The exact transmission coefficient of a one-dimensional potential barrier, V(x)=${\mathit{V}}_0$(1-[{1-exp(x/a)}/{1+c exp(x/a)}${]}^2$), of which the Morse and the Eckart barriers are special cases, has been obtained. Comparing the exact and WKB transmission coefficients, the limitation of the semiclassical method has been quantified. Another branch of this potential (c<0) has been shown useful in nucleus-nucleus fusion. Using the transmission coefficients, the exact energy eigenvalues of the inverted (${\mathit{V}}_0$→-${\mathit{V}}_0$) potential (oscillator) have been derived.

• Received 29 July 1992

DOI:https://doi.org/10.1103/PhysRevA.47.4761