Editing Instanton (section)

=== WKB approximation ===
One way to calculate this probability is by means of the semi-classical [[WKB approximation]], which requires the value of <math>\hbar</math> to be small. The [[Schrödinger equation#Time-independent equation|time independent Schrödinger equation]] for the particle reads

:<math>\frac{d^2\psi}{dx^2}=\frac{2m(V(x)-E)}{\hbar^2}\psi.</math>

If the potential were constant, the solution would be a plane wave, up to a proportionality factor,

:<math>\psi = \exp(-\mathrm{i}kx)\,</math>

with

:<math>k=\frac{\sqrt{2m(E-V)}}{\hbar}.</math>

This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to

:<math>e^{-\frac{1}{\hbar}\int_a^b\sqrt{2m(V(x)-E)} \, dx},</math>

where ''a'' and ''b'' are the beginning and endpoint of the tunneling trajectory.