Editing WKB approximation (section)

=== 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>