143 research outputs found

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

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

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

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

It is shown a complex function $\Phi$ 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 $\Phi$-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
$\Phi$-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 $\Phi$-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

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