{"id":4425,"date":"2024-09-22T17:33:33","date_gmt":"2024-09-22T17:33:33","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4425"},"modified":"2024-09-22T17:33:33","modified_gmt":"2024-09-22T17:33:33","slug":"matter-waves","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/22\/matter-waves\/","title":{"rendered":"MATTER WAVES"},"content":{"rendered":"\n<p>Even in the light of the Compton Effect, critics of the early single-photon interference experiments dismissed the importance of the observation by noting that a photon doesn\u2019t have mass. Through some fancy hand-waving, they argued that the low-light interference could be caused through splitting and recombining the light quanta\u2019s wavefront. Decisive proof would come when particles with mass would show interference.<\/p>\n\n\n\n<p>In 1924, French physicist Prince Louis-Victor de Broglie (pronounced \u201cde Broy,\u201d and by the way, he did belong to the French royalty) proposed that maybe it\u2019s not just light that has this dual personality, maybe it\u2019s everything! He reasoned that if the quanta of light could be both a wave and a particle, then maybe the same could be true of electrons.<\/p>\n\n\n\n<p>Remember that although a photon doesn\u2019t have mass, it does have momentum&nbsp;<em>p<\/em>&nbsp;given by:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781118170700\/files\/OEBPS\/images\/137-1.gif\" alt=\"equation\"\/><\/figure>\n\n\n\n<p>De Broglie\u2019s hypothesis was that matter\u2014which is commonly described by our perception as \u201csolid\u201d\u2014can also behave as a wave. He took the concept used to find the momentum of a photon, and applied it to particles, proposing that the wavelength associated with a particle is:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781118170700\/files\/OEBPS\/images\/137-2.gif\" alt=\"equation\"\/><\/figure>\n\n\n\n<p>where the momentum&nbsp;<em>p<\/em>&nbsp;of a particle equals the product of its mass&nbsp;<em>m<\/em>, and velocity&nbsp;<em>v<\/em>. In the same way, he used Planck\u2019s relationship to propose that the frequency&nbsp;<em>f<\/em>&nbsp;of the matter wave is related to the energy&nbsp;<em>E<\/em>&nbsp;of the particle by:<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781118170700\/files\/OEBPS\/images\/137-3.gif\" alt=\"equation\"\/><\/figure>\n\n\n\n<h1 class=\"wp-block-heading\" id=\"c05-s7\">MATTER WAVES AND THE BOHR ATOM<\/h1>\n\n\n\n<p>As you may remember from chapter 4, Bohr was able to explain the discrete spectral lines emitted by the hydrogen atom by forcing the electrons into a limited number of permitted orbits (Figure 94a). However, like Planck before him, he did this without having a physical justification.<\/p>\n\n\n\n<p id=\"fig94\">Figure 94\u00a0De Broglie used his proposed matter waves to explain why Bohr\u2019s atomic orbits would be quantized.\u00a0<strong>(a)<\/strong>\u00a0Bohr explained the discrete spectral lines of hydrogen by enforcing a limited number of orbits. However, he did this without having a physical justification.\u00a0<strong>(b)<\/strong>\u00a0De Broglie proposed that, like in a guitar string fastened at the ends to rigid supports, allowed orbits would be those where a complete number of electron waves would fit on the orbit.\u00a0<strong>(c)<\/strong>\u00a0A higher-energy orbit would thus fit a larger number of complete vibrations than a lower-energy orbit.<img loading=\"lazy\" decoding=\"async\" width=\"510\" height=\"136\" src=\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781118170700\/files\/OEBPS\/images\/138-1.jpg\" alt=\"\"><\/p>\n\n\n\n<p>The first theoretical success of de Broglie\u2019s matter waves came when he used them to explain why certain electron orbits are allowed, and why others are not. He proposed that the allowed orbits for an electron are those that support a standing wave of specific wavelength, energy, and frequency (i.e., Bohr\u2019s energy levels); much like a guitar string sets up a standing wave when plucked.<\/p>\n\n\n\n<p>As shown in\u00a0Figure 94b, de Broglie proposed that if one were to straighten an orbit into a string and fix the ends to rigid supports, it could be set to vibrate. However, since the ends are fixed, the only vibrations that it can support are those where full wavelengths fit between the ends. De Broglie proposed that these vibrations would correspond to his matter waves, so a higher frequency vibration would correspond to an electron at a higher energy state. As shown in\u00a0Figure 94c, de Broglie\u2019s view of Bohr\u2019s atom consisted of electron waves of different wavelengths. An electron wave would transition between these wavelengths (each at one of Bohr\u2019s energy levels), giving off or absorbing photons with an energy\u00a0<em>hf<\/em>\u00a0equal to the difference in energy \u0394<em>E<\/em>\u00a0between the electron waves.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Even in the light of the Compton Effect, critics of the early single-photon interference experiments dismissed the importance of the observation by noting that a photon doesn\u2019t have mass. Through some fancy hand-waving, they argued that the low-light interference could be caused through splitting and recombining the light quanta\u2019s wavefront. Decisive proof would come when [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":4186,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[515],"tags":[],"class_list":["post-4425","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-wave-particle-duality"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/evaporation.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4425","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4425"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4425\/revisions"}],"predecessor-version":[{"id":4426,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4425\/revisions\/4426"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4186"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4425"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4425"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}