In some molecules such as ammonia, double vibrations are observed at small energies. These small vibrations are thought to arise from double well potentials. To explain these small energy gaps, double potential wells are investigated, and complex calculations are made. Until recently, since the exact solution of the Schrödinger equation was not known, approximate solutions were always obtained. The exact solution of the Schrödinger equation is now known. In previous studies, a simple procedure for the general solution of the radial Schrödinger Equation has been found for spherical symmetric potentials without making any approximation. In this article, the Schrodinger equation was solved with this new solution method. It has been applied to the oscillations through a barrier tunneling potential for a particle in a central quadratic deformed potential well (double potential well). It is seen that there is no need to look for complex potentials to explain the double wells observed in small vibrations in ammonia and similar molecules, they were compared with experimentally measured energies. The results were found to be fully compatible. It has been seen that there can always be double well potentials at small energies for all molecules, atomic nuclei, and similar particles with angular momentum other than zero.
Nelson's stochastic mechanics formulates quantum dynamics as a real-time conservative diffusion process in which a particle undergoes Brownian-like motion with a fluctuation amplitude fixed by Planck's constant. While being mathematically equivalent to the Schrödinger formulation, this approach provides an alternative dynamical framework that enables the study of time-resolved quantities that are not straightforwardly defined within the standard operator-based approach. In the present work, Nelson's stochastic mechanics is employed to investigate tunneling-time statistics for bound states in double-well potentials. Using first-passage time theory within this framework, both the mean tunneling time,
τˉ, and the full probability distribution,
p(τ), are computed. The theoretical predictions are validated through extensive numerical simulations of stochastic trajectories for two representative potentials. For the square double-well potential, analytical expressions for
τˉ are derived and are shown to be in excellent agreement with simulations. In the high-barrier limit, the results reveal a direct relation between the stochastic-mechanical and quantum-mechanical tunneling times, expressed as
τQM=(π/2)τˉ, where
τQM corresponds to half the oscillation period of the probability of finding the particle in either well. This relation is further confirmed for generic double-well systems through a WKB analysis. As a concrete application, the inversion dynamics of the ammonia molecule is analyzed, yielding an inversion frequency of approximately 24 GHz, in close agreement with experimental observations. These results highlight the potential of stochastic mechanics as a conceptually coherent and quantitatively consistent framework for analyzing tunneling phenomena in quantum systems.
Highly accurate closed-form approximations are given for the ground state and first excited state wavefunctions and energies for a nonrelativistic particle in a one-dimensional double square well potential with a square barrier in between (so that the potential is a sequence of five constant pieces that alternate in value from being above and below the ground state and first excited state energies), under the assumption that the barrier is sufficiently wide that the tunneling across it is very small.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.
Chapter 5 familiarizes the student with some of the most basic features of quantum systems, by applying the Schrödinger equation to simple, one-dimensional problems. It shows that it is possible to obtain qualitatively correct conclusions about the behavior of quantum particles in the presence of more or less arbitrary potentials by replacing them with piecewise-constant potentials, which greatly simplifies the mathematics. Replacing the real potential with a simpler one that retains its basic characteristics has the great advantage of allowing us to study the essence of the physical situation, as illustrated by the examples of the one-dimensional potential step, the square well, the square potential barrier and the symmetric double well. These examples are used to discuss specific features such as quantum tunneling, and applications such as masers and molecular clocks.