{"title":"A real-valued description of quantum mechanics with Schrödinger's 4th-order matter-wave equation","authors":"Nicos Makris , Gary F. Dargush","doi":"10.1016/j.physo.2025.100262","DOIUrl":null,"url":null,"abstract":"<div><div>Using a variational formulation, we show that Schrödinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span>, produces the precise eigenvalues of Schrödinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. The variational forms of the matter-wave equations are computed numerically with a Ritz-spline method and we show how this method handles accurately discontinuous potentials with singular derivatives. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative, mirror energy levels. Schrödinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow></mrow></math></span>, at the expense of missing the negative, mirror energy levels.</div></div>","PeriodicalId":36067,"journal":{"name":"Physics Open","volume":"23 ","pages":"Article 100262"},"PeriodicalIF":0.0000,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physics Open","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2666032625000122","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0
Abstract
Using a variational formulation, we show that Schrödinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential , produces the precise eigenvalues of Schrödinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. The variational forms of the matter-wave equations are computed numerically with a Ritz-spline method and we show how this method handles accurately discontinuous potentials with singular derivatives. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative, mirror energy levels. Schrödinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential , at the expense of missing the negative, mirror energy levels.