FEYNMAN, RICHARD P. 2 PP AM ON QUANTUM PROBLEM USING HIS FAVORITE VARIATIONAL METHOD, CA 1978

Autograph manuscript, 2 pp (8 1/2 x 11 in) in black ink on plain white laid paper. Creases where previously folded, a couple of small marginal closed tears.

FEYNMAN, RICHARD. "Difficulties in Applying the Variational Principle to Quantum Field Theories." pp. 28-40 IN: Proceedings of the International Workshop on Variational Calculations. (L. Polley and D.E.L. Pottinger, eds.) Singapore: World Scientific Publishing, 1988.

FEYNMAN WORKS OUT A QUIRKY QUANTUM PROBLEM WITH THE SINUSOIDAL POTENTIAL, USING THE VARIATIONAL METHOD, A FAVORITE EARLY TRICK OF HIS, TO WHICH HE OFTEN RETURNED IN HIS RESEARCH.   In early September 1987, Feynman presented gave a talk entitled "Difficulties in Applying the Variational Principle to Quantum Field Theories" at the International Workshop on Variational Calculations in Quantum Field Theory in West Germany. The present manuscript is likely notes made while preparing for his talk. The printed proceedings of the International Workshop were dedicated to the memory of Feynman who died just 5 short months after the workshop took place.

In the calculation, Feynman considers a trial wavefunction consisting of Gaussians centered in the valleys of the given sinusoidal confining potential; he then Fourier expands this periodic waveform in term of cosine harmonics, obtaining a recursion relation connecting Fourier coefficients a_k in successive rungs of the ladder.  This gains him a nested, continued fraction representation for the energy e as it relates to amplitude, V_o, of the confining potential.