{"id":343,"date":"2023-06-30T16:46:31","date_gmt":"2023-06-30T16:46:31","guid":{"rendered":"https:\/\/www.quantum-classical-physics.com\/qcp\/?p=343"},"modified":"2023-07-26T17:01:15","modified_gmt":"2023-07-26T17:01:15","slug":"dirac-equation-from-stereographic-projection-of-the-momentum-sphere","status":"publish","type":"post","link":"https:\/\/www.quantum-classical-physics.com\/qcp\/dirac-equation-from-stereographic-projection-of-the-momentum-sphere\/","title":{"rendered":"Dirac equation from stereographic projection of the momentum sphere"},"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><span style=\"font-size: 14pt;\">The Dirac equation is commonly demonstrated under stringent hypotheses and after considerable math work made in relativistic quantum mechanics and quantum field theory. Here, a purely geometric approach free from hypotheses is suggested. The suggestion draws inspiration from the technique of stereographic projection that was developed before the quantum era to solve gyroscopic problems of classical mechanics. The projected variable is the generalized (or canonical) momentum vector. Its undetermined geometric orientations define a sphere in the momentum space and the projection onto the equatorial plane generates the Pauli matrices as soon as the conventional stereographic matrix is introduced. The associated eigenvalue problem results in the Dirac equation and the eigenvector (bispinor) has components that are related to geometric elements of the momentum space. The procedure has the advantage of revealing the correct form of the Dirac matrices without the mathematical effort that characterizes the presentation of the equation in traditional approaches. The other remarkable advantage is that, unlike the common reduction to the case of free space, the spatial inhomogeneity due to interaction potentials is included in the demonstration from the very beginning&#8230;.<\/span><\/p>\n<h5 style=\"text-align: center;\"><a href=\"https:\/\/iopscience.iop.org\/article\/10.1088\/1361-6404\/acdb0e\" target=\"_blank\" rel=\"noopener noreferrer\">FULL TEXT IN OPEN ACCESS (PDF)<\/a><\/h5>\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\/dirac-equation-from-stereographic-projection-of-the-momentum-sphere\/\" class=\"more-link style1-button\">Read 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