143 research outputs found

    Exact Solution of the Klein-Gordon Equation for the Hydrogen Atom Including Electron Spin

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    The term describing the coupling between total angular momentum and energy-momentum in the hydrogen atom is isolated from the radial Dirac equation and used to replace the corresponding orbital angular momentum coupling term in the radial K-G equation. The resulting spin-corrected K-G equation is a second order differential equation that contains no matrices. It is solved here to generate the same energy eigenvalues for the hydrogen atom as the Dirac equation.Comment: 6 page

    Conformal Transformation of the Schr\"{o}dinger Equation for the Harmonic Oscillator into a Simpler Form

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    The Schr\"{o}dinger equation and ladder operators for the harmonic oscillator are shown to simplify through the use of an isometric conformal transformation. These results are discussed in relation to the Bargmann representation. It is further demonstrated that harmonic interactions can be introduced into quantum mechanics as an imaginary component of time equivalent to adding the oscillator potential into the hamiltonian for the confined particle.Comment: 7 page

    Conformal Transformation of the Schr\"{o}dinger Equation for Central Potential Problems in Three-Dimensions

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    In a recent paper, it has been shown the Schr\"{o}dinger equation for the three-dimensional harmonic oscillator can be simplified through the use of an isometric conformal transformation. Here, it is demonstrated that the same transformation technique is also applicable to the Schr\"{o}dinger equation for the hydrogen atom. This approach has two interesting features. Firstly, it eliminates potential fields from the Schr\"{o}dinger equation. The Coulomb and harmonic binding terms are instead represented as imaginary parts of complex time. Secondly, the method leads to a general relationship between potential energy and ground state energy that encompasses both the hydrogen atom and the harmonic oscillator as special cases.Comment: 8 page

    A Derivation of the Quantized Electromagnetic Field Using Complex Dirac Delta Functions

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    It is shown a complex function Φ\Phi defined to be the product of a real Gaussian function and a complex Dirac delta function satisfies the Cauchy-Riemann equations. It is also shown these harmonic Φ\Phi-functions can be included in the solution of the classical electromagnetic field equations to generate the quantum field as a many-particle solution such that the Φ\Phi-functions represent the particle states. Creation and destruction operators are defined as usual to add or subtract photons from the particle states. The orbital angular momentum of the Φ\Phi-states is interpreted as spin since it emerges from a point source that must be circularly polarized as a requirement of the gauge condition.Comment: 10 page