Approximate solutions to the one-dimensional Schroedinger equation by the method of comparison equations

The method of "comparison equations" for obtaining approximate solutions to the one-dimensional Schrödinger equation is discussed. In this method, one Schrödinger equation, ψ''(x) + k²(x) ψ(x) = 0, is transformed into another, v''(z) + K²(z) v(z) = 0, by a simultaneous change of independent and dependent variables, x → z, ψ → v. Then ψ(x) and v(z) are related by ψ(x) = v(z) (dz/dx)^(−1/2) whenever k²(x) and K²(z) satisfy the relation K²(z) (dz/dx)² = k²(x) − ½ ⟨z; x⟩. Here ⟨z; x⟩ = z'''/z' − (3/2) (z''/z')² is the Schwarzian derivative of z with respect to x, and the prime indicates d/dx. A set of "best" criteria for the transformed potential K²(z) is obtained, where by "best" is meant that we can completely neglect ⟨z; x⟩ in first approximation and yet not have the turning-point problems that plague the WKB method (which is a special case of the comparison-equation method). The WKB method sets K²(x) ≡ 1 and neglects ⟨z; x⟩; the result is that the transformation x → z, ψ → ν is singular at the turning points, where k²(x) = 0. We choose K²(z) to have the proper asymptotic behavior far from the turning points, so that z' ≈ 1; hence ⟨z; x⟩ ≈ 0 in these regions, and we match the zeroes of k²(x) and K²(z) in order to keep the transformation regular. This method is applied to various potentials with one and two turning points. Transmission and reflection coefficients T and R and transmitted and reflected phase shifts μ and ν are calculated for potentials with one turning point and potential barriers, and expressed in terms of the energy E and the quantity $W=|{\int }_{x_1}^{x_2}kdx|$, where x₁,₂ are the possibly complex turning points, k²(x₁,₂) = 0. Quantization rules, in terms of the classical action, are derived for various types of potential wells.