Abstract The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.
For a nonrelativistic quantal particle moving with the angular momentum
l=0 in a spherically symmetric, smooth potential well, which in the neighborhood of the center of force gives rise to an attractive Coulomb force, a formula for the probability density of the particle at the center of force is derived by means of certain phase-integral approximations of an arbitrary order. From this formula, which is expected to be very accurate, the non-relativistic approximation of the Fermi-Segr\`e formula for the probability density of finding an
s electron at the nucleus of an atom or ion is obtained. The limits of validity of the Fermi-Segr\`e formula in the nonrelativistic approximation are clarified by this derivation.