Editing WKB approximation (section)

== Examples in quantum mechanics ==
Although WKB potential only applies to smoothly varying potentials,<ref name=":2" /> in the examples where rigid walls produce infinities for potential, the WKB approximation can still be used to approximate wavefunctions in regions of smoothly varying potentials. Since the rigid walls have highly discontinuous potential, the connection condition cannot be used at these points and the results obtained can also differ from that of the above treatment.<ref name=":1" />

=== Bound states for 1 rigid wall ===
The potential of such systems can be given in the form:

<math>V(x) = \begin{cases}
V(x) & \text{if }  x \geq x_1\\
  \infty & \text{if } x < x_1 \\
 \end{cases}</math>

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


Finding wavefunction in bound region, ie. within classical turning points <math display="inline">x_1 </math> and <math display="inline">x_2 </math>, by considering approximations far from <math display="inline">x_1 </math> and <math display="inline">x_2 </math> respectively we have two solutions:

<math>\Psi_{\text{WKB}}(x) = 
\frac{A}{\sqrt{|p(x)|}}\sin{\left(\frac 1 \hbar \int_{x}^{x_1} |p(x)| dx +\alpha \right)}   </math>

<math>\Psi_{\text{WKB}}(x) = 
\frac{B}{\sqrt{|p(x)|}}\cos{\left(\frac 1 \hbar \int_{x}^{x_2} |p(x)| dx +\beta \right)}     </math>

Since wavefunction must vanish near <math display="inline">x_1 </math>, we conclude <math display="inline">\alpha = 0 </math>. For airy functions near <math display="inline">x_2 </math>, we require <math display="inline">\beta = - \frac \pi 4    </math>. We require that angles within these functions have a phase difference <math>\pi(n+1/2)</math> where the <math>\frac \pi 2</math> phase difference accounts for changing sine to cosine and <math>n \pi</math> allowing <math>B= (-1)^n A </math>.

<math display="block">\frac 1 \hbar \int_{x_1}^{x_2} |p(x)| dx = \pi \left(n + \frac 3 4\right)  </math>Where ''n'' is a non-negative integer.<ref name=":1" /> Note that the right hand side of this would instead be <math>\pi(n-1/4)</math> if n was only allowed to non-zero natural numbers. 


Thus we conclude that, for <math display="inline">n = 1,2,3,\cdots      </math><math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = \left(n-\frac 1 4\right)\pi \hbar </math>In 3 dimensions with spherically symmetry, the same condition holds where the position x is replaced by radial distance r, due to its similarity with this problem.<ref>{{Cite book |last=Weinberg |first=Steven |url=http://dx.doi.org/10.1017/cbo9781316276105 |title=Lectures on Quantum Mechanics |date=2015-09-10 |publisher=Cambridge University Press |isbn=978-1-107-11166-0 |edition=2nd |pages=204|doi=10.1017/cbo9781316276105 }}</ref>

=== Bound states within 2 rigid wall ===
The potential of such systems can be given in the form:

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

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


For <math display="inline">E \geq V(x)          </math> between <math display="inline">x_1 </math> and <math display="inline">x_2 </math> which are thus the classical turning points, by considering approximations far from <math display="inline">x_1 </math> and <math display="inline">x_2 </math> respectively we have two solutions:

<math>\Psi_{\text{WKB}}(x) = 
\frac{A}{\sqrt{|p(x)|}}\sin{\left(\frac 1 \hbar \int_{x}^{x_1} |p(x)| dx \right)}    </math>

<math>\Psi_{\text{WKB}}(x) = 
\frac{B}{\sqrt{|p(x)|}}\sin{\left(\frac 1 \hbar \int_{x}^{x_2} |p(x)| dx \right)}     </math>

Since wavefunctions must vanish at <math display="inline">x_1 </math> and <math display="inline">x_2 </math>. Here, the phase difference only needs to account for <math>n \pi</math> which allows <math>B= (-1)^n A </math>. Hence the condition becomes:

<math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = n\pi \hbar  </math>where <math display="inline">n = 1,2,3,\cdots      </math> but not equal to zero since it makes the wavefunction zero everywhere.<ref name=":1" />

=== Quantum bouncing ball ===
Consider the following potential a bouncing ball is subjected to:

<math>V(x) = \begin{cases}
mgx & \text{if }  x \geq 0\\
  \infty & \text{if } x < 0 \\
 \end{cases}</math>

The wavefunction solutions of the above can be solved using the WKB method by considering only odd parity solutions of the alternative potential <math>V(x) = mg|x|</math>. The classical turning points are identified <math display="inline">x_1 = - {E \over mg}   </math> and <math display="inline">x_2 = {E \over mg}    </math>. Thus applying the quantization condition obtained in WKB:

<math display="block">\int_{x_1}^{x_2} \sqrt{2m \left( E-V(x)\right)}\,dx = (n_{\text{odd}}+1/2)\pi \hbar</math>

Letting <math display="inline">n_{\text{odd}}=2n-1 </math> where <math display="inline">n = 1,2,3,\cdots      </math>, solving for <math display="inline">E </math> with given <math>V(x) = mg|x|</math>, we get the quantum mechanical energy of a bouncing ball:<ref>{{Cite book |last1=Sakurai |first1=Jun John |title=Modern quantum mechanics |last2=Napolitano |first2=Jim |date=2021 |publisher=Cambridge University Press |isbn=978-1-108-47322-4 |edition=3rd |location=Cambridge}}</ref>

<math display="block">E = {\left(3\left(n-\frac 1 4\right)\pi\right)^{\frac 2 3} \over 2}(mg^2\hbar^2)^{\frac 1 3}. </math>

This result is also consistent with the use of equation from bound state of one rigid wall without needing to consider an alternative potential.

=== 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" />