Revisiting Schrodinger's fourth-order, real-valued wave equation and its implications to energy levels
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| Format: | Preprint |
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2023
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| _version_ | 1866910308287643648 |
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| author | Makris, Nicos |
| author_facet | Makris, Nicos |
| contents | In his seminal part IV, Ann. der Phys. Vol 81, 1926 paper, Schrodinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is 4th-order in space and 2nd-order in time. In view of the mathematical difficulties associated with the eigenvalue analysis of a 4th-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrodinger splits the 4th-order real operator into the product of two, 2nd-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic 2nd-order, complex-valued wave equation. In this paper we show that Schrodinger's original 4th-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his 2nd-order, complex-valued wave equation that predicted with remarkable success the visible energy levels observed in the visible atomic line-spectra of the chemical elements. Accordingly, the 4th-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that Quantum Mechanics can only be described with the less stiff, 2nd-order complex-valued wave equation; unless in addition to the emitted visible energy there is also dark energy emitted. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_13416 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Revisiting Schrodinger's fourth-order, real-valued wave equation and its implications to energy levels Makris, Nicos Quantum Physics In his seminal part IV, Ann. der Phys. Vol 81, 1926 paper, Schrodinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is 4th-order in space and 2nd-order in time. In view of the mathematical difficulties associated with the eigenvalue analysis of a 4th-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrodinger splits the 4th-order real operator into the product of two, 2nd-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic 2nd-order, complex-valued wave equation. In this paper we show that Schrodinger's original 4th-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his 2nd-order, complex-valued wave equation that predicted with remarkable success the visible energy levels observed in the visible atomic line-spectra of the chemical elements. Accordingly, the 4th-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that Quantum Mechanics can only be described with the less stiff, 2nd-order complex-valued wave equation; unless in addition to the emitted visible energy there is also dark energy emitted. |
| title | Revisiting Schrodinger's fourth-order, real-valued wave equation and its implications to energy levels |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2302.13416 |