{"id":173,"date":"2018-11-14T08:49:50","date_gmt":"2018-11-14T08:49:50","guid":{"rendered":"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=173"},"modified":"2019-04-30T09:36:08","modified_gmt":"2019-04-30T09:36:08","slug":"absorption-and-emission-of-light-by-atomic-electrons","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/chapter\/absorption-and-emission-of-light-by-atomic-electrons\/","title":{"rendered":"Absorption and Emission of light by atomic electrons"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/3mZEWJXH9wk\" 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&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>\u00a0 \u00a0 1.\u00a0<\/strong><strong>Learning Outcomes<\/strong>\r\n<ul>\r\n \t<li>Learn how the probabilities for absorption and emission of light by electrons in an atom are calculated<\/li>\r\n \t<li>Understand how the 1 in ( ??,? +1) corresponds to spontaneous emission and ??,?<span style=\"text-align: initial;font-size: 1em\">\u00a0corresponds to induced or stimulated emission parts.<\/span><\/li>\r\n \t<li>Calculate the quantum expressions for the radiated energy and compare it with that of classical energy.<\/li>\r\n \t<li>Learn to evaluate the binding energy of hydrogen atom in ev<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\r\n \t<li>Learn to calculate in natural units ,<span style=\"text-align: initial;font-size: 1em\"> where \u0127 = 1, c=1.<\/span><\/li>\r\n<\/ul>\r\n<strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 2. Introduction<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In the previous <\/span>module<span style=\"text-align: initial;font-size: 1em\"> we had studied the induced emission, absorption and spontaneous emission of <\/span>photon<span style=\"text-align: initial;font-size: 1em\"> by the non-relativistic electron in a broader sense. We have so far not considered the details of the approximations involved in obtaining transitions probabilities per unit for these processes. We will now study in details the spontaneous emission process in the Dipole approximations. We shall study other interactions like spin interaction term, the quadrupole <\/span>and<span style=\"text-align: initial;font-size: 1em\"> magnetic dipole approximations later on. In studying these processes we shall see how certain selection rules emerge which allow some transitions and forbid other transitions. Closely related to the transition probability per unit time is the <\/span>life time<span style=\"text-align: initial;font-size: 1em\"> of emitted states and we shall take up an example of the same.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">We are treating electron <\/span>non-relativistically ,<span style=\"text-align: initial;font-size: 1em\"> so the kinetic energy of the electron is less than its rest mass. <\/span>Also<span style=\"text-align: initial;font-size: 1em\"> the radiation energy of photons is in the visible range. Thus, we may give some justification in retaining some terms and ignoring others. In the dipole <\/span>approximation<span style=\"text-align: initial;font-size: 1em\"> we <\/span>expand,<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-177 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99.png\" alt=\"\" width=\"675\" height=\"97\" \/>\r\n\r\n<\/div>\r\n<p style=\"text-align: justify\"><strong>\u00a0<\/strong><span style=\"text-align: initial;font-size: 1em\">This is justified as the value of <\/span>radius<span style=\"text-align: initial;font-size: 1em\"> of atom R is 10<\/span><sup style=\"text-align: initial\">\u22128\u00a0<\/sup><span style=\"text-align: initial;font-size: 1em\">cm while the <\/span>wave length<span style=\"text-align: initial;font-size: 1em\"> of the photon <\/span>is ,<span style=\"text-align: initial;font-size: 1em\"> in the visible range is ~ 10<\/span><sup style=\"text-align: initial\">\u22125<\/sup> cm . <em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\"> ~ 1\/ and so <\/span><em style=\"text-align: initial;font-size: 1em\">kR.<\/em><span style=\"text-align: initial;font-size: 1em\"> the maximum value of <\/span><em style=\"text-align: initial;font-size: 1em\">kx<\/em><span style=\"text-align: initial;font-size: 1em\"> is ~ 10<\/span><sup style=\"text-align: initial\">\u22123<\/sup><span style=\"text-align: initial;font-size: 1em\"> &lt;&lt; 1. So the terms in <\/span>eq<span style=\"text-align: initial;font-size: 1em\">. 2.1 are successively smaller.<\/span><\/p>\r\n\r\n<div>\r\n\r\n|???| \u2243 105 \u00d7 10\u22128 \u2243 10\u22123 \u226a 1. \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (2.2)\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">On the other hand if a certain transition is not allowed in dipole approximation due to the selection rules then we shall include the ( ?? \u20d7 \u2219 ? ) term also.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In section 3, we study the absorption and emission of Light. We study the spontaneous emission process in the dipole approximation in section 4. We then consider the classical formula for radiated energy of an electron and compare it with the quantum one. In section 5 we learn to calculate in natural units in which \u0127=1, c=1. These are very useful in quantum field theory.<\/p>\r\n&nbsp;\r\n\r\n<strong>3.<\/strong>\u00a0\u00a0\u00a0 <strong>Absorption and Emission of Light<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">We now have the requisite mechanism to deal with the absorption of photons by non-relativistic atomic electrons. In absorption process an atom in a state A absorbs a is absorbed the term 2 \u2219 in the interaction term does not contribute as it involves the participation of two photons. We shall assume that photons of one kind (\u00a0 ? \u20d7 , \u2130 ? ) are present. If there are \u00a0,\u00a0 \u00a0photons in the initial state, then there are , ??,? \u2212 1\u00a0photons in the final state. Thus \u00a0,\u00a0 \u00a0in the expression for ? (? , ?) contributes to the matrix element of HI. Denoting the initial state by | |?; ??,?\u232a and the final state by |?; ??,? \u2212 1\u232a, we have from equation (3.5) of Module 6, (dropping 2 term)\u00a0<span style=\"text-align: initial;font-size: 1em\">From Module 6 eq. 3.14 and <\/span>3.15 ,<span style=\"text-align: initial;font-size: 1em\"> we have the matrix element for absorption<\/span><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-178 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100.png\" alt=\"\" width=\"680\" height=\"572\" \/>\r\n\r\n&nbsp;\r\n\r\nFor emission the procedure is quite similar and we have using eq. (3.2) the transition probability\r\n\r\n<img class=\"alignnone wp-image-180 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101.png\" alt=\"\" width=\"672\" height=\"129\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>4.\u00a0<\/strong><strong>Spontaneous Emission in Dipole approximation<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">In spontaneous <\/span>emission<span style=\"text-align: initial;font-size: 1em\"> there is no radiation\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">present initially and in the absence of <\/span>the any<span style=\"text-align: initial;font-size: 1em\"> initial radiation i.e. = 0, the transition probability per unit time into the solid angle \u00a0\u03a9 is given by<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-181 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102.png\" alt=\"\" width=\"457\" height=\"114\" \/>\r\n\r\nAs explained in the introduction using dipole approximation we can put\u00a0 ?\u2212?? \u20d7 \u2219? = 1.\r\n\r\n<img class=\"alignnone wp-image-182 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103.png\" alt=\"\" width=\"668\" height=\"51\" \/>\r\n\r\n<strong>4.1 Dipole Approximation<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">When the wavelength of the emitting photon (? \u20d7 , ?) is much larger than the range R, we may use the equation (4.2) for the transition probability. We can simplify this equation further by the following procedure. From uncertainty relation<\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-183 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104.png\" alt=\"\" width=\"682\" height=\"444\" \/>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-184 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105.png\" alt=\"\" width=\"673\" height=\"99\" \/>\r\n\r\nLet us take \u20d7 vector along z axis and let ? makes an angle ? with respect to ? \u20d7 axis. As \u2130\u00a0\u00a0 1 and \u2130 2 polarization vectors are in theplane, let us choose them along x-axis and y-axis respectively as shown in the diagram\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone wp-image-185 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106.png\" alt=\"\" width=\"574\" height=\"575\" \/>\r\n\r\n<\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-186 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107.png\" alt=\"\" width=\"705\" height=\"399\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone wp-image-187 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108.png\" alt=\"\" width=\"675\" height=\"521\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone wp-image-188 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109.png\" alt=\"\" width=\"690\" height=\"551\" \/>\r\n\r\n<img class=\"alignnone wp-image-189 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110.png\" alt=\"\" width=\"677\" height=\"253\" \/>\r\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If we want the time corresponding to this we have to\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">divide by c which is length by time. This <\/span>time ,<span style=\"text-align: initial;font-size: 1em\"> which is the time taken by light to cross \/<\/span>2\u03c0 ,<span style=\"text-align: initial;font-size: 1em\"> is equal to 1.29 x 10\u221221 s or sec. We have taken \u0127 and c in CGS units.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">For MKS (Metre, Kilogram Second ) units we have to take the values\u00a0 h = 6.6 x 10=34 J\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">s and c = 3x 108\u00a0 m\/sec . We shall mostly use CGS units.<\/span><\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">5.2 Hydrogen spectral energy values \/ Binding Energy<\/strong><\/p>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-190\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111.png\" alt=\"\" width=\"684\" height=\"266\" \/>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">For atoms with nuclear charge Ze the energy level is multiplied by 2. So the electron in Helium is much stronger, 54.4 ev. We shall use some of these values in the next module.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">Often mass of electron or proton is quoted in Mev. This is not mass but energy units. The rest mass is multiplied by 2 to get th energy unit. So if energy unit is used to get length we have to multiply by \u0127c. ( For mass we multiply by \u0127\/c. ). Energy units are often more convenient to use.<\/p>\r\n\r\n<\/div>\r\n<ol start=\"6\">\r\n \t<li><strong>Summary<\/strong><\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">We have obtained the matrix elements for the absorption and emission of light by electrons in atoms by using the plane wave expansion for the vector potential and identifying the number eigen states. In the case of emission there is emission even in the absence of radiation (?? = 0). This is called Spontaneous emission and was firsr identified by Einstein as we shall discuss later in module 9.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We then expressed the matrix element in the dipole approximation where it becomes the expectation value of <strong>x. ? .<\/strong>\u00a0We recall the classical expression for radiation by an accelerated electron and compare it with the dipole approximation expression. Finally we discussed how to work in natural units with \u0127 =c = 1.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on Absorption and Emission of light by atomic electrons<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/3mZEWJXH9wk\" 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<strong>Learn More<\/strong>\r\n\r\n&nbsp;\r\n\r\nBooks:\r\n<ol>\r\n \t<li>Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education, Singapore 1998)<\/li>\r\n<\/ol>\r\n<ol start=\"2\">\r\n \t<li>Quantum Mechanics Vol. 3 by L.D. Landau and E.M. Lifshitz (Pergamon Press, Oxford, Reprinted 1981)<\/li>\r\n<\/ol>\r\n<ol start=\"3\">\r\n \t<li>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<strong>\u00a0 \u00a0 Web Links<\/strong>\r\n\r\n&nbsp;\r\n\r\nLifetime of excited state;\u00a0 https\/\/en\/wikipedia.org\/wiki\/Excited state\r\n\r\n&nbsp;\r\n\r\nfarsideph.utexas.edu\/teaching\/qmech\/Quantum\/node122.html\r\n\r\n&nbsp;\r\n\r\n<strong>Interesting Facts<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">The calculations in quantum mechanics are largely in perturbation theory. In classical mechanics we solve differential equations and imposing initial and boundary conditions predict the actual path of a particle. Perturbation method would take an actual path and calculate the perturbation or small change in path due to small perturbations.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">In quantum mechanics also, the differential equation is solved for getting the unperturbed eigen functions, which are then used for calculating the change due to perturbations.<\/p>\r\n&nbsp;","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/3mZEWJXH9wk\" 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>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>\u00a0 \u00a0 1.\u00a0<\/strong><strong>Learning Outcomes<\/strong><\/p>\n<ul>\n<li>Learn how the probabilities for absorption and emission of light by electrons in an atom are calculated<\/li>\n<li>Understand how the 1 in ( ??,? +1) corresponds to spontaneous emission and ??,?<span style=\"text-align: initial;font-size: 1em\">\u00a0corresponds to induced or stimulated emission parts.<\/span><\/li>\n<li>Calculate the quantum expressions for the radiated energy and compare it with that of classical energy.<\/li>\n<li>Learn to evaluate the binding energy of hydrogen atom in ev<span style=\"text-align: initial;font-size: 1em\">.<\/span><\/li>\n<li>Learn to calculate in natural units ,<span style=\"text-align: initial;font-size: 1em\"> where \u0127 = 1, c=1.<\/span><\/li>\n<\/ul>\n<p><strong style=\"text-align: initial;font-size: 1em\">\u00a0 \u00a0 2. Introduction<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">In the previous <\/span>module<span style=\"text-align: initial;font-size: 1em\"> we had studied the induced emission, absorption and spontaneous emission of <\/span>photon<span style=\"text-align: initial;font-size: 1em\"> by the non-relativistic electron in a broader sense. We have so far not considered the details of the approximations involved in obtaining transitions probabilities per unit for these processes. We will now study in details the spontaneous emission process in the Dipole approximations. We shall study other interactions like spin interaction term, the quadrupole <\/span>and<span style=\"text-align: initial;font-size: 1em\"> magnetic dipole approximations later on. In studying these processes we shall see how certain selection rules emerge which allow some transitions and forbid other transitions. Closely related to the transition probability per unit time is the <\/span>life time<span style=\"text-align: initial;font-size: 1em\"> of emitted states and we shall take up an example of the same.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">We are treating electron <\/span>non-relativistically ,<span style=\"text-align: initial;font-size: 1em\"> so the kinetic energy of the electron is less than its rest mass. <\/span>Also<span style=\"text-align: initial;font-size: 1em\"> the radiation energy of photons is in the visible range. Thus, we may give some justification in retaining some terms and ignoring others. In the dipole <\/span>approximation<span style=\"text-align: initial;font-size: 1em\"> we <\/span>expand,<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-177 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99.png\" alt=\"\" width=\"675\" height=\"97\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99.png 675w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99-300x43.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99-65x9.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99-225x32.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-99-350x50.png 350w\" sizes=\"auto, (max-width: 675px) 100vw, 675px\" \/><\/p>\n<\/div>\n<p style=\"text-align: justify\"><strong>\u00a0<\/strong><span style=\"text-align: initial;font-size: 1em\">This is justified as the value of <\/span>radius<span style=\"text-align: initial;font-size: 1em\"> of atom R is 10<\/span><sup style=\"text-align: initial\">\u22128\u00a0<\/sup><span style=\"text-align: initial;font-size: 1em\">cm while the <\/span>wave length<span style=\"text-align: initial;font-size: 1em\"> of the photon <\/span>is ,<span style=\"text-align: initial;font-size: 1em\"> in the visible range is ~ 10<\/span><sup style=\"text-align: initial\">\u22125<\/sup> cm . <em style=\"text-align: initial;font-size: 1em\">k<\/em><span style=\"text-align: initial;font-size: 1em\"> ~ 1\/ and so <\/span><em style=\"text-align: initial;font-size: 1em\">kR.<\/em><span style=\"text-align: initial;font-size: 1em\"> the maximum value of <\/span><em style=\"text-align: initial;font-size: 1em\">kx<\/em><span style=\"text-align: initial;font-size: 1em\"> is ~ 10<\/span><sup style=\"text-align: initial\">\u22123<\/sup><span style=\"text-align: initial;font-size: 1em\"> &lt;&lt; 1. So the terms in <\/span>eq<span style=\"text-align: initial;font-size: 1em\">. 2.1 are successively smaller.<\/span><\/p>\n<div>\n<p>|???| \u2243 105 \u00d7 10\u22128 \u2243 10\u22123 \u226a 1. \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (2.2)<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">On the other hand if a certain transition is not allowed in dipole approximation due to the selection rules then we shall include the ( ?? \u20d7 \u2219 ? ) term also.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In section 3, we study the absorption and emission of Light. We study the spontaneous emission process in the dipole approximation in section 4. We then consider the classical formula for radiated energy of an electron and compare it with the quantum one. In section 5 we learn to calculate in natural units in which \u0127=1, c=1. These are very useful in quantum field theory.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.<\/strong>\u00a0\u00a0\u00a0 <strong>Absorption and Emission of Light<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We now have the requisite mechanism to deal with the absorption of photons by non-relativistic atomic electrons. In absorption process an atom in a state A absorbs a is absorbed the term 2 \u2219 in the interaction term does not contribute as it involves the participation of two photons. We shall assume that photons of one kind (\u00a0 ? \u20d7 , \u2130 ? ) are present. If there are \u00a0,\u00a0 \u00a0photons in the initial state, then there are , ??,? \u2212 1\u00a0photons in the final state. Thus \u00a0,\u00a0 \u00a0in the expression for ? (? , ?) contributes to the matrix element of HI. Denoting the initial state by | |?; ??,?\u232a and the final state by |?; ??,? \u2212 1\u232a, we have from equation (3.5) of Module 6, (dropping 2 term)\u00a0<span style=\"text-align: initial;font-size: 1em\">From Module 6 eq. 3.14 and <\/span>3.15 ,<span style=\"text-align: initial;font-size: 1em\"> we have the matrix element for absorption<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-178 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100.png\" alt=\"\" width=\"680\" height=\"572\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100.png 680w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100-300x252.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100-65x55.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100-225x189.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-100-350x294.png 350w\" sizes=\"auto, (max-width: 680px) 100vw, 680px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>For emission the procedure is quite similar and we have using eq. (3.2) the transition probability<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-180 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101.png\" alt=\"\" width=\"672\" height=\"129\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101.png 672w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101-300x58.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101-65x12.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101-225x43.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-101-350x67.png 350w\" sizes=\"auto, (max-width: 672px) 100vw, 672px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.\u00a0<\/strong><strong>Spontaneous Emission in Dipole approximation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">In spontaneous <\/span>emission<span style=\"text-align: initial;font-size: 1em\"> there is no radiation\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">present initially and in the absence of <\/span>the any<span style=\"text-align: initial;font-size: 1em\"> initial radiation i.e. = 0, the transition probability per unit time into the solid angle \u00a0\u03a9 is given by<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-181 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102.png\" alt=\"\" width=\"457\" height=\"114\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102.png 457w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102-300x75.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102-65x16.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102-225x56.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-102-350x87.png 350w\" sizes=\"auto, (max-width: 457px) 100vw, 457px\" \/><\/p>\n<p>As explained in the introduction using dipole approximation we can put\u00a0 ?\u2212?? \u20d7 \u2219? = 1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-182 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103.png\" alt=\"\" width=\"668\" height=\"51\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103.png 668w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103-300x23.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103-65x5.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103-225x17.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-103-350x27.png 350w\" sizes=\"auto, (max-width: 668px) 100vw, 668px\" \/><\/p>\n<p><strong>4.1 Dipole Approximation<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">When the wavelength of the emitting photon (? \u20d7 , ?) is much larger than the range R, we may use the equation (4.2) for the transition probability. We can simplify this equation further by the following procedure. From uncertainty relation<\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-183 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104.png\" alt=\"\" width=\"682\" height=\"444\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104.png 682w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104-300x195.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104-65x42.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104-225x146.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-104-350x228.png 350w\" sizes=\"auto, (max-width: 682px) 100vw, 682px\" \/><\/p>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-184 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105.png\" alt=\"\" width=\"673\" height=\"99\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105.png 673w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105-300x44.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105-65x10.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105-225x33.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-105-350x51.png 350w\" sizes=\"auto, (max-width: 673px) 100vw, 673px\" \/><\/p>\n<p>Let us take \u20d7 vector along z axis and let ? makes an angle ? with respect to ? \u20d7 axis. As \u2130\u00a0\u00a0 1 and \u2130 2 polarization vectors are in theplane, let us choose them along x-axis and y-axis respectively as shown in the diagram<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-185 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106.png\" alt=\"\" width=\"574\" height=\"575\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106.png 574w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106-150x150.png 150w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106-300x300.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106-65x65.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106-225x225.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-106-350x351.png 350w\" sizes=\"auto, (max-width: 574px) 100vw, 574px\" \/><\/p>\n<\/div>\n<div><\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-186 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107.png\" alt=\"\" width=\"705\" height=\"399\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107.png 705w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107-300x170.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107-65x37.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107-225x127.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-107-350x198.png 350w\" sizes=\"auto, (max-width: 705px) 100vw, 705px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-187 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108.png\" alt=\"\" width=\"675\" height=\"521\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108.png 675w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108-300x232.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108-65x50.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108-225x174.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-108-350x270.png 350w\" sizes=\"auto, (max-width: 675px) 100vw, 675px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-188 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109.png\" alt=\"\" width=\"690\" height=\"551\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109.png 690w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109-300x240.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109-65x52.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109-225x180.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-109-350x279.png 350w\" sizes=\"auto, (max-width: 690px) 100vw, 690px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-189 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110.png\" alt=\"\" width=\"677\" height=\"253\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110.png 677w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110-300x112.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110-65x24.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110-225x84.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-110-350x131.png 350w\" sizes=\"auto, (max-width: 677px) 100vw, 677px\" \/><\/p>\n<p style=\"text-align: justify\"><span style=\"text-align: initial;font-size: 1em\">If we want the time corresponding to this we have to\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">divide by c which is length by time. This <\/span>time ,<span style=\"text-align: initial;font-size: 1em\"> which is the time taken by light to cross \/<\/span>2\u03c0 ,<span style=\"text-align: initial;font-size: 1em\"> is equal to 1.29 x 10\u221221 s or sec. We have taken \u0127 and c in CGS units.\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">For MKS (Metre, Kilogram Second ) units we have to take the values\u00a0 h = 6.6 x 10=34 J\u00a0<\/span><span style=\"text-align: initial;font-size: 1em\">s and c = 3x 108\u00a0 m\/sec . We shall mostly use CGS units.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\"><strong style=\"text-align: initial;font-size: 1em\">5.2 Hydrogen spectral energy values \/ Binding Energy<\/strong><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-190\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111.png\" alt=\"\" width=\"684\" height=\"266\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111.png 684w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111-300x117.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111-225x88.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-111-350x136.png 350w\" sizes=\"auto, (max-width: 684px) 100vw, 684px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">For atoms with nuclear charge Ze the energy level is multiplied by 2. So the electron in Helium is much stronger, 54.4 ev. We shall use some of these values in the next module.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">Often mass of electron or proton is quoted in Mev. This is not mass but energy units. The rest mass is multiplied by 2 to get th energy unit. So if energy unit is used to get length we have to multiply by \u0127c. ( For mass we multiply by \u0127\/c. ). Energy units are often more convenient to use.<\/p>\n<\/div>\n<ol start=\"6\">\n<li><strong>Summary<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify\">We have obtained the matrix elements for the absorption and emission of light by electrons in atoms by using the plane wave expansion for the vector potential and identifying the number eigen states. In the case of emission there is emission even in the absence of radiation (?? = 0). This is called Spontaneous emission and was firsr identified by Einstein as we shall discuss later in module 9.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We then expressed the matrix element in the dipole approximation where it becomes the expectation value of <strong>x. ? .<\/strong>\u00a0We recall the classical expression for radiation by an accelerated electron and compare it with the dipole approximation expression. Finally we discussed how to work in natural units with \u0127 =c = 1.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on Absorption and Emission of light by atomic electrons<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/3mZEWJXH9wk\" 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<p><strong>Learn More<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Books:<\/p>\n<ol>\n<li>Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education, Singapore 1998)<\/li>\n<\/ol>\n<ol start=\"2\">\n<li>Quantum Mechanics Vol. 3 by L.D. Landau and E.M. Lifshitz (Pergamon Press, Oxford, Reprinted 1981)<\/li>\n<\/ol>\n<ol start=\"3\">\n<li>Quantum Electrodynamics, Vol. 4 by V.B. Berestetskii, E.M. Lifshitz and L.P. Pitaevskii (Pergamon Press, Oxford, 1982)<\/li>\n<\/ol>\n<p><strong>\u00a0 \u00a0 Web Links<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>Lifetime of excited state;\u00a0 https\/\/en\/wikipedia.org\/wiki\/Excited state<\/p>\n<p>&nbsp;<\/p>\n<p>farsideph.utexas.edu\/teaching\/qmech\/Quantum\/node122.html<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Interesting Facts<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The calculations in quantum mechanics are largely in perturbation theory. In classical mechanics we solve differential equations and imposing initial and boundary conditions predict the actual path of a particle. Perturbation method would take an actual path and calculate the perturbation or small change in path due to small perturbations.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In quantum mechanics also, the differential equation is solved for getting the unperturbed eigen functions, which are then used for calculating the change due to perturbations.<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":7,"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-173","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\/173","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":7,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/173\/revisions"}],"predecessor-version":[{"id":263,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/173\/revisions\/263"}],"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\/173\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/media?parent=173"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapter-type?post=173"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/contributor?post=173"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/license?post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}