372(2): Analytical Solutions

The analytical solutions of the Lagrangian quantum mechanics of H, (Eq (1) of the attached) are related to the associated Laguerre polynomials as in Eq. (16). Maxima can probably work out all the analytical radial wavefunctions of atomic H for any valid n and l. This has been done many times by Horst Eckardt in previous UFT papers using the hydrogenic wavefunctions. This note checks that the Lagrangian and Hamiltonian quantum mechanics give the same result for H. The Lagrangian method may have computational advantages over the well known Hamiltonian method. That is the purpose of this work. To check the Runge Kutta integration of Eq. (1) with Maxima use the analytical results. The purpose of this is to check that the code works for H, then apply and develop the code for other atoms and molecules. Eq. (1) also holds for hydrogen like atoms, a term used in computational quantum chemistry (for example the alkali metals sodium, potassium and so on). I have used some interpretative material from Atkins, and I hope that he has not made an error. His calculations can be checked by computer algebra.


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