Editing WKB approximation (section)

=== Quantum Tunneling ===
{{Main|Quantum tunnelling}}
The potential of such systems can be given in the form:

<math display="block">V(x) = \begin{cases}
0 & \text{if } x < x_1 \\
V(x) & \text{if } x_2 \geq x \geq x_1\\
0 & \text{if } x > x_2 \\ 
 \end{cases} </math>

where <math display="inline">x_1 < x_2 </math>. 


Its solutions for an incident wave is given as

<math display="block">\psi(x) = \begin{cases}
A \exp({ i p_0 x \over \hbar} ) + B \exp({- i p_0 x \over \hbar}) & \text{if } x < x_1 \\
 \frac{C}{\sqrt{|p(x)|}}\exp{(-\frac 1 \hbar \int_{x_1}^{x}  |p(x)| dx )} & \text{if } x_2 \geq x \geq x_1\\ 
D \exp({ i p_0 x \over \hbar} )  & \text{if } x > x_2 \\

 \end{cases}  </math>

where the wavefunction in the classically forbidden region is the WKB approximation but neglecting the growing exponential. This is a fair assumption for wide potential barriers through which the wavefunction is not expected to grow to high magnitudes. 


By the requirement of continuity of wavefunction and its derivatives, the following relation can be shown:<math display="block">\frac {|D|^2} {|A|^2} = \frac{4}{(1+{a_1^2}/{p_0^2} )} \frac{a_1}{a_2}\exp\left(-\frac 2 \hbar \int_{x_1}^{x_2} |p(x')| dx'\right)    </math>

where <math>a_1 = |p(x_1)|</math> and <math>a_2 = |p(x_2)|    </math>.


Using <math display="inline">\mathbf J(\mathbf x,t) = \frac{i\hbar}{2m}(\psi^* \nabla\psi-\psi\nabla\psi^*) </math> we express the values without signs as:

<math display="inline">J_{\text{inc.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|A|^2) </math>

<math display="inline">J_{\text{ref.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|B|^2)  </math>

<math display="inline">J_{\text{trans.}} = \frac{\hbar}{2m}(\frac{2p_0}{\hbar}|D|^2)  </math>


Thus, the [[transmission coefficient]] is found to be:

<math display="block">T = \frac {|D|^2} {|A|^2} = \frac{4}{(1+{a_1^2}/{p_0^2} )} \frac{a_1}{a_2}\exp\left(-\frac 2 \hbar \int_{x_1}^{x_2} |p(x')| dx'\right)    </math>

where <math display="inline">p(x) = \sqrt {2m( E - V(x))}        </math>, <math>a_1 = |p(x_1)|</math> and <math>a_2 = |p(x_2)|    </math>. The result can be stated as <math display="inline">T \sim ~ e^{-2\gamma} </math> where <math display="inline">\gamma = \int_{x_1}^{x_2} |p(x')| dx' </math>.<ref name=":1" />