Closed-form solutions for the modified potential

Abstract

Heretofore, the modified potential had to be described by approximation. Now, closed-form solutions may be developed for the modified potential. Closed-form solutions may now prove the hypotheses, which heretofore could only be shown to be plausible, that the quantum-action-variable quantization is precisely consistent with quantization of wave mechanics and that the Schradinger wave function has microstates. An alternate representation of the set of nonlocal hidden variables for determining quantum continuous motion may be developed in terms of the closed-form solutions. The relationship between this alternate representation for the set of nonlocal hidden variables and Ermakov's invariant may now be developed.The modified potential, which is the keystone for describing continuous quantum r n~t i o n , ' .~ has been determined heretofore by either numerical ~o m~u t a t i o n s ,~~~ perturbation expansion,' or power series.' The onerous numerical computations have obscured insight while the ponderous asymptotic expansions diverge in the classically "forbidden" region. Consequently, the Bohr-Sommerfeld quantization of the (effective) action variable for continuous quantum motion for a stationary bound state has been inferred numerically for only a particular set of potentials, and a general proof has yet to be ~f f e r e d .~ The existence of microstates has only been inferred numerically and by power-series expansion also for only a particular set of potentials. l Herein, we present closed-form solutions for the modified potential. These solutions are combinations of products of the pair of solutions to the time-independent Schrbdinger equation. The closed-form solutions render insight into the foundations of quantum mechanics. With the closed-form solutions for the modified potential, we can prove that the action variable quantization for continuous quantum motion' is consistent with eigenvalue quantization of the time-independent Schrljdinger equation. These solutions for the modified potential are not unique, which is a manifestation of hidden variables. We present an alternate set of hidden variables formed, in part, from the coefficients of the closed-form solutions. With the closed-form solutions, we can present a general proof that the Schrodinger wave function has microstates. We also present the relationship of these coefficients to the Ermakov invariant. One dimension suffices for this exposition.For stationary states in one dimension, x, the Hamilton-Jacobi equation for continuous quantum motion is given by2where W is Hamilton's characteristic function, U is the modified potential where the dependence upon the entire set of hidden variablesIs2 [ x , i , x ] is made explicit here, E is energy, and p is mass. The modified potential, U, is determined in one dimension, x, by the auxiliary equation'where fi is Planck's constant and V is the potential. A closed-form solution for U and the conjugate momentum P may be given by and where 4 and 0 are the pair of independent solutions to the time-independent S...

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