{"id":192,"date":"2018-11-14T10:55:23","date_gmt":"2018-11-14T10:55:23","guid":{"rendered":"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=192"},"modified":"2019-04-30T09:42:52","modified_gmt":"2019-04-30T09:42:52","slug":"application-to-h-atom","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/chapter\/application-to-h-atom\/","title":{"rendered":"Application to H atom"},"content":{"raw":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/esqEl3s_e9I\" 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 style=\"text-align: justify\">Learn how selection rules for transitions are obtained by evaluating matrix elements and finding when they are non zero.<\/li>\r\n \t<li>Learn how the life time<span style=\"text-align: initial;font-size: 1em\"> of an excited state of <\/span>hydrogen<span style=\"text-align: initial;font-size: 1em\"> atom is calculated. <\/span><\/li>\r\n \t<li><span style=\"text-align: initial;font-size: 1em\">Become familiar with <\/span>energy<span style=\"text-align: initial;font-size: 1em\"> values of the states of <\/span>hydrogen<span style=\"text-align: initial;font-size: 1em\"> atom.<\/span><\/li>\r\n \t<li>Learn a bit more about calculating with natural units.<\/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\">You have studied the energy spectrum of hydrogen atom. We would like to apply the framework developed so far to the concrete case of hydrogen atom. We shall consider first when a transition from the one state to another state is allowed. The states are labeled by total quantum number n, the orbital quantum number and the azimuthal or magnetic quantum number m, If the initial state has values of n, and m and final state has values of n<em>\u2019<\/em>, \u00a0\u00a0\u2032 and m\u2019 , we find that change =\u00b1 1 and m =0. \u00b1 1 are only allowed. These are called selection rules and are derived by using the properties of spherical harmonics or associated Legendre functions.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We next study the specific transition in hydrogen from the state 2P ( =1, the first excited state ) to the ground state 1S ( =0 ). As = -1 this transition is allowed, so also for change in m. We carry out the calculations using the well known expressions for the wave functions for both the states and obtain the transition probability. The lifetime of the excited state is just the inverse of this value.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">The calculations are specially simple if we use the natural units in which \u0127= c=1. We discuss the use of such units in calculations in the last section more fully.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center\">??????????? ?? ???????? ????<\/p>\r\n&nbsp;\r\n\r\n<strong>3.1 Selection Rules<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">In this section we apply our formalism to the electron bound in the hydrogen atom. We consider the structure of the matrix element<\/p>\r\n&nbsp;\r\n\r\n\u27e8?|?|?\u27e9 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (3.1)\r\n\r\n&nbsp;\r\n\r\nWhere the initial state A is characterized by the wave function\r\n\r\n&nbsp;\r\n\r\n\u27e8?|?\u27e9 = ???(?)???(?, ?) \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (3.2)\r\n\r\n&nbsp;\r\n\r\nThe final sate is characterized by\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">\u27e8?|?\u27e9 = ??\u2032?\u2032 (?)??\u2032?\u2032 (?, ?)\u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (3.3)<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone wp-image-195 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112.png\" alt=\"\" width=\"674\" height=\"462\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-196\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113.png\" alt=\"\" width=\"675\" height=\"399\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-197\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114.png\" alt=\"\" width=\"671\" height=\"592\" \/>\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-198\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115.png\" alt=\"\" width=\"679\" height=\"317\" \/>\r\n\r\n<img class=\"alignnone size-full wp-image-199\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116.png\" alt=\"\" width=\"673\" height=\"529\" \/>\r\n\r\n<\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-200\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117.png\" alt=\"\" width=\"645\" height=\"378\" \/>\r\n\r\n<img class=\"alignnone size-full wp-image-202\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119.png\" alt=\"\" width=\"664\" height=\"582\" \/>\r\n\r\n<\/div>\r\n<img class=\"alignnone size-full wp-image-203\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120.png\" alt=\"\" width=\"677\" height=\"321\" \/>\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">\u00a0 <\/span>\r\n<div><img class=\"alignnone size-full wp-image-205\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121.png\" alt=\"\" width=\"685\" height=\"538\" \/><\/div>\r\n<div><\/div>\r\n<div>\r\n\r\n<img class=\"alignnone size-full wp-image-206\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122.png\" alt=\"\" width=\"685\" height=\"396\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong style=\"text-align: initial;font-size: 1em\">3.3 Life Time of Excited states:<\/strong>\r\n\r\n&nbsp;\r\n\r\n<span style=\"text-align: initial;font-size: 1em\">As the transition probability has a dimension of <\/span>inverse<span style=\"text-align: initial;font-size: 1em\"> of time, we can find the l<\/span>ife time of<span style=\"text-align: initial;font-size: 1em\"> the excited state. We can thus calculate the <\/span>life time<span style=\"text-align: initial;font-size: 1em\"> of ? state\u00a0??\u2192?+?<\/span>\r\n\r\n<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<strong>4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Calculations using natural units<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe \u03bc meson ,or muon as it is often called, decays into three particles. These are the electron and two neutrinos one of electron type and the other of the muon type.\u00a0by the inverse of ??\u2192?+? of equation (3.34). Thus for mass of electron ? =\r\n\r\n<img class=\"alignnone size-full wp-image-207\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123.png\" alt=\"\" width=\"675\" height=\"148\" \/>\r\n\r\n<\/div>\r\nThis is called life time of the excited state 2p.\r\n\r\n&nbsp;\r\n\r\n<strong>4. Calculations using natural units<\/strong>\r\n\r\n&nbsp;\r\n\r\nThe \u03bc meson ,or muon as it is often called, decays into three particles. These are the electron and two neutrinos one of electron type ?? and the other of the muon type ??.\r\n\r\nIf life time ? of muon decay is given by the formula\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-208\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124.png\" alt=\"\" width=\"663\" height=\"336\" \/>\r\n\r\n? ~ 2 x ??<sup>\u2212?<\/sup> s. ~ 2 \u03bcs.\r\n<ol start=\"5\">\r\n \t<li><strong> Summary<\/strong><\/li>\r\n<\/ol>\r\n<p style=\"text-align: justify\">\u00a0 \u00a0 In this module we have applied our formalism to the hydrogen atom. We have obtained the selection rule by considering the wave functions of the atom as product of radial function and spherical harmonics. The properties of these functions give us the selection rules or permitted changes of ? and m . These selection rules are ?? =\u00b1 1 and m =0 \u00b1 1 .<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We studied the transition from the 2P to 1S state of hydrogen atom and calculated the transition probability per unit time. The final integral became a gamma function integral. The inverse of the transition probability gave us the life time of the excited state 2P ?. ? \u00d7 ??<sup>\u2212?<\/sup>?. The calculation was done in the natural units.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: justify\">We then , learnt a bit more about working in natural units and calculated the life time of muon decay.<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>you can view video on  Application to H atom<\/strong><\/td>\r\n<td><a href=\"https:\/\/youtu.be\/esqEl3s_e9I\" 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>8. Learn More Books<\/strong>\r\n<ol>\r\n \t<li>Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education Singapore (1998)<\/li>\r\n \t<li>Quantum Mechanics Vol.3 by L.D. Landau and Lifshitz (Pergramon Press Oxford, Reprinted<\/li>\r\n \t<li>\u201cSubtle is the Lord , The science and Life of Albert Einstein\u201d , Oxford University Press, 1982<\/li>\r\n<\/ol>\r\n<strong>\u00a0 \u00a0 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>Interesting Facts<\/strong>\r\n\r\n&nbsp;\r\n<p style=\"text-align: justify\">P.A.M.Dirac was the first to apply quantum mechanics to absorption and emission of light in 1927. So the theory was often called Dirac theory of radiation. This was the crucial step beyond Bohr theory.<\/p>\r\n&nbsp;\r\n\r\n&nbsp;","rendered":"<div><span style=\"float: right\"><a href=\"https:\/\/youtu.be\/esqEl3s_e9I\" 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 style=\"text-align: justify\">Learn how selection rules for transitions are obtained by evaluating matrix elements and finding when they are non zero.<\/li>\n<li>Learn how the life time<span style=\"text-align: initial;font-size: 1em\"> of an excited state of <\/span>hydrogen<span style=\"text-align: initial;font-size: 1em\"> atom is calculated. <\/span><\/li>\n<li><span style=\"text-align: initial;font-size: 1em\">Become familiar with <\/span>energy<span style=\"text-align: initial;font-size: 1em\"> values of the states of <\/span>hydrogen<span style=\"text-align: initial;font-size: 1em\"> atom.<\/span><\/li>\n<li>Learn a bit more about calculating with natural units.<\/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\">You have studied the energy spectrum of hydrogen atom. We would like to apply the framework developed so far to the concrete case of hydrogen atom. We shall consider first when a transition from the one state to another state is allowed. The states are labeled by total quantum number n, the orbital quantum number and the azimuthal or magnetic quantum number m, If the initial state has values of n, and m and final state has values of n<em>\u2019<\/em>, \u00a0\u00a0\u2032 and m\u2019 , we find that change =\u00b1 1 and m =0. \u00b1 1 are only allowed. These are called selection rules and are derived by using the properties of spherical harmonics or associated Legendre functions.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We next study the specific transition in hydrogen from the state 2P ( =1, the first excited state ) to the ground state 1S ( =0 ). As = -1 this transition is allowed, so also for change in m. We carry out the calculations using the well known expressions for the wave functions for both the states and obtain the transition probability. The lifetime of the excited state is just the inverse of this value.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">The calculations are specially simple if we use the natural units in which \u0127= c=1. We discuss the use of such units in calculations in the last section more fully.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">??????????? ?? ???????? ????<\/p>\n<p>&nbsp;<\/p>\n<p><strong>3.1 Selection Rules<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">In this section we apply our formalism to the electron bound in the hydrogen atom. We consider the structure of the matrix element<\/p>\n<p>&nbsp;<\/p>\n<p>\u27e8?|?|?\u27e9 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (3.1)<\/p>\n<p>&nbsp;<\/p>\n<p>Where the initial state A is characterized by the wave function<\/p>\n<p>&nbsp;<\/p>\n<p>\u27e8?|?\u27e9 = ???(?)???(?, ?) \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (3.2)<\/p>\n<p>&nbsp;<\/p>\n<p>The final sate is characterized by<\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">\u27e8?|?\u27e9 = ??\u2032?\u2032 (?)??\u2032?\u2032 (?, ?)\u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 \u2212 (3.3)<\/span><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-195 size-full\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112.png\" alt=\"\" width=\"674\" height=\"462\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112.png 674w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112-300x206.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112-65x45.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112-225x154.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-112-350x240.png 350w\" sizes=\"auto, (max-width: 674px) 100vw, 674px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-196\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113.png\" alt=\"\" width=\"675\" height=\"399\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113.png 675w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113-300x177.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113-225x133.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-113-350x207.png 350w\" sizes=\"auto, (max-width: 675px) 100vw, 675px\" \/><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-197\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114.png\" alt=\"\" width=\"671\" height=\"592\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114.png 671w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114-300x265.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114-65x57.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114-225x199.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-114-350x309.png 350w\" sizes=\"auto, (max-width: 671px) 100vw, 671px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-198\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115.png\" alt=\"\" width=\"679\" height=\"317\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115.png 679w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115-300x140.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115-65x30.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115-225x105.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-115-350x163.png 350w\" sizes=\"auto, (max-width: 679px) 100vw, 679px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-199\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116.png\" alt=\"\" width=\"673\" height=\"529\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116.png 673w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116-300x236.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116-65x51.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116-225x177.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-116-350x275.png 350w\" sizes=\"auto, (max-width: 673px) 100vw, 673px\" \/><\/p>\n<\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-200\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117.png\" alt=\"\" width=\"645\" height=\"378\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117.png 645w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117-300x176.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117-225x132.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-117-350x205.png 350w\" sizes=\"auto, (max-width: 645px) 100vw, 645px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-202\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119.png\" alt=\"\" width=\"664\" height=\"582\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119.png 664w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119-300x263.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119-65x57.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119-225x197.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-119-350x307.png 350w\" sizes=\"auto, (max-width: 664px) 100vw, 664px\" \/><\/p>\n<\/div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-203\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120.png\" alt=\"\" width=\"677\" height=\"321\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120.png 677w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120-300x142.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120-65x31.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120-225x107.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-120-350x166.png 350w\" sizes=\"auto, (max-width: 677px) 100vw, 677px\" \/><\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">\u00a0 <\/span><\/p>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-205\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121.png\" alt=\"\" width=\"685\" height=\"538\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121.png 685w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121-300x236.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121-65x51.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121-225x177.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-121-350x275.png 350w\" sizes=\"auto, (max-width: 685px) 100vw, 685px\" \/><\/div>\n<div><\/div>\n<div>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-206\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122.png\" alt=\"\" width=\"685\" height=\"396\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122.png 685w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122-300x173.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122-65x38.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122-225x130.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-122-350x202.png 350w\" sizes=\"auto, (max-width: 685px) 100vw, 685px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong style=\"text-align: initial;font-size: 1em\">3.3 Life Time of Excited states:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"text-align: initial;font-size: 1em\">As the transition probability has a dimension of <\/span>inverse<span style=\"text-align: initial;font-size: 1em\"> of time, we can find the l<\/span>ife time of<span style=\"text-align: initial;font-size: 1em\"> the excited state. We can thus calculate the <\/span>life time<span style=\"text-align: initial;font-size: 1em\"> of ? state\u00a0??\u2192?+?<\/span><\/p>\n<\/div>\n<div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4.\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong><strong>Calculations using natural units<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The \u03bc meson ,or muon as it is often called, decays into three particles. These are the electron and two neutrinos one of electron type and the other of the muon type.\u00a0by the inverse of ??\u2192?+? of equation (3.34). Thus for mass of electron ? =<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-207\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123.png\" alt=\"\" width=\"675\" height=\"148\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123.png 675w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123-300x66.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123-65x14.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123-225x49.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-123-350x77.png 350w\" sizes=\"auto, (max-width: 675px) 100vw, 675px\" \/><\/p>\n<\/div>\n<p>This is called life time of the excited state 2p.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>4. Calculations using natural units<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p>The \u03bc meson ,or muon as it is often called, decays into three particles. These are the electron and two neutrinos one of electron type ?? and the other of the muon type ??.<\/p>\n<p>If life time ? of muon decay is given by the formula<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-208\" src=\"http:\/\/phyp11.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124.png\" alt=\"\" width=\"663\" height=\"336\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124.png 663w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124-300x152.png 300w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124-65x33.png 65w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124-225x114.png 225w, https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-content\/uploads\/sites\/92\/2018\/11\/Untitled-124-350x177.png 350w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><\/p>\n<p>? ~ 2 x ??<sup>\u2212?<\/sup> s. ~ 2 \u03bcs.<\/p>\n<ol start=\"5\">\n<li><strong> Summary<\/strong><\/li>\n<\/ol>\n<p style=\"text-align: justify\">\u00a0 \u00a0 In this module we have applied our formalism to the hydrogen atom. We have obtained the selection rule by considering the wave functions of the atom as product of radial function and spherical harmonics. The properties of these functions give us the selection rules or permitted changes of ? and m . These selection rules are ?? =\u00b1 1 and m =0 \u00b1 1 .<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We studied the transition from the 2P to 1S state of hydrogen atom and calculated the transition probability per unit time. The final integral became a gamma function integral. The inverse of the transition probability gave us the life time of the excited state 2P ?. ? \u00d7 ??<sup>\u2212?<\/sup>?. The calculation was done in the natural units.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">We then , learnt a bit more about working in natural units and calculated the life time of muon decay.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>you can view video on  Application to H atom<\/strong><\/td>\n<td><a href=\"https:\/\/youtu.be\/esqEl3s_e9I\" 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>8. Learn More Books<\/strong><\/p>\n<ol>\n<li>Advanced Quantum Mechanics by J.J. Sakurai (Pearson Education Singapore (1998)<\/li>\n<li>Quantum Mechanics Vol.3 by L.D. Landau and Lifshitz (Pergramon Press Oxford, Reprinted<\/li>\n<li>\u201cSubtle is the Lord , The science and Life of Albert Einstein\u201d , Oxford University Press, 1982<\/li>\n<\/ol>\n<p><strong>\u00a0 \u00a0 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>Interesting Facts<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: justify\">P.A.M.Dirac was the first to apply quantum mechanics to absorption and emission of light in 1927. So the theory was often called Dirac theory of radiation. This was the crucial step beyond Bohr theory.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"author":3,"menu_order":8,"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-192","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\/192","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\/192\/revisions"}],"predecessor-version":[{"id":265,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapters\/192\/revisions\/265"}],"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\/192\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/media?parent=192"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/pressbooks\/v2\/chapter-type?post=192"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/contributor?post=192"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/phyp11\/wp-json\/wp\/v2\/license?post=192"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}