{"id":1566,"date":"2017-10-27T16:32:25","date_gmt":"2017-10-27T16:32:25","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/quantization-of-energy\/"},"modified":"2017-11-08T03:27:31","modified_gmt":"2017-11-08T03:27:31","slug":"quantization-of-energy","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/quantization-of-energy\/","title":{"raw":"Quantization of Energy","rendered":"Quantization of Energy"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain Max Planck\u2019s contribution to the development of quantum mechanics.<\/li>\n<li>Explain why atomic spectra indicate quantization.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1255905\">\n<h1 data-type=\"title\">Planck\u2019s Contribution<\/h1>\n<p id=\"import-auto-id1889484\">Energy is quantized in some systems, meaning that the system can have only certain energies and not a continuum of energies, unlike the classical case. This would be like having only certain speeds at which a car can travel because its kinetic energy can have only certain values. We also find that some forms of energy transfer take place with discrete lumps of energy. While most of us are familiar with the quantization of matter into lumps called atoms, molecules, and the like, we are less aware that energy, too, can be quantized. Some of the earliest clues about the necessity of quantum mechanics over classical physics came from the quantization of energy.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1458382\">\n<div class=\"bc-figcaption figcaption\">Graphs of blackbody radiation (from an ideal radiator) at three different radiator temperatures. The intensity or rate of radiation emission increases dramatically with temperature, and the peak of the spectrum shifts toward the visible and ultraviolet parts of the spectrum. The shape of the spectrum cannot be described with classical physics.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1919373\" data-alt=\"The blackbody radiation graph of E M radiation intensity versus wavelengths is shown, with the visible band represented as vertical colored strip marked on x axis. Wavelength is along x axis and E M radiation intensity is along y axis. The variation of E M radiation intensity is shown by three curves that start at origin, rise up to their highest point and then drop toward the x axis smoothly, and finally extend parallel to the x axis. There are three curves for three different temperatures, and each has a different peak for radiation intensity. As the temperature decreases, the peak of the black body radiation curves moves to a lower radiation intensity and longer wavelength.\"><img src=\"\/resources\/506cc8150f2dd8a78cb76992182de2e26134b913\/Figure 30_01_01a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The blackbody radiation graph of E M radiation intensity versus wavelengths is shown, with the visible band represented as vertical colored strip marked on x axis. Wavelength is along x axis and E M radiation intensity is along y axis. The variation of E M radiation intensity is shown by three curves that start at origin, rise up to their highest point and then drop toward the x axis smoothly, and finally extend parallel to the x axis. There are three curves for three different temperatures, and each has a different peak for radiation intensity. As the temperature decreases, the peak of the black body radiation curves moves to a lower radiation intensity and longer wavelength.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1352686\">Where is the quantization of energy observed? Let us begin by considering the emission and absorption of electromagnetic (EM) radiation. The EM spectrum radiated by a hot solid is linked directly to the solid\u2019s temperature. (See <a href=\"#import-auto-id1458382\" class=\"autogenerated-content\">(Figure)<\/a>.) An ideal radiator is one that has an emissivity of 1 at all wavelengths and, thus, is jet black. Ideal radiators are therefore called <span data-type=\"term\" id=\"import-auto-id1881547\">blackbodies<\/span>, and their EM radiation is called <span data-type=\"term\" id=\"import-auto-id1973496\">blackbody radiation<\/span>. It was discussed that the total intensity of the radiation varies as [latex]{T}^{4},[\/latex] the fourth power of the absolute temperature of the body, and that the peak of the spectrum shifts to shorter wavelengths at higher temperatures. All of this seems quite continuous, but it was the curve of the spectrum of intensity versus wavelength that gave a clue that the energies of the atoms in the solid are quantized. In fact, providing a theoretical explanation for the experimentally measured shape of the spectrum was a mystery at the turn of the century. When this \u201cultraviolet catastrophe\u201d was eventually solved, the answers led to new technologies such as computers and the sophisticated imaging techniques described in earlier chapters. Once again, physics as an enabling science changed the way we live.<\/p>\n<p id=\"import-auto-id2907455\">The German physicist Max Planck (1858\u20131947) used the idea that atoms and molecules in a body act like oscillators to absorb and emit radiation. The energies of the oscillating atoms and molecules had to be quantized to correctly describe the shape of the blackbody spectrum. Planck deduced that the energy of an oscillator having a frequency [latex]f[\/latex] is given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]E=\\left(n+\\frac{1}{2}\\right)\\text{hf}.[\/latex]<\/div>\n<p id=\"import-auto-id1751018\">Here [latex]n[\/latex] is any nonnegative integer (0, 1, 2, 3, \u2026). The symbol [latex]h[\/latex] stands for <span data-type=\"term\" id=\"import-auto-id1986691\">Planck\u2019s constant<\/span>, given by<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]h=6\\text{.}\\text{626}\u00d7{\\text{10}}^{\\text{\u201334}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot \\text{s}.[\/latex]<\/div>\n<p id=\"import-auto-id2219008\">The equation [latex]E=\\left(n+\\frac{1}{2}\\right)\\text{hf}[\/latex] means that an oscillator having a frequency [latex]f[\/latex] (emitting and absorbing EM radiation of frequency [latex]f[\/latex]) can have its energy increase or decrease only in <em data-effect=\"italics\">discrete<\/em> steps of size<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\Delta E=\\text{hf}.[\/latex]<\/div>\n<p id=\"import-auto-id1639233\">It might be helpful to mention some macroscopic analogies of this quantization of energy phenomena. This is like a pendulum that has a characteristic oscillation frequency but can swing with only certain amplitudes. Quantization of energy also resembles a standing wave on a string that allows only particular harmonics described by integers. It is also similar to going up and down a hill using discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step. <\/p>\n<p id=\"import-auto-id1451886\">Using the quantization of oscillators, Planck was able to correctly describe the experimentally known shape of the blackbody spectrum. This was the first indication that energy is sometimes quantized on a small scale and earned him the Nobel Prize in Physics in 1918. Although Planck\u2019s theory comes from observations of a macroscopic object, its analysis is based on atoms and molecules. It was such a revolutionary departure from classical physics that Planck himself was reluctant to accept his own idea that energy states are not continuous. The general acceptance of Planck\u2019s energy quantization was greatly enhanced by Einstein\u2019s explanation of the photoelectric effect (discussed in the next section), which took energy quantization a step further. Planck was fully involved in the development of both early quantum mechanics and relativity. He quickly embraced Einstein\u2019s special relativity, published in 1905, and in 1906 Planck was the first to suggest the correct formula for relativistic momentum, [latex]p=\\text{\u03b3mu}[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1532199\">\n<div class=\"bc-figcaption figcaption\">The German physicist Max Planck had a major influence on the early development of quantum mechanics, being the first to recognize that energy is sometimes quantized. Planck also made important contributions to special relativity and classical physics. (credit: Library of Congress, Prints and Photographs Division via Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" data-alt=\"A photo of German physicist Max Plank is shown.\"><img src=\"\/resources\/0ac69e9cff999ae47179c7fa688b407568b5dc6c\/Figure 30_01_02a.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"A photo of German physicist Max Plank is shown.\" width=\"300\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2182118\">Note that Planck\u2019s constant [latex]h[\/latex] is a very small number. So for an infrared frequency of [latex]{\\text{10}}^{\\text{14}}\\phantom{\\rule{0.25em}{0ex}}\\text{Hz}[\/latex] being emitted by a blackbody, for example, the difference between energy levels is only [latex]\\Delta E=\\text{hf}\\text{=}\\left(6\\text{.}\\text{63}\u00d7{\\text{10}}^{\\text{\u201334}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\u00b7s}\\right)\\left({\\text{10}}^{\\text{14}}\\phantom{\\rule{0.25em}{0ex}}\\text{Hz}\\right)\\text{= 6}\\text{.}\\text{63}\u00d7{\\text{10}}^{\\text{\u201320}}\\phantom{\\rule{0.25em}{0ex}}\\text{J,}[\/latex] or about 0.4 eV. This 0.4 eV of energy is significant compared with typical atomic energies, which are on the order of an electron volt, or thermal energies, which are typically fractions of an electron volt. But on a macroscopic or classical scale, energies are typically on the order of joules. Even if macroscopic energies are quantized, the quantum steps are too small to be noticed. This is an example of the correspondence principle. For a large object, quantum mechanics produces results indistinguishable from those of classical physics.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1273466\">\n<h1 data-type=\"title\">Atomic Spectra<\/h1>\n<p id=\"import-auto-id1796879\">Now let us turn our attention to the <em data-effect=\"italics\">emission and absorption of EM radiation by gases<\/em>. The Sun is the most common example of a body containing gases emitting an EM spectrum that includes visible light. We also see examples in neon signs and candle flames. Studies of emissions of hot gases began more than two centuries ago, and it was soon recognized that these emission spectra contained huge amounts of information. The type of gas and its temperature, for example, could be determined. We now know that these EM emissions come from electrons transitioning between energy levels in individual atoms and molecules; thus, they are called <span data-type=\"term\" id=\"import-auto-id1921365\">atomic spectra<\/span>. Atomic spectra remain an important analytical tool today. <a href=\"#import-auto-id1826718\" class=\"autogenerated-content\">(Figure)<\/a> shows an example of an emission spectrum obtained by passing an electric discharge through a material. One of the most important characteristics of these spectra is that they are discrete. By this we mean that only certain wavelengths, and hence frequencies, are emitted. This is called a line spectrum. If frequency and energy are associated as [latex]\\Delta E=\\text{hf}\\text{,}[\/latex] the energies of the electrons in the emitting atoms and molecules are quantized. This is discussed in more detail later in this chapter.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1826718\">\n<div class=\"bc-figcaption figcaption\">Emission spectrum of oxygen. When an electrical discharge is passed through a substance, its atoms and molecules absorb energy, which is reemitted as EM radiation. The discrete nature of these emissions implies that the energy states of the atoms and molecules are quantized. Such atomic spectra were used as analytical tools for many decades before it was understood why they are quantized. (credit: Teravolt, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2037720\" data-alt=\"Emission spectrum of oxygen is shown as a band containing all colors with some distinct colors as discrete bold lines.\"><img src=\"\/resources\/886fa2b13f2ea0ff3636fe1b5b425f1f968eff53\/Figure 30_01_03a.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"Emission spectrum of oxygen is shown as a band containing all colors with some distinct colors as discrete bold lines.\" width=\"550\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2096770\">It was a major puzzle that atomic spectra are quantized. Some of the best minds of 19th-century science failed to explain why this might be. Not until the second decade of the 20th century did an answer based on quantum mechanics begin to emerge. Again a macroscopic or classical body of gas was involved in the studies, but the effect, as we shall see, is due to individual atoms and molecules.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-521\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Models of the Hydrogen Atom<\/div>\n<p id=\"eip-id2664469\">How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1989619\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/3ace2745d2434765a75ebe5d34835609f0aa5ad7\/hydrogen-atom_en.jar\">Models of the Hydrogen Atom<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_30.1\" data-alt=\"\"><a href=\"\/resources\/3ace2745d2434765a75ebe5d34835609f0aa5ad7\/hydrogen-atom_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1187682\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1234357\">\n<li id=\"import-auto-id1320924\">The first indication that energy is sometimes quantized came from blackbody radiation, which is the emission of EM radiation by an object with an emissivity of 1. <\/li>\n<li id=\"import-auto-id1809104\">Planck recognized that the energy levels of the emitting atoms and molecules were quantized, with only the allowed values of [latex]E=\\left(n+\\frac{1}{2}\\right)\\text{hf}\\text{,}[\/latex] where [latex]n[\/latex] is any non-negative integer (0, 1, 2, 3, \u2026).<\/li>\n<li id=\"import-auto-id1561986\">[latex]h[\/latex] is Planck\u2019s constant, whose value is [latex]h=6\\text{.}\\text{626}\u00d7{\\text{10}}^{\\text{\u201334}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot \\text{s.}[\/latex]<\/li>\n<li id=\"import-auto-id2319049\">Thus, the oscillatory absorption and emission energies of atoms and molecules in a blackbody could increase or decrease only in steps of size [latex]\\Delta E=\\text{hf}[\/latex] where [latex]f[\/latex] is the frequency of the oscillatory nature of the absorption and emission of EM radiation.<\/li>\n<li id=\"import-auto-id1295259\">Another indication of energy levels being quantized in atoms and molecules comes from the lines in atomic spectra, which are the EM emissions of individual atoms and molecules.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1254593\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1254787\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id689241\">\n<p id=\"import-auto-id1567423\">Give an example of a physical entity that is quantized. State specifically what the entity is and what the limits are on its values.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1268120\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1271120\">\n<p id=\"import-auto-id1780947\">Give an example of a physical entity that is not quantized, in that it is continuous and may have a continuous range of values.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1250135\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1257034\">\n<p id=\"import-auto-id1858390\">What aspect of the blackbody spectrum forced Planck to propose quantization of energy levels in its atoms and molecules?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id977492\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1237938\">\n<p id=\"import-auto-id2009213\">If Planck\u2019s constant were large, say [latex]{\\text{10}}^{\\text{34}}[\/latex] times greater than it is, we would observe macroscopic entities to be quantized. Describe the motions of a child\u2019s swing under such circumstances.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1267708\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1292391\">\n<p id=\"import-auto-id1607412\">Why don\u2019t we notice quantization in everyday events?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1296326\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1296980\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1297102\">\n<p id=\"import-auto-id2223128\">A LiBr molecule oscillates with a frequency of [latex]1\\text{.}7\u00d7{\\text{10}}^{\\text{13}}\\phantom{\\rule{0.25em}{0ex}}\\text{Hz.}[\/latex] (a) What is the difference in energy in eV between allowed oscillator states? (b) What is the approximate value of [latex]n[\/latex] for a state having an energy of 1.0 eV?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1310061\">\n<p id=\"import-auto-id2275909\">(a) 0.070 eV<\/p>\n<p id=\"import-auto-id2205881\">(b) 14 <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1316461\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1317385\">\n<p id=\"import-auto-id1638440\">The difference in energy between allowed oscillator states in HBr molecules is 0.330 eV. What is the oscillation frequency of this molecule?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1319731\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1249514\">\n<p id=\"import-auto-id1479865\">A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized. (a) What is the difference in energy in joules between allowed oscillator states? (b) What is the value of [latex]n[\/latex] for a state where the energy is 5.00 J? (c) Can the quantization be observed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1254818\">\n<p id=\"import-auto-id2099236\">(a) [latex]2\\text{.}\\text{21}\u00d7{\\text{10}}^{\\text{34}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]<\/p>\n<p id=\"import-auto-id2116800\">(b) [latex]2\\text{.}\\text{26}\u00d7{\\text{10}}^{\\text{34}}[\/latex]<\/p>\n<p id=\"import-auto-id2240162\">(c) No <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2889977\">\n<dt>blackbody<\/dt>\n<dd id=\"fs-id1315765\">an ideal radiator, which can radiate equally well at all wavelengths<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1957973\">\n<dt>blackbody radiation<\/dt>\n<dd id=\"fs-id1324996\">the electromagnetic radiation from a blackbody<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1814861\">\n<dt>Planck\u2019s constant<\/dt>\n<dd id=\"fs-id1323775\">[latex]h=6\\text{.}\\text{626}\u00d7{\\text{10}}^{\\text{\u201334}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\cdot \\text{s}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1931942\">\n<dt>atomic spectra<\/dt>\n<dd id=\"fs-id1290278\">the electromagnetic emission from atoms and molecules<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain Max Planck\u2019s contribution to the development of quantum mechanics.<\/li>\n<li>Explain why atomic spectra indicate quantization.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1255905\">\n<h1 data-type=\"title\">Planck\u2019s Contribution<\/h1>\n<p id=\"import-auto-id1889484\">Energy is quantized in some systems, meaning that the system can have only certain energies and not a continuum of energies, unlike the classical case. This would be like having only certain speeds at which a car can travel because its kinetic energy can have only certain values. We also find that some forms of energy transfer take place with discrete lumps of energy. While most of us are familiar with the quantization of matter into lumps called atoms, molecules, and the like, we are less aware that energy, too, can be quantized. Some of the earliest clues about the necessity of quantum mechanics over classical physics came from the quantization of energy.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1458382\">\n<div class=\"bc-figcaption figcaption\">Graphs of blackbody radiation (from an ideal radiator) at three different radiator temperatures. The intensity or rate of radiation emission increases dramatically with temperature, and the peak of the spectrum shifts toward the visible and ultraviolet parts of the spectrum. The shape of the spectrum cannot be described with classical physics.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1919373\" data-alt=\"The blackbody radiation graph of E M radiation intensity versus wavelengths is shown, with the visible band represented as vertical colored strip marked on x axis. Wavelength is along x axis and E M radiation intensity is along y axis. The variation of E M radiation intensity is shown by three curves that start at origin, rise up to their highest point and then drop toward the x axis smoothly, and finally extend parallel to the x axis. There are three curves for three different temperatures, and each has a different peak for radiation intensity. As the temperature decreases, the peak of the black body radiation curves moves to a lower radiation intensity and longer wavelength.\"><img decoding=\"async\" src=\"\/resources\/506cc8150f2dd8a78cb76992182de2e26134b913\/Figure 30_01_01a.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The blackbody radiation graph of E M radiation intensity versus wavelengths is shown, with the visible band represented as vertical colored strip marked on x axis. Wavelength is along x axis and E M radiation intensity is along y axis. The variation of E M radiation intensity is shown by three curves that start at origin, rise up to their highest point and then drop toward the x axis smoothly, and finally extend parallel to the x axis. There are three curves for three different temperatures, and each has a different peak for radiation intensity. As the temperature decreases, the peak of the black body radiation curves moves to a lower radiation intensity and longer wavelength.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1352686\">Where is the quantization of energy observed? Let us begin by considering the emission and absorption of electromagnetic (EM) radiation. The EM spectrum radiated by a hot solid is linked directly to the solid\u2019s temperature. (See <a href=\"#import-auto-id1458382\" class=\"autogenerated-content\">(Figure)<\/a>.) An ideal radiator is one that has an emissivity of 1 at all wavelengths and, thus, is jet black. Ideal radiators are therefore called <span data-type=\"term\" id=\"import-auto-id1881547\">blackbodies<\/span>, and their EM radiation is called <span data-type=\"term\" id=\"import-auto-id1973496\">blackbody radiation<\/span>. It was discussed that the total intensity of the radiation varies as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-75bd821f6f0184d97ccebbdb481d06f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#84;&#125;&#94;&#123;&#52;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"25\" style=\"vertical-align: -4px;\" \/> the fourth power of the absolute temperature of the body, and that the peak of the spectrum shifts to shorter wavelengths at higher temperatures. All of this seems quite continuous, but it was the curve of the spectrum of intensity versus wavelength that gave a clue that the energies of the atoms in the solid are quantized. In fact, providing a theoretical explanation for the experimentally measured shape of the spectrum was a mystery at the turn of the century. When this \u201cultraviolet catastrophe\u201d was eventually solved, the answers led to new technologies such as computers and the sophisticated imaging techniques described in earlier chapters. Once again, physics as an enabling science changed the way we live.<\/p>\n<p id=\"import-auto-id2907455\">The German physicist Max Planck (1858\u20131947) used the idea that atoms and molecules in a body act like oscillators to absorb and emit radiation. The energies of the oscillating atoms and molecules had to be quantized to correctly describe the shape of the blackbody spectrum. Planck deduced that the energy of an oscillator having a frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cc1669297632ee38b47851023c16c537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"119\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id1751018\">Here <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is any nonnegative integer (0, 1, 2, 3, \u2026). The symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> stands for <span data-type=\"term\" id=\"import-auto-id1986691\">Planck\u2019s constant<\/span>, given by<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea36ce8a3987ff45b1928fcbbbd635ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#50;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"148\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id2219008\">The equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9ee046a75bba272c95e6465284dd8553_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"117\" style=\"vertical-align: -7px;\" \/> means that an oscillator having a frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> (emitting and absorbing EM radiation of frequency <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/>) can have its energy increase or decrease only in <em data-effect=\"italics\">discrete<\/em> steps of size<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-594f00d7e4304c8f1beaea9f30f529bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#69;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"72\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id1639233\">It might be helpful to mention some macroscopic analogies of this quantization of energy phenomena. This is like a pendulum that has a characteristic oscillation frequency but can swing with only certain amplitudes. Quantization of energy also resembles a standing wave on a string that allows only particular harmonics described by integers. It is also similar to going up and down a hill using discrete stair steps rather than being able to move up and down a continuous slope. Your potential energy takes on discrete values as you move from step to step. <\/p>\n<p id=\"import-auto-id1451886\">Using the quantization of oscillators, Planck was able to correctly describe the experimentally known shape of the blackbody spectrum. This was the first indication that energy is sometimes quantized on a small scale and earned him the Nobel Prize in Physics in 1918. Although Planck\u2019s theory comes from observations of a macroscopic object, its analysis is based on atoms and molecules. It was such a revolutionary departure from classical physics that Planck himself was reluctant to accept his own idea that energy states are not continuous. The general acceptance of Planck\u2019s energy quantization was greatly enhanced by Einstein\u2019s explanation of the photoelectric effect (discussed in the next section), which took energy quantization a step further. Planck was fully involved in the development of both early quantum mechanics and relativity. He quickly embraced Einstein\u2019s special relativity, published in 1905, and in 1906 Planck was the first to suggest the correct formula for relativistic momentum, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7144463ef4f57b11048cfe6e7db56f74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#112;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&gamma;&#109;&#117;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1532199\">\n<div class=\"bc-figcaption figcaption\">The German physicist Max Planck had a major influence on the early development of quantum mechanics, being the first to recognize that energy is sometimes quantized. Planck also made important contributions to special relativity and classical physics. (credit: Library of Congress, Prints and Photographs Division via Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" data-alt=\"A photo of German physicist Max Plank is shown.\"><img decoding=\"async\" src=\"\/resources\/0ac69e9cff999ae47179c7fa688b407568b5dc6c\/Figure 30_01_02a.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"A photo of German physicist Max Plank is shown.\" width=\"300\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2182118\">Note that Planck\u2019s constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> is a very small number. So for an infrared frequency of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-64e1ed81edb629a8d50caa145079fc49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -1px;\" \/> being emitted by a blackbody, for example, the difference between energy levels is only <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0bc75bd545708648e570750097587de5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#69;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&middot;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#122;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#32;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#50;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"375\" style=\"vertical-align: -7px;\" \/> or about 0.4 eV. This 0.4 eV of energy is significant compared with typical atomic energies, which are on the order of an electron volt, or thermal energies, which are typically fractions of an electron volt. But on a macroscopic or classical scale, energies are typically on the order of joules. Even if macroscopic energies are quantized, the quantum steps are too small to be noticed. This is an example of the correspondence principle. For a large object, quantum mechanics produces results indistinguishable from those of classical physics.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1273466\">\n<h1 data-type=\"title\">Atomic Spectra<\/h1>\n<p id=\"import-auto-id1796879\">Now let us turn our attention to the <em data-effect=\"italics\">emission and absorption of EM radiation by gases<\/em>. The Sun is the most common example of a body containing gases emitting an EM spectrum that includes visible light. We also see examples in neon signs and candle flames. Studies of emissions of hot gases began more than two centuries ago, and it was soon recognized that these emission spectra contained huge amounts of information. The type of gas and its temperature, for example, could be determined. We now know that these EM emissions come from electrons transitioning between energy levels in individual atoms and molecules; thus, they are called <span data-type=\"term\" id=\"import-auto-id1921365\">atomic spectra<\/span>. Atomic spectra remain an important analytical tool today. <a href=\"#import-auto-id1826718\" class=\"autogenerated-content\">(Figure)<\/a> shows an example of an emission spectrum obtained by passing an electric discharge through a material. One of the most important characteristics of these spectra is that they are discrete. By this we mean that only certain wavelengths, and hence frequencies, are emitted. This is called a line spectrum. If frequency and energy are associated as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6daf219a3f7aee14e7d93167d7e3bc6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#69;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -3px;\" \/> the energies of the electrons in the emitting atoms and molecules are quantized. This is discussed in more detail later in this chapter.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1826718\">\n<div class=\"bc-figcaption figcaption\">Emission spectrum of oxygen. When an electrical discharge is passed through a substance, its atoms and molecules absorb energy, which is reemitted as EM radiation. The discrete nature of these emissions implies that the energy states of the atoms and molecules are quantized. Such atomic spectra were used as analytical tools for many decades before it was understood why they are quantized. (credit: Teravolt, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2037720\" data-alt=\"Emission spectrum of oxygen is shown as a band containing all colors with some distinct colors as discrete bold lines.\"><img decoding=\"async\" src=\"\/resources\/886fa2b13f2ea0ff3636fe1b5b425f1f968eff53\/Figure 30_01_03a.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"Emission spectrum of oxygen is shown as a band containing all colors with some distinct colors as discrete bold lines.\" width=\"550\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2096770\">It was a major puzzle that atomic spectra are quantized. Some of the best minds of 19th-century science failed to explain why this might be. Not until the second decade of the 20th century did an answer based on quantum mechanics begin to emerge. Again a macroscopic or classical body of gas was involved in the studies, but the effect, as we shall see, is due to individual atoms and molecules.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"eip-521\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Models of the Hydrogen Atom<\/div>\n<p id=\"eip-id2664469\">How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental results.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id1989619\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/3ace2745d2434765a75ebe5d34835609f0aa5ad7\/hydrogen-atom_en.jar\">Models of the Hydrogen Atom<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_30.1\" data-alt=\"\"><a href=\"\/resources\/3ace2745d2434765a75ebe5d34835609f0aa5ad7\/hydrogen-atom_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1187682\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1234357\">\n<li id=\"import-auto-id1320924\">The first indication that energy is sometimes quantized came from blackbody radiation, which is the emission of EM radiation by an object with an emissivity of 1. <\/li>\n<li id=\"import-auto-id1809104\">Planck recognized that the energy levels of the emitting atoms and molecules were quantized, with only the allowed values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3dc284c8b117cbb14299c5b3be49dc36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"119\" style=\"vertical-align: -7px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is any non-negative integer (0, 1, 2, 3, \u2026).<\/li>\n<li id=\"import-auto-id1561986\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"10\" style=\"vertical-align: 0px;\" \/> is Planck\u2019s constant, whose value is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b7c1b6581f12d68e98ef8f8a86a24f3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#50;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"148\" style=\"vertical-align: -1px;\" \/><\/li>\n<li id=\"import-auto-id2319049\">Thus, the oscillatory absorption and emission energies of atoms and molecules in a blackbody could increase or decrease only in steps of size <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-069367cb616097f20e370265699c2858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#69;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#104;&#102;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"70\" style=\"vertical-align: -1px;\" \/> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9c09a708375fde2676da319bcdfe8b24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"10\" style=\"vertical-align: -4px;\" \/> is the frequency of the oscillatory nature of the absorption and emission of EM radiation.<\/li>\n<li id=\"import-auto-id1295259\">Another indication of energy levels being quantized in atoms and molecules comes from the lines in atomic spectra, which are the EM emissions of individual atoms and molecules.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1254593\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1254787\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id689241\">\n<p id=\"import-auto-id1567423\">Give an example of a physical entity that is quantized. State specifically what the entity is and what the limits are on its values.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1268120\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1271120\">\n<p id=\"import-auto-id1780947\">Give an example of a physical entity that is not quantized, in that it is continuous and may have a continuous range of values.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1250135\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1257034\">\n<p id=\"import-auto-id1858390\">What aspect of the blackbody spectrum forced Planck to propose quantization of energy levels in its atoms and molecules?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id977492\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1237938\">\n<p id=\"import-auto-id2009213\">If Planck\u2019s constant were large, say <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f3ce03e4de7c63930be6f6ee61dc8c1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"31\" style=\"vertical-align: -1px;\" \/> times greater than it is, we would observe macroscopic entities to be quantized. Describe the motions of a child\u2019s swing under such circumstances.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1267708\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1292391\">\n<p id=\"import-auto-id1607412\">Why don\u2019t we notice quantization in everyday events?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1296326\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1296980\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1297102\">\n<p id=\"import-auto-id2223128\">A LiBr molecule oscillates with a frequency of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-91ca53c61547ef031c5c8c7b9bca8634_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#122;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"83\" style=\"vertical-align: -1px;\" \/> (a) What is the difference in energy in eV between allowed oscillator states? (b) What is the approximate value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> for a state having an energy of 1.0 eV?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1310061\">\n<p id=\"import-auto-id2275909\">(a) 0.070 eV<\/p>\n<p id=\"import-auto-id2205881\">(b) 14 <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1316461\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1317385\">\n<p id=\"import-auto-id1638440\">The difference in energy between allowed oscillator states in HBr molecules is 0.330 eV. What is the oscillation frequency of this molecule?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1319731\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1249514\">\n<p id=\"import-auto-id1479865\">A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized. (a) What is the difference in energy in joules between allowed oscillator states? (b) What is the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> for a state where the energy is 5.00 J? (c) Can the quantization be observed?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1254818\">\n<p id=\"import-auto-id2099236\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c4b435e111e96bcc295bc835aebf5ba7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2116800\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d41d2122b2b4d857c93e4e02b6a78477_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2240162\">(c) No <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2889977\">\n<dt>blackbody<\/dt>\n<dd id=\"fs-id1315765\">an ideal radiator, which can radiate equally well at all wavelengths<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1957973\">\n<dt>blackbody radiation<\/dt>\n<dd id=\"fs-id1324996\">the electromagnetic radiation from a blackbody<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1814861\">\n<dt>Planck\u2019s constant<\/dt>\n<dd id=\"fs-id1323775\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-891afa9a4e2d2b2a1f88ea9b6247f0f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#104;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#50;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#51;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"144\" style=\"vertical-align: -1px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id1931942\">\n<dt>atomic spectra<\/dt>\n<dd id=\"fs-id1290278\">the electromagnetic emission from atoms and molecules<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1566","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1561,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1566","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1566\/revisions"}],"predecessor-version":[{"id":1567,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1566\/revisions\/1567"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1561"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1566\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1566"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1566"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1566"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}