Editing Quantum tunnelling (section)

=== Schrödinger equation ===
The [[Schrödinger equation#Time independent equation 2|time-independent Schrödinger equation]] for one particle in one [[dimension]] can be written as
<math display="block">-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x)</math>
or
<math display="block">\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) ,</math>
where
* <math>\hbar </math> is the [[reduced Planck constant]],
* ''m'' is the particle mass,
* ''x'' represents distance measured in the direction of motion of the particle,
* Ψ is the Schrödinger wave function,
* ''V'' is the [[potential energy]] of the particle (measured relative to any convenient reference level),
* ''E'' is the energy of the particle that is associated with motion in the ''x''-axis (measured relative to ''V''),
* ''M''(''x'') is a quantity defined by ''V''(''x'') − ''E'', which has no accepted name in physics.

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
<math display="block">\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. </math>

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
<math display="block">\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. </math>

The solutions of this equation are rising and falling exponentials in the form of [[evanescent wave]]s. 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.