{"id":70,"date":"2008-09-03T22:21:34","date_gmt":"2008-09-04T03:21:34","guid":{"rendered":"http:\/\/www.fruechtetheory.com\/blog\/2008\/09\/03\/the-graviton-as-a-momentum-operator\/"},"modified":"2009-05-26T18:52:31","modified_gmt":"2009-05-26T23:52:31","slug":"the-graviton-as-a-momentum-operator","status":"publish","type":"post","link":"https:\/\/www.fruechtetheory.com\/blog\/2008\/09\/03\/the-graviton-as-a-momentum-operator\/","title":{"rendered":"The Graviton as a Momentum Operator"},"content":{"rendered":"<p>The fundamental commutator relation [<strong>x<\/strong>, <strong>p<\/strong>] = i\u045b, between the operators of coordinate and momentum, provides a way to show how a graviton can add linear momentum to an electron.<br \/>\nAn energy relation for a synchronous encounter by a graviton with an oncoming electron in an atomic orbital can start with:<br \/>\n[<strong>x<\/strong>, <strong>p<sub>g<\/sub><\/strong>]<sup>2<\/sup> = i<sup>2<\/sup>\u045b<sup>2<\/sup> = -\u045b<sup>2<\/sup>,<br \/>\nthe added kinetic energy being \u045b<sup>2<\/sup>k<sub>i<\/sub><sup>2<\/sup>\/2m<sub>i<\/sub>, and the added momentum -\u221a(\u045b<sup>2<\/sup>k<sub>i<\/sub><sup>2<\/sup>).<br \/>\nThe added momentum, as shown by the minus sign, is in the opposite direction of that in which the graviton was traveling at the speed of light in a vacuum before it was absorbed by the electron.<br \/>\nInternal to the electron we can use the {N} representation to form the basis of a set of wavefunctions forming orthonormal vectors: \u25020 >, \u25021 >, \u2026, \u2502n >, \u2026, with eigenvalues of N: 0, 1, \u2026, n, \u2026 [Messiah, XII.16, pg 436].\u00a0 The graviton in the process of being absorbed by an electron in a quantum atomic orbital can then be seen as a raising operator, where<br \/>\n<strong>a<\/strong><sup>\u2020<\/sup>\u03c6<sub>n<\/sub> = \u03c6<sub>n+1<\/sub> and <strong>a<\/strong><sup>\u2020<\/sup>\u03c6<sub>n+1<\/sub> = \u03c6<sub>n+2<\/sub> ,<br \/>\nand the release of a graviton a lowering operator, with<br \/>\n<strong>a<\/strong>\u03c6<sub>n<\/sub> = \u03c6<sub>n-1<\/sub> and <strong>a<\/strong>\u03c6<sub>n-1<\/sub> = \u03c6<sub>n-2<\/sub> .<br \/>\nThe Hamiltonian for such a system is represented as:<br \/>\n<strong>H<\/strong> \u03c6<sub>n<\/sub> = \u045b\u03c9<sub>0<\/sub>(<strong>a<\/strong><sup>\u2020<\/sup><strong>a<\/strong> + \u00bd) \u03c6<sub>n<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0[Liboff, Section 7.2],<br \/>\nwith energy eigenvalues<br \/>\nE<sub>n<\/sub> = \u045b\u03c9<sub>0<\/sub>(n + \u00bd)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 n = (0, 1, 2, \u2026, 68, \u2026n<sub>max<\/sub>).<br \/>\nHere 68 represents the average n value at the face of the earth, and n<sub>max<\/sub> depends on the orbital.<br \/>\nIf the mass of the electron diminishes as the gravitational field diminishes, the characteristic wavenumber <em>\u03b2<\/em> of the electron also diminishes.\u00a0 For each graviton internal to the electron <em>\u03b2<\/em><sub>i<\/sub><sup>2<\/sup> = m<sub>i<\/sub>\u03c9<sub>0<\/sub>\/\u045b, and for the mass of the electron at the face of the earth we have \u03a3m<sub>i<\/sub> = m<sub>e<\/sub> = 9.1095 x 10<sup>-31<\/sup> kg.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The fundamental commutator relation [x, p] = i\u045b, between the operators of coordinate and momentum, provides a way to show how a graviton can add linear momentum to an electron. An energy relation for a synchronous encounter by a graviton with an oncoming electron in an atomic orbital can start with: [x, pg]2 = i2\u045b2 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-70","post","type-post","status-publish","format-standard","hentry","category-quantum-mechanics"],"_links":{"self":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/70","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/comments?post=70"}],"version-history":[{"count":0,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/posts\/70\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/media?parent=70"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/categories?post=70"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fruechtetheory.com\/blog\/wp-json\/wp\/v2\/tags?post=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}