structural:summarization, quality:correction
Schrödinger equation

The time-independent Schrödinger equation for one particle in one dimension can be written as − ℏ 2 2 m d 2 d x 2 Ψ (x) + V (x) Ψ (x) = E Ψ (x) {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi (x)+V(x)\Psi (x)=E\Psi (x)} or d 2 d x 2 Ψ (x) = 2 m ℏ 2 (V (x) − E) Ψ (x) ≡ 2 m ℏ 2 M (x) Ψ (x), {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}\left(V(x)-E\right)\Psi (x)\equiv {\frac {2m}{\hbar ^{2}}}M(x)\Psi (x),} where

The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M (x) is positive or negative. When M (x) is constant and negative, then the Schrödinger equation can be written in the form d 2 d x 2 Ψ (x) = 2 m ℏ 2 M (x) Ψ (x) = − k 2 Ψ (x), where k 2 = − 2 m ℏ 2 M. {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)=-k^{2}\Psi (x),\qquad {\text{where}}\quad k^{2}=-{\frac {2m}{\hbar ^{2}}}M.}

The solutions of this equation represent travelling waves, with phase-constant + k or − k. Alternatively, if M (x) is constant and positive, then the Schrödinger equation can be written in the form d 2 d x 2 Ψ (x) = 2 m ℏ 2 M (x) Ψ (x) = κ 2 Ψ (x), where κ 2 = 2 m ℏ 2 M. {\displaystyle {\frac {d^{2}}{dx^{2}}}\Psi (x)={\frac {2m}{\hbar ^{2}}}M(x)\Psi (x)={\kappa }^{2}\Psi (x),\qquad {\text{where}}\quad {\kappa }^{2}={\frac {2m}{\hbar ^{2}}}M.}

The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M (x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M (x) determines the nature of the medium, with negative M (x) corresponding to medium A and positive M (x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M (x) is sandwiched between two regions of negative M (x), hence creating a potential barrier.

The mathematics of dealing with the situation where M (x) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A full mathematical treatment appears in the 1965 monograph by Fröman and Fröman. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.