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In section 6 and 7 we discuss two land mark works by Einstein and Satyendra Nath Bose. The first is Einstein\u2019s derivation of the Planck Law by considering trasitions between two states of an atom A and b as mentioned earlier. Apart from two terms for absorption and emission in the presence of radiation he had to consider a term for spontaneous emission even in the absence of <\/span>radiation .<span style=\"text-align: initial;font-size: 1em\"> This is the 1 part of + 1, we have already seen. This was not known in 1917 when Einstein did his calculation. This discussed in section 6.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In 1924, Bose derived the Planck\u2019s law in a totally revolutionary way. He could not get it published and so took the unusual step of sending it to Einstein, who <\/span>recognizing<span style=\"text-align: initial;font-size: 1em\"> its importance got it translated into German, and published. Einstein also showed that it had applications to other particles than photons, which could be having a mass unlike photon, which is massless. This we know now as Bose-Einstein statistics applicable to particles of integral, including zero, spin. Such particles are now called Bosons, a name <\/span>given<span style=\"text-align: initial;font-size: 1em\"> by Dirac.<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<strong>3. Spin Interaction Term:<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">So far we have neglected the spin and hence the magnetic moment of the electron in its interaction with the radiation field. If we consider the spin of the electron then we have one additional term in the interaction ??;<\/p>\r\n<img class=\"alignnone size-full wp-image-215\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125.png\" alt=\"\" width=\"676\" height=\"89\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-216\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126.png\" alt=\"\" width=\"690\" height=\"552\" \/>\r\n\r\n<img class=\"alignnone size-full wp-image-217\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127.png\" alt=\"\" width=\"677\" height=\"197\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">4.1 Magnetic Dipole and Quadrupole Transitions<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">Occasionally we may have (?\u0305)BA = <\/span>0 ,<span style=\"text-align: initial;font-size: 1em\"> for every state B with energy lower than that of the state A. This may be due to the selection rules. In these <\/span>cases<span style=\"text-align: initial;font-size: 1em\"> we then go back to the plane wave expansion<\/span>\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-218\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128.png\" alt=\"\" width=\"668\" height=\"533\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The first term of equation ( 4.3 ) has (k\u20d7\u20d7 \u00d7\u2107\u20d7\u20d7\u20d7\u03bb)) which is the first term in the expansion of curl A and hence denotes the magnetic field B. L is related to the angular momentum and hence can be expressed in terms of the magnetic moment \u03bc also. The\u00a0first term becomes<\/p>\r\n\r\n<\/div>\r\n<img class=\"alignnone size-full wp-image-220\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129.png\" alt=\"\" width=\"435\" height=\"48\" \/>\r\n<div>\r\n<p style=\"text-align: justify\">where ?\u20d7 is the magnetic moment of the electron. Thus ( 4.4 ) is the interaction of magnetic moment of the charged particle with the magnetic field ?\u20d7\u20d7. This term is called M1 or magnetic dipole term.<\/p>\r\n\r\n<\/div>\r\n<p style=\"text-align: justify\">The second term of equation (4.3 ) is called E2 or Electric quadrupole term. To simplify this term we now use the relation [?\u20d72, ?\u20d7] = \u22122??\u20d7 ??, 2?[?0, ?\u20d7] = \u22122??\u20d7 , where ?0 is the free Hamiltonian.<\/p>\r\n&nbsp;\r\n\r\nThus ?\u20d7 = ??[?0, ?\u20d7] \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(4.5)\r\n\r\n&nbsp;\r\n\r\nUsing (4.5) we write the second term of eq. (4.3) as\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-221\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130.png\" alt=\"\" width=\"622\" height=\"244\" \/>\r\n\r\nThis term gives rise to the electric quadrupole or E2 term. To see this in another way we go back to another expansion for plane wave .\r\n\r\n&nbsp;\r\n\r\n<strong>4.2 Another Expansion of Plane Wave<\/strong>\r\n\r\n&nbsp;\r\n\r\nWe can write\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-222\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131.png\" alt=\"\" width=\"651\" height=\"168\" \/>\r\n<div>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a0with the <\/span><strong style=\"text-align: initial;font-size: 1em\">p.<\/strong><span style=\"text-align: initial;font-size: 1em\"> term this will give a product of two\u00a0 <\/span><em style=\"text-align: initial;font-size: 1em\">l=<\/em><span style=\"text-align: initial;font-size: 1em\">1\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">terms. The resulting term will have <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\">=2,1 or 0. The quadrupole term is the one with\u00a0 <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\">=2.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">This results in <\/span>emission<span style=\"text-align: initial;font-size: 1em\"> of quadrupole term E2.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">5.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Derivation of Planck\u2019s Law from field theory.<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Consider a system of atoms in thermodynamic equilibrium<\/span>. .<span style=\"text-align: initial;font-size: 1em\"> The atom in a state A emits a photon of energy \u03c9 and goes to state B. The atom in state A absorbs a photon of\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">energy\u00a0\u00a0 and goes to state B.\u00a0\u00a0 The processes are<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n\u00a0 \u00a0 ? \u2192 ? + ?, ? + ? \u2192 ? , ?? ? \u21c4 ? + ? \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(5.1)\r\n\r\n&nbsp;\r\n\r\nLet EA be the energy of the atom A and EB that of B then\r\n\r\n&nbsp;\r\n\r\n?? = ?? + ?\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Let the population (number) of atoms in state A and B be N(A) and N(B) respectively. The transition probability ??\u2212? is given by Golden rule ( Module 6) which for a transition is<\/p>\r\n&nbsp;\r\n\r\n??\u2192? = 2?|??? |<sup>2<\/sup> \u03b4(?? \u2212 ?? \u2212 ?) \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (5.2)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In the above expression ??? is the matrix element of the interaction Hamiltonian (i.e. perturbation theory is assumed to be valid ). In the equilibrium state<\/p>\r\n\r\n<\/div>\r\n<img class=\"alignnone size-full wp-image-223\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132.png\" alt=\"\" width=\"590\" height=\"125\" \/>\r\n<div>\r\n\r\nIn the above ?? is Boltzman Constant and T the equilibrium temperature. If \u2130\u20d7? is the polarization vector of the photon of energy ? and momentum k then,\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-225\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133.png\" alt=\"\" width=\"577\" height=\"546\" \/>\r\n\r\nThe number of states dn in a black body filled with radiation is obtained using the same arguments as for ?? and is\r\n\r\n? 2.4??<sup>2<\/sup> ??\/(2?)<sup>3<\/sup>. = 8??<sup>2<\/sup>???\/(2?)<sup>3<\/sup> \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(5.9)\r\n<p style=\"text-align: justify\">The factor 2 is due to the availability of two possible directions in which light can be polarized. Thus the\u00a0 number of photons with an energy from ? and ? + ?? in the volume V is<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-226\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134.png\" alt=\"\" width=\"603\" height=\"108\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-227\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135.png\" alt=\"\" width=\"687\" height=\"299\" \/>\r\n\r\nwhich is Planck\u2019s radiation Law for determining the energy density ?( \u03c9, ?) or U (\u03bd, ?) of black body radiation,\r\n\r\n&nbsp;\r\n\r\n<strong>6.\u00a0\u00a0\u00a0 <\/strong><strong>Einstein\u2019s Derivation of Planck\u2019s Law and Spontaneous emission<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">This derivation was given by Einstein in 1917 where he first identified the factor for spontaneous emission. Let the initial state of the electron of the atom be n with energy En . After absorbing a photon of energy it has a state m with energy Em , then<\/p>\r\n&nbsp;\r\n\r\n?? = ?? + ? \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(6.1)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Let population (number) of the atoms in state m and n be Nm and Nn respectively.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">Einstein realized that the probability per unit time per atom for the emission transition\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">must have two parts. We denote the spontaneous emission by A, which does not need\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">any radiation to be present and the stimulated emission by ??(?), which is proportional\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">to the amount of radiation ?(?) present. The absorbed emission is denoted by ?\u2032?(?)\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">and depends on the radiation present. In thermodynamic equilibrium we have<\/span><\/p>\r\n\r\n<\/div>\r\n???\u2032?(?) = ??[? + ??(?)] \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (6.2)\r\n\r\n&nbsp;\r\n\r\n<strong>\u00a0<img class=\"alignnone size-full wp-image-228\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136.png\" alt=\"\" width=\"662\" height=\"500\" \/><\/strong>\r\n<div>\r\n\r\nEinstein argued that from the principle of detailed balance ?\u2032 = ?. To derive the law from field theory we had used hermiticity of the Hamiltonian . Comparing (6.5) with Planck\u2019s law (5.10) we get A = B ( ?3 \/ ?2 )\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">7. Bose Derivation of Planck Law<\/strong>\r\n\r\n<\/div>\r\n<p style=\"text-align: justify\">Using statistical mechanics Bose calculated the probability for distributing ?? photons over ?? cells, all having energy ?? . Since there can be any number of photons in one cell this corresponds to inserting ?? -1 partitions between ?? photons. If we have ?? =4 we have<\/p>\r\n<p style=\"text-align: center\">\u2026\u2026.|\u2026|\u2026..|\u2026\u2026\u2026.<\/p>\r\n<p style=\"text-align: center\">?1 ?2 ?3<\/p>\r\n<p style=\"text-align: justify\">with dots showing photons and vertical lines partitions ?1 , ?2and ?3.<\/p>\r\n\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-230\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137.png\" alt=\"\" width=\"665\" height=\"409\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">Bose\u2019s paper was rejected by a journal for publication. So he sent it in 1924 to Prof. Einstein requesting him to forward it for publication if he thought it was correct. Einstein translated the paper into German language and sent it for publication and wrote a few more papers applying the Bose method of counting to normal gases. This was the birth of systems obeying Bose-Einstein statistics. We now know that integral (including zero spin ) particles obey Bose statistics and are called Bosons . We saw in the earlier module that bosons, scalar particles satisfy commutation relation when quantized. Photons have spin 1 and so are bosons. They do not have 2S + 1 or 3 components, but have only two components or polarisations as they have zero mass. Zero mass implies gauge constraints and we have seen how gauge constraints reduce the degrees of freedom to two polarisations.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">There were criticisms that the counting of states by Bose and Einstein violated the independent nature of particles. Einstein accepted this criticism but said that\u00a0<span style=\"text-align: initial;font-size: 1em\">as the result was right they have to find out the reason for\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">this mysterious attraction between <\/span>photons .<span style=\"text-align: initial;font-size: 1em\"> We now know that symmetry of boson wave function brings in an attraction just as that of anti-symmetry of fermions (particles with <\/span>half integral<span style=\"text-align: initial;font-size: 1em\"> spins) brings in a repulsion leading to Pauli principle. This was finally explained by P.A.M.Dirac in 1926. The statistics obeyed by Fermions is called Fermi-Dirac Statistics.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Planck introduced the photon in 1900 and Einstein gave the <\/span>photo electric<span style=\"text-align: initial;font-size: 1em\"> effect in 1905. But even in 1924, <\/span>photon<span style=\"text-align: initial;font-size: 1em\"> was still not accepted as a particle and Einstein expressed surprise at the way Bose had derived the number of <\/span>states .<span style=\"text-align: initial;font-size: 1em\"> It took a long time 27 years for dual nature of light to be accepted. Abraham Pais in his book on <\/span>Eistein<span style=\"text-align: initial;font-size: 1em\"> and his work mentions four <\/span>path breaking<span style=\"text-align: initial;font-size: 1em\"> papers published before <\/span>quantum<span style=\"text-align: initial;font-size: 1em\"> revolution in 1926. They are by Max Planck (1900), Albert Einstein (1905), Niels Bohr (1911) and Satyendra Nath Bose (1924).<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"font-size: 1em\">8. Summary<\/strong><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In this <\/span>module<span style=\"text-align: initial;font-size: 1em\"> we have studied the higher order interactions like spin interaction, electric quadrupole <\/span>and<span style=\"text-align: initial;font-size: 1em\"> magnetic dipole interactions. We find these higher order terms are thousand times smaller as they are proportional to the ratio of the radius of the atom to the <\/span>wave length<span style=\"text-align: initial;font-size: 1em\"> of light. We have <\/span>learnt<span style=\"text-align: initial;font-size: 1em\"> how Planck\u2019s Law can be derived from quantum field theory. We have also <\/span>learnt<span style=\"text-align: initial;font-size: 1em\"> how Einstein derived the Planck\u2019s law from Bohr\u2019s theory in 1917, and Satyendra Nath Bose derived it from statistical arguments in 1924 before the discovery of quantum mechanics in <\/span>1926,<span style=\"text-align: initial;font-size: 1em\"> and became the first author on quantum statistics.<\/span><\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Higher Order Interactions_Landmarks on the way to Quantum Field Theory<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/KYt2lAHiwjU\" target=\"_blank\" rel=\"noopener\"><img class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<strong>\u00a0 \u00a0 Learn More Books<\/strong>\r\n\r\n&nbsp;\r\n\r\n1.\u00a0 Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education Singapore (1998)\r\n\r\n2.\u00a0 Quantum Mechanics Vol.3 by L.D. Landau and Lifshitz (Pergramon Press Oxford, Reprinted\r\n\r\n3.\u00a0 \u201cSubtle is the Lord , The science and Life of Albert Einstein\u201d , Oxford University Press, 1982\r\n\r\n&nbsp;\r\n\r\n<strong>Web sites<\/strong>\r\n\r\n&nbsp;\r\n\r\nFor history see\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.physics.udel.edu\/~msafrono\/626\/Lectures%2013-14.pdf\">www.physics.udel.edu\/~msafrono\/626\/Lectures%2013-14.pdf<\/a>\r\n\r\n&nbsp;\r\n\r\n<a href=\"http:\/\/www.colorado.edu\/entities\/quantum-field-theory\/qft.history.html\">www.colorado.edu\/entities\/quantum-field-theory\/qft.history.html<\/a>\r\n\r\n&nbsp;\r\n\r\n<strong>Photons not being independent<\/strong>\r\n\r\n&nbsp;\r\n\r\nDirac, in his book on Quantum Mechanics (1947 edition) explains this very simply.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">I describe this in my own words. If you have two coins and toss them. The probability for two heads to appear is one quarter or \u00bc . This is because it is one possibility among four.<\/p>\r\n&nbsp;\r\n\r\n2 heads, 2 tails, one head one tail or one tail and one head.\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">When they have to be symmetric wave functions, then only the sum of the last two is symmetric. The difference is antisymmetric.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">So there are only three possibilities. Hence the probability for two heads or two tails is one third or 1\/3. This is more than \u00bc, what it was earlier. The probability has gone up due to attraction due to Bose-Einstein Statistics.<\/p>\r\n&nbsp;\r\n\r\n<strong>True Story<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">When S.N.Bose was taking Mr. and Mrs, Dirac, in car , home for lunch, the Diracs sat at the back and Bose and driver were in front. When another student wanted also to come Bose took him in front. Mrs Dirac objected and said he should come back. Dirac remarked :<span style=\"text-align: initial;font-size: 1em\"> it is all a question of statistics.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">( Dirac statistics repels and Bose statistics attracts.)<\/span><\/p>\r\n\r\n<\/div>","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/KYt2lAHiwjU\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"epgp books\" width=\"75px\" height=\"75px;\" \/><\/a><br \/>\n<\/span><\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Learning outcomes<\/strong><\/p>\n<ul>\n<li>Estimate the value of higher order interactions like spin, quadrupole and magnetic dipole.<\/li>\n<li>Learn to estimate the power by which each order of perturbation decreases.<\/li>\n<li>Learn the derivation of Planck\u2019s Law from QED with hermiticity playing a crucial role.<\/li>\n<li>Learn about two landmarks before quantum theory of radiation :\u00a0 Planck\u2019s<\/li>\n<li>Law derivation by (1) Einstein in 1917 which first identified Spontaneous Emission and (2) by Satyendra Nath Bose in 1924 which led to Bose-Einstein statistics.<\/li>\n<li>Understand , that the principle of detailed balance used by Einstein is a consequence of hermiticity of the Hamiltonian used in Quantum mechanics.<\/li>\n<\/ul>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 2. Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The interaction due to the magnetic moment of the electron, not included in our interaction Hamiltonian so far is next studied and shown to be of higher order and hence smaller by thousand times in the amplitude . The momentm <strong>k<\/strong> of photon in <strong>k.x<\/strong> is of the order of (~1<em>\/<\/em> <em>)<\/em> and <strong>x<\/strong> is limited in the integral over the radius of the atom <em>a<\/em> . The amplitude is smaller by <strong>|kx|<\/strong> (~<em>a\/<\/em> <em>~<\/em> 10\u22123). and the transition probability is smaller by a\u00a0factor of , square of that (10\u22126). where <em>a<\/em> is the radius of the atom and is the wave length of light emitted. We shall study this in section 3.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Some times the first term 1 , in the expansion of plane wave , leading to dipole emission or absorption is forbidden by selection rules. The next term in the expansion the leads to the quadrupole and the magnetic dipole terms. These are of higher order and hence smaller as seen above . In studying them it is convenient to expand the plane wave part \u00a0\u00a0\u00a0.\u00a0 \u00a0in terms of spherical Bessel function and Legender polynomials. We do this in section 4.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We can apply our results obtained to calculate Planck\u2019s law for black body radiation, the law which started off the quantum revolution and brought Planck\u2019s constant h into physics. We make use of the hermiticity of the interaction Hamiltonian in doing this. A charged particle like electron emits radiations as well as absorbs electromagnetic radiations. So in deriving Planck\u2019s Law inside a back body, one can consider a system of atoms in thermodynamic equilibrium. In emitting and absorption of photons the electrons of the atoms can be treated non-relativistically. The atom in a state A emits a photon of energy \u03c9 and goes to state B. The atom in state A absorbs a photon of energy\u00a0<span style=\"text-align: initial;font-size: 1em\">and goes to state B.\u00a0 By treating <\/span>these system<span style=\"text-align: initial;font-size: 1em\"> of atoms\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">in thermodynamic equilibrium, we derive the Planck\u2019s Law of radiation in section 5.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In section 6 and 7 we discuss two land mark works by Einstein and Satyendra Nath Bose. The first is Einstein\u2019s derivation of the Planck Law by considering trasitions between two states of an atom A and b as mentioned earlier. Apart from two terms for absorption and emission in the presence of radiation he had to consider a term for spontaneous emission even in the absence of <\/span>radiation .<span style=\"text-align: initial;font-size: 1em\"> This is the 1 part of + 1, we have already seen. This was not known in 1917 when Einstein did his calculation. This discussed in section 6.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In 1924, Bose derived the Planck\u2019s law in a totally revolutionary way. He could not get it published and so took the unusual step of sending it to Einstein, who <\/span>recognizing<span style=\"text-align: initial;font-size: 1em\"> its importance got it translated into German, and published. Einstein also showed that it had applications to other particles than photons, which could be having a mass unlike photon, which is massless. This we know now as Bose-Einstein statistics applicable to particles of integral, including zero, spin. Such particles are now called Bosons, a name <\/span>given<span style=\"text-align: initial;font-size: 1em\"> by Dirac.<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><strong>3. Spin Interaction Term:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">So far we have neglected the spin and hence the magnetic moment of the electron in its interaction with the radiation field. If we consider the spin of the electron then we have one additional term in the interaction ??;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-215\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125.png\" alt=\"\" width=\"676\" height=\"89\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125.png 676w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125-300x39.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125-225x30.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-125-350x46.png 350w\" sizes=\"auto, (max-width: 676px) 100vw, 676px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-216\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126.png\" alt=\"\" width=\"690\" height=\"552\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126.png 690w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126-300x240.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126-65x52.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126-225x180.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-126-350x280.png 350w\" sizes=\"auto, (max-width: 690px) 100vw, 690px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-217\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127.png\" alt=\"\" width=\"677\" height=\"197\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127.png 677w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127-300x87.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127-65x19.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127-225x65.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-127-350x102.png 350w\" sizes=\"auto, (max-width: 677px) 100vw, 677px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">4.1 Magnetic Dipole and Quadrupole Transitions<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">Occasionally we may have (?\u0305)BA = <\/span>0 ,<span style=\"text-align: initial;font-size: 1em\"> for every state B with energy lower than that of the state A. This may be due to the selection rules. In these <\/span>cases<span style=\"text-align: initial;font-size: 1em\"> we then go back to the plane wave expansion<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-218\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128.png\" alt=\"\" width=\"668\" height=\"533\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128.png 668w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128-300x239.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128-65x52.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128-225x180.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-128-350x279.png 350w\" sizes=\"auto, (max-width: 668px) 100vw, 668px\" \/><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The first term of equation ( 4.3 ) has (k\u20d7\u20d7 \u00d7\u2107\u20d7\u20d7\u20d7\u03bb)) which is the first term in the expansion of curl A and hence denotes the magnetic field B. L is related to the angular momentum and hence can be expressed in terms of the magnetic moment \u03bc also. The\u00a0first term becomes<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-220\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129.png\" alt=\"\" width=\"435\" height=\"48\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129.png 435w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129-300x33.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129-65x7.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129-225x25.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-129-350x39.png 350w\" sizes=\"auto, (max-width: 435px) 100vw, 435px\" \/><\/p>\n<div>\n<p style=\"text-align: justify\">where ?\u20d7 is the magnetic moment of the electron. Thus ( 4.4 ) is the interaction of magnetic moment of the charged particle with the magnetic field ?\u20d7\u20d7. This term is called M1 or magnetic dipole term.<\/p>\n<\/div>\n<p style=\"text-align: justify\">The second term of equation (4.3 ) is called E2 or Electric quadrupole term. To simplify this term we now use the relation [?\u20d72, ?\u20d7] = \u22122??\u20d7 ??, 2?[?0, ?\u20d7] = \u22122??\u20d7 , where ?0 is the free Hamiltonian.<\/p>\n<p>&nbsp;<\/p>\n<p>Thus ?\u20d7 = ??[?0, ?\u20d7] \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(4.5)<\/p>\n<p>&nbsp;<\/p>\n<p>Using (4.5) we write the second term of eq. (4.3) as<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-221\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130.png\" alt=\"\" width=\"622\" height=\"244\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130.png 622w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130-300x118.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130-225x88.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-130-350x137.png 350w\" sizes=\"auto, (max-width: 622px) 100vw, 622px\" \/><\/p>\n<p>This term gives rise to the electric quadrupole or E2 term. To see this in another way we go back to another expansion for plane wave .<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.2 Another Expansion of Plane Wave<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>We can write<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-222\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131.png\" alt=\"\" width=\"651\" height=\"168\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131.png 651w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131-300x77.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131-65x17.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131-225x58.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-131-350x90.png 350w\" sizes=\"auto, (max-width: 651px) 100vw, 651px\" \/><\/p>\n<div>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 \u00a0with the <\/span><strong style=\"text-align: initial;font-size: 1em\">p.<\/strong><span style=\"text-align: initial;font-size: 1em\"> term this will give a product of two\u00a0 <\/span><em style=\"text-align: initial;font-size: 1em\">l=<\/em><span style=\"text-align: initial;font-size: 1em\">1\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">terms. The resulting term will have <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\">=2,1 or 0. The quadrupole term is the one with\u00a0 <\/span><em style=\"text-align: initial;font-size: 1em\">l<\/em><span style=\"text-align: initial;font-size: 1em\">=2.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">This results in <\/span>emission<span style=\"text-align: initial;font-size: 1em\"> of quadrupole term E2.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">5.\u00a0 <\/strong><strong style=\"text-align: initial;font-size: 1em\">Derivation of Planck\u2019s Law from field theory.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Consider a system of atoms in thermodynamic equilibrium<\/span>. .<span style=\"text-align: initial;font-size: 1em\"> The atom in a state A emits a photon of energy \u03c9 and goes to state B. The atom in state A absorbs a photon of\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">energy\u00a0\u00a0 and goes to state B.\u00a0\u00a0 The processes are<\/span><\/p>\n<\/div>\n<div>\n<p>\u00a0 \u00a0 ? \u2192 ? + ?, ? + ? \u2192 ? , ?? ? \u21c4 ? + ? \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(5.1)<\/p>\n<p>&nbsp;<\/p>\n<p>Let EA be the energy of the atom A and EB that of B then<\/p>\n<p>&nbsp;<\/p>\n<p>?? = ?? + ?<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Let the population (number) of atoms in state A and B be N(A) and N(B) respectively. The transition probability ??\u2212? is given by Golden rule ( Module 6) which for a transition is<\/p>\n<p>&nbsp;<\/p>\n<p>??\u2192? = 2?|??? |<sup>2<\/sup> \u03b4(?? \u2212 ?? \u2212 ?) \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (5.2)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In the above expression ??? is the matrix element of the interaction Hamiltonian (i.e. perturbation theory is assumed to be valid ). In the equilibrium state<\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-223\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132.png\" alt=\"\" width=\"590\" height=\"125\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132.png 590w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132-300x64.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132-65x14.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132-225x48.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-132-350x74.png 350w\" sizes=\"auto, (max-width: 590px) 100vw, 590px\" \/><\/p>\n<div>\n<p>In the above ?? is Boltzman Constant and T the equilibrium temperature. If \u2130\u20d7? is the polarization vector of the photon of energy ? and momentum k then,<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-225\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133.png\" alt=\"\" width=\"577\" height=\"546\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133.png 577w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133-300x284.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133-65x62.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133-225x213.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-133-350x331.png 350w\" sizes=\"auto, (max-width: 577px) 100vw, 577px\" \/><\/p>\n<p>The number of states dn in a black body filled with radiation is obtained using the same arguments as for ?? and is<\/p>\n<p>? 2.4??<sup>2<\/sup> ??\/(2?)<sup>3<\/sup>. = 8??<sup>2<\/sup>???\/(2?)<sup>3<\/sup> \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(5.9)<\/p>\n<p style=\"text-align: justify\">The factor 2 is due to the availability of two possible directions in which light can be polarized. Thus the\u00a0 number of photons with an energy from ? and ? + ?? in the volume V is<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-226\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134.png\" alt=\"\" width=\"603\" height=\"108\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134.png 603w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134-300x54.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134-225x40.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-134-350x63.png 350w\" sizes=\"auto, (max-width: 603px) 100vw, 603px\" \/><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-227\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135.png\" alt=\"\" width=\"687\" height=\"299\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135.png 687w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135-300x131.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135-65x28.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135-225x98.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-135-350x152.png 350w\" sizes=\"auto, (max-width: 687px) 100vw, 687px\" \/><\/p>\n<p>which is Planck\u2019s radiation Law for determining the energy density ?( \u03c9, ?) or U (\u03bd, ?) of black body radiation,<\/p>\n<p>&nbsp;<\/p>\n<p><strong>6.\u00a0\u00a0\u00a0 <\/strong><strong>Einstein\u2019s Derivation of Planck\u2019s Law and Spontaneous emission<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">This derivation was given by Einstein in 1917 where he first identified the factor for spontaneous emission. Let the initial state of the electron of the atom be n with energy En . After absorbing a photon of energy it has a state m with energy Em , then<\/p>\n<p>&nbsp;<\/p>\n<p>?? = ?? + ? \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212(6.1)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Let population (number) of the atoms in state m and n be Nm and Nn respectively.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">Einstein realized that the probability per unit time per atom for the emission transition\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">must have two parts. We denote the spontaneous emission by A, which does not need\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">any radiation to be present and the stimulated emission by ??(?), which is proportional\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">to the amount of radiation ?(?) present. The absorbed emission is denoted by ?\u2032?(?)\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">and depends on the radiation present. In thermodynamic equilibrium we have<\/span><\/p>\n<\/div>\n<p>???\u2032?(?) = ??[? + ??(?)] \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (6.2)<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-228\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136.png\" alt=\"\" width=\"662\" height=\"500\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136.png 662w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136-300x227.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136-65x49.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136-225x170.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-136-350x264.png 350w\" sizes=\"auto, (max-width: 662px) 100vw, 662px\" \/><\/strong><\/p>\n<div>\n<p>Einstein argued that from the principle of detailed balance ?\u2032 = ?. To derive the law from field theory we had used hermiticity of the Hamiltonian . Comparing (6.5) with Planck\u2019s law (5.10) we get A = B ( ?3 \/ ?2 )<\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">7. Bose Derivation of Planck Law<\/strong><\/p>\n<\/div>\n<p style=\"text-align: justify\">Using statistical mechanics Bose calculated the probability for distributing ?? photons over ?? cells, all having energy ?? . Since there can be any number of photons in one cell this corresponds to inserting ?? -1 partitions between ?? photons. If we have ?? =4 we have<\/p>\n<p style=\"text-align: center\">\u2026\u2026.|\u2026|\u2026..|\u2026\u2026\u2026.<\/p>\n<p style=\"text-align: center\">?1 ?2 ?3<\/p>\n<p style=\"text-align: justify\">with dots showing photons and vertical lines partitions ?1 , ?2and ?3.<\/p>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-230\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137.png\" alt=\"\" width=\"665\" height=\"409\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137.png 665w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137-300x185.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137-65x40.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137-225x138.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-137-350x215.png 350w\" sizes=\"auto, (max-width: 665px) 100vw, 665px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Bose\u2019s paper was rejected by a journal for publication. So he sent it in 1924 to Prof. Einstein requesting him to forward it for publication if he thought it was correct. Einstein translated the paper into German language and sent it for publication and wrote a few more papers applying the Bose method of counting to normal gases. This was the birth of systems obeying Bose-Einstein statistics. We now know that integral (including zero spin ) particles obey Bose statistics and are called Bosons . We saw in the earlier module that bosons, scalar particles satisfy commutation relation when quantized. Photons have spin 1 and so are bosons. They do not have 2S + 1 or 3 components, but have only two components or polarisations as they have zero mass. Zero mass implies gauge constraints and we have seen how gauge constraints reduce the degrees of freedom to two polarisations.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">There were criticisms that the counting of states by Bose and Einstein violated the independent nature of particles. Einstein accepted this criticism but said that\u00a0<span style=\"text-align: initial;font-size: 1em\">as the result was right they have to find out the reason for\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">this mysterious attraction between <\/span>photons .<span style=\"text-align: initial;font-size: 1em\"> We now know that symmetry of boson wave function brings in an attraction just as that of anti-symmetry of fermions (particles with <\/span>half integral<span style=\"text-align: initial;font-size: 1em\"> spins) brings in a repulsion leading to Pauli principle. This was finally explained by P.A.M.Dirac in 1926. The statistics obeyed by Fermions is called Fermi-Dirac Statistics.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">Planck introduced the photon in 1900 and Einstein gave the <\/span>photo electric<span style=\"text-align: initial;font-size: 1em\"> effect in 1905. But even in 1924, <\/span>photon<span style=\"text-align: initial;font-size: 1em\"> was still not accepted as a particle and Einstein expressed surprise at the way Bose had derived the number of <\/span>states .<span style=\"text-align: initial;font-size: 1em\"> It took a long time 27 years for dual nature of light to be accepted. Abraham Pais in his book on <\/span>Eistein<span style=\"text-align: initial;font-size: 1em\"> and his work mentions four <\/span>path breaking<span style=\"text-align: initial;font-size: 1em\"> papers published before <\/span>quantum<span style=\"text-align: initial;font-size: 1em\"> revolution in 1926. They are by Max Planck (1900), Albert Einstein (1905), Niels Bohr (1911) and Satyendra Nath Bose (1924).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"font-size: 1em\">8. Summary<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In this <\/span>module<span style=\"text-align: initial;font-size: 1em\"> we have studied the higher order interactions like spin interaction, electric quadrupole <\/span>and<span style=\"text-align: initial;font-size: 1em\"> magnetic dipole interactions. We find these higher order terms are thousand times smaller as they are proportional to the ratio of the radius of the atom to the <\/span>wave length<span style=\"text-align: initial;font-size: 1em\"> of light. We have <\/span>learnt<span style=\"text-align: initial;font-size: 1em\"> how Planck\u2019s Law can be derived from quantum field theory. We have also <\/span>learnt<span style=\"text-align: initial;font-size: 1em\"> how Einstein derived the Planck\u2019s law from Bohr\u2019s theory in 1917, and Satyendra Nath Bose derived it from statistical arguments in 1924 before the discovery of quantum mechanics in <\/span>1926,<span style=\"text-align: initial;font-size: 1em\"> and became the first author on quantum statistics.<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Higher Order Interactions_Landmarks on the way to Quantum Field Theory<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/KYt2lAHiwjU\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-120\" src=\"http:\/\/epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/2018\/11\/download.png\" alt=\"\" width=\"36\" height=\"36\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div>\n<p><strong>\u00a0 \u00a0 Learn More Books<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>1.\u00a0 Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education Singapore (1998)<\/p>\n<p>2.\u00a0 Quantum Mechanics Vol.3 by L.D. Landau and Lifshitz (Pergramon Press Oxford, Reprinted<\/p>\n<p>3.\u00a0 \u201cSubtle is the Lord , The science and Life of Albert Einstein\u201d , Oxford University Press, 1982<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Web sites<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>For history see<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.physics.udel.edu\/~msafrono\/626\/Lectures%2013-14.pdf\">www.physics.udel.edu\/~msafrono\/626\/Lectures%2013-14.pdf<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"http:\/\/www.colorado.edu\/entities\/quantum-field-theory\/qft.history.html\">www.colorado.edu\/entities\/quantum-field-theory\/qft.history.html<\/a><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Photons not being independent<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Dirac, in his book on Quantum Mechanics (1947 edition) explains this very simply.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">I describe this in my own words. If you have two coins and toss them. The probability for two heads to appear is one quarter or \u00bc . This is because it is one possibility among four.<\/p>\n<p>&nbsp;<\/p>\n<p>2 heads, 2 tails, one head one tail or one tail and one head.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">When they have to be symmetric wave functions, then only the sum of the last two is symmetric. The difference is antisymmetric.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">So there are only three possibilities. Hence the probability for two heads or two tails is one third or 1\/3. This is more than \u00bc, what it was earlier. The probability has gone up due to attraction due to Bose-Einstein Statistics.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>True Story<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">When S.N.Bose was taking Mr. and Mrs, Dirac, in car , home for lunch, the Diracs sat at the back and Bose and driver were in front. When another student wanted also to come Bose took him in front. Mrs Dirac objected and said he should come back. Dirac remarked :<span style=\"text-align: initial;font-size: 1em\"> it is all a question of statistics.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">( Dirac statistics repels and Bose statistics attracts.)<\/span><\/p>\n<\/div>\n","protected":false},"author":3,"menu_order":9,"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-211","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\/211","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\/211\/revisions"}],"predecessor-version":[{"id":267,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/211\/revisions\/267"}],"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\/211\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/media?parent=211"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapter-type?post=211"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/contributor?post=211"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/license?post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}