Motion of the proton in an asymmetric double minimum potential

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Abstract
A method different from the semiclassical WKB method was developed for calculating the energy levels of a proton and its penetration through a barrier under an asymmetric double minimum potential. The potential was written as the sum of a parabolic term with minimum at X = 0 and a Gaussian with maximum at X0. The stationary states were found by the variational method using the eigenfunctions of the harmonic oscillator for the parabolic potential. The time dependence of the system was found by expanding the eigenstate of the proton at t = 0 in terms of the stationary states. A Gaussian centered at the minimum of the left well and with an exponent consistent with the ground state of the proton in this well represented the proton at t = 0. The proton is found to oscillate anharmonically between the two wells. The penetration through the barrier is conveniently described in terms of the average frequency of oscillation and in terms of the extreme of the expectation value of the proton's position in the second well. Both the average frequency of oscillation and the extent of penetration decrease as the height of the barrier increases from 3.0 to 5.0 kcal/mole. At 6.2 kcal/mole no penetration was observed.

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