371(6): Wavefunctions from a Lagrange Method

This note begins a new numerical development of quantum mechanics and relativistic quantum mechanics starting from the classical lagrangian, then quantizing the results. This first note is a simple idealized atom modelled by an electron orbiting a proton in a plane. It could also be applied to quantize the usual conic section orbit of a mass m orbiting a mass M in a plane. After quantization, two simultaneous equations (17) and (18) are found. These can be solved for psi(r) and psi(phi). In the Born Oppenheimer approximation the complete wavefunction is psi = psi(phi)psi(r). It may be argued that these can be found analytically with well known methods, but the advantage of this numerical method is that it can be extended to three dimensions and to a new development of relativisic quantum mechanics. It gives a new method for computational quantum chemistry in general, given the supercomputer power. The Dirac equation, for example, can be solved in a new way. The Maxima code is very powerful and contemporary desktops are also very powerful. We begin with this baseline problem so that the numerical results can be checked against known analytical results, notably the non relativistic orbitals of the lodestone of quantum mechanics, the H atom. Numerical mthods such as these can also be applied ot the ECE wave equation inferred in 2003.


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