{"id":196,"date":"2015-09-23T13:11:44","date_gmt":"2015-09-23T13:11:44","guid":{"rendered":"https:\/\/www.quantum-classical-physics.com\/qcp\/?p=196"},"modified":"2019-09-24T09:12:13","modified_gmt":"2019-09-24T09:12:13","slug":"simple-derivation-of-schrodinger-equation-from-newtonian-dynamics","status":"publish","type":"post","link":"https:\/\/www.quantum-classical-physics.com\/qcp\/simple-derivation-of-schrodinger-equation-from-newtonian-dynamics\/","title":{"rendered":"Simple derivation of Schr\u00f6dinger equation from Newtonian dynamics"},"content":{"rendered":"<div  class=\"tatsu-g7apowtr2v7qcctc tatsu-section  tatsu-bg-overlay   tatsu-clearfix\" data-title=\"\"  data-headerscheme=\"background--dark\"><div class='tatsu-section-pad clearfix' data-padding='{\"d\":\"15px 0px 15px 0px\"}' data-padding-top='15px'><div class=\"tatsu-row-wrap  tatsu-wrap tatsu-row-one-col tatsu-row-has-one-cols tatsu-medium-gutter tatsu-reg-cols  tatsu-clearfix tatsu-g7apowtr987vhqoh\" ><div  class=\"tatsu-row \" ><div  class=\"tatsu-column  tatsu-bg-overlay tatsu-one-col tatsu-column-image-none tatsu-column-effect-none  tatsu-g7apowtrdra9kr51\"  data-parallax-speed=\"0\" style=\"\"><div class=\"tatsu-column-inner \" ><div class=\"tatsu-column-pad-wrap\"><div class=\"tatsu-column-pad\" ><div  class=\"tatsu-module tatsu-text-block-wrap tatsu-g7apowtrhlabwa9h  \"><div class=\"tatsu-text-inner tatsu-align-center  clearfix\" ><style>.tatsu-g7apowtrhlabwa9h.tatsu-text-block-wrap .tatsu-text-inner{width: 100%;text-align: left;}<\/style>\n<p><span style=\"color: #ffffff; font-size: 8pt;\">Simple derivation of Schr\u00f6dinger equation from Newtonian dynamics<\/span> Michele Marrocco Dipartimento di Fisica, Universit\u00e0 di Roma \u2018La Sapienza\u2019 P.le Aldo Moro 5, I-00185 Rome, Italy &amp; ENEA (Italian National Agency for New Technologies, Energies and Sustainable Economic Development) via Anguillarese 301, I-00123 Rome, Italy<\/p>\n<h3>Abstract<\/h3>\n<p>If Schr\u00f6dinger representation of quantum mechanicsreproduces Newton\u2019s laws of motion in terms of expectation values (Ehrenfest theorem), the contraryis considered elusive. Against this opinion, we present here a simple method to make Newtonian dynamics develop into Schr\u00f6dinger representation. The proofis laid outin two steps. First, Newton\u2019s laws of motion are used to determine a classical wave equation whose similarity with the Schr\u00f6dinger equation is mediatedby a parameter that plays the identical role of the constant Kintroduced by Schr\u00f6dinger in the original formulation of his theory. In the second step, the classical wave equation becomes exactly the Schr\u00f6dinger equationthanks to the numerical value of the parameter obtained from the identification of the classical momentum with de Broglie momentum of matter waves.<\/p>\n<h3>I. INTRODUCTION<\/h3>\n<p>One of the main routes to quantum mechanics runs through the Schr\u00f6dinger equationand the concept of wave function. Despite theundoubted importanceof this cornerstone of modern physics, the subject is ordinarily introduced in courses on quantum mechanics without much detail aboutits conceptual foundations, with the result that, according to prominent physicists, the derivation ofthe Schr\u00f6dinger equation conveysa senseof dissatisfaction. In reality, the disappointmenthas something to do with the heuristic explanation reportedby Schr\u00f6dingerin his original conjecture. Its weaknessis, for instance, underlinedby Feynmanin the third volume of the Lectureswhere one can read the following comment about the founder of wave mechanics: \u201csome of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature\u201d. Against this compliant attitude&#8230; <a href=\"https:\/\/www.quantum-classical-physics.com\/qcp\/articles\/NtoS_v1.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">read more<\/a><\/p>\n<h3><a href=\"https:\/\/www.quantum-classical-physics.com\/qcp\/articles\/NtoS_v1.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Read the Abstract <\/a> (PDF)<\/h3>\n<\/div><\/div><\/div><\/div><div class = \"tatsu-column-bg-image-wrap\"><div class = \"tatsu-column-bg-image\" ><\/div><\/div><div class=\"tatsu-overlay tatsu-column-overlay tatsu-animate-none\" ><\/div><\/div><style>.tatsu-row > .tatsu-g7apowtrdra9kr51.tatsu-column{width: 100%;}.tatsu-g7apowtrdra9kr51.tatsu-column > .tatsu-column-inner > .tatsu-column-overlay{mix-blend-mode: normal;}.tatsu-g7apowtrdra9kr51 > .tatsu-column-inner > .tatsu-top-divider{z-index: 9999;}.tatsu-g7apowtrdra9kr51 > .tatsu-column-inner > .tatsu-bottom-divider{z-index: 9999;}.tatsu-g7apowtrdra9kr51 > .tatsu-column-inner > .tatsu-left-divider{z-index: 9999;}.tatsu-g7apowtrdra9kr51 > .tatsu-column-inner > .tatsu-right-divider{z-index: 9999;}<\/style><\/div><\/div><\/div><\/div><div class=\"tatsu-section-background-wrap\"><div class = \"tatsu-section-background\" ><\/div><\/div><div class=\"tatsu-overlay tatsu-section-overlay\"><\/div><style>.tatsu-g7apowtr2v7qcctc .tatsu-section-pad{padding: 15px 0px 15px 0px;}.tatsu-g7apowtr2v7qcctc > .tatsu-bottom-divider{z-index: 9999;}.tatsu-g7apowtr2v7qcctc > .tatsu-top-divider{z-index: 9999;}.tatsu-g7apowtr2v7qcctc .tatsu-section-overlay{mix-blend-mode: normal;}<\/style><\/div>\n","protected":false},"excerpt":{"rendered":"<p><a href=\"https:\/\/www.quantum-classical-physics.com\/qcp\/simple-derivation-of-schrodinger-equation-from-newtonian-dynamics\/\" class=\"more-link style1-button\">Read More<\/a><\/p>","protected":false},"author":1,"featured_media":197,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[4,3],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Simple derivation of Schr\u00f6dinger equation... - Quantum Classical Physics by Michele Marrocco<\/title>\n<meta name=\"description\" content=\"Simple derivation of Schr\u00f6dinger equation from Newtonian dynamics, Michele Marrocco. 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