WKB approximation: Difference between revisions

Revision as of 01:07, 4 December 2023

Revision as of 02:13, 4 December 2023

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===Connection conditions===

It now remains to construct a global (approximate) solution to the Schrödinger equation. For the wave function to be square-integrable, we must take only the exponentially decaying solution in the two classically forbidden regions. These must then "connect" properly through the turning points to the classically allowed region. For most values of {{math|''E''}}, this matching procedure will not work: The function obtained by connecting the solution near <math>+\infty</math> to the classically allowed region will not agree with the function obtained by connecting the solution near <math>-\infty</math> to the classically allowed region. The requirement that the two functions agree imposes a condition on the energy {{math|''E''}}, which will give an approximation to the exact quantum energy levels.[[File:WKB approximation example.svg|thumb|WKB approximation to the indicated potential. Vertical lines show the energy level and its intersection with potential shows the turning points with dotted lines. The problem has two classical turning points with <math>U_1 < 0</math> at <math>x=ax_1

</math> and <math>U_1 > 0</math> at <math>x=b x_2

</math>.]]The wavefunction's coefficients can be calculated for a simple problem shown in the figure. Let the first turning point, where the potential is decreasing over x, beoccur labelledat as point<math>x=x_1 a
</math> and the second turning point, where potential is increasing over x, beoccur labelledat as point b<math>x=x_2
</math>. Given that we expect wavefunctions to be of the following form, we can calculate their coefficients by connecting the different regions using Airy and Bairy functions.

<math display="block">\begin{align}

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==== First classical turning point ====

For <math>U_1 < 0</math> ie. decreasing potential condition or <math>x=ax_1

</math> in the given example shown by the figure, we require the exponential function to decay for negative values of x so that wavefunction for it to go to zero. Considering Bairy functions to be the required connection formula, we get:

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<math>\Psi_{\text{WKB}}(x) = \begin{cases}

-\frac{N}{\sqrt{|p(x)|}}\exp{(-\frac 1 \hbar \int_{ax}^{xx_1} |p(x)| dx )} & \text{if } x < ax_1\\

\frac{N}{\sqrt{|p(x)|}} \sin{(\frac 1 \hbar \int_{x}^{ax_1} |p(x)| dx - \frac \pi 4)} & \text{if } bx_2 > x > ax_1 \\

\end{cases} </math>

==== Second classical turning point ====

For <math>U_1 > 0</math> ie. increasing potential condition or <math>x=b x_2

</math> in the given example shown by the figure, we require the exponential function to decay for positive values of x so that wavefunction for it to go to zero. Considering [[Airy function|Airy functions]] to be the required connection formula, we get:

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<math>\Psi_{\text{WKB}}(x) = \begin{cases}

\frac{N'}{\sqrt{|p(x)|}} \cos{(\frac 1 \hbar \int_{x}^{bx_2} |p(x)| dx - \frac \pi 4)} & \text{if } ax_1 < x < bx_2 \\

\frac{N'}{2\sqrt{|p(x)|}}\exp{(-\frac 1 \hbar \int_{bx_2}^{x} |p(x)| dx )} & \text{if } x > bx_2\\

\end{cases}</math>

==== Common oscillating wavefunction ====

Matching the two solutions for region <math>ax_1<x<bx_2 </math>, it is required that the difference between the angles in these functions is <math>\pi(n+1/2)</math> where the <math>\frac \pi 2</math> phase difference accounts for changing sinecosine to cosinesine for the wavefunction and <math>n \pi</math> difference since negation of the function can occur by letting <math>N= (-1)^n N' </math>. Thus:

<math display="block">\int_{bx_1}^{ax_2} \sqrt{2m \left( E-V(x)\right)}\,dx = (n+1/2)\pi \hbar ,</math>

Where ''n'' is a non-negative integer. This condition can also be rewritten as saying that:

::The area enclosed by the classical energy curve is <math>2\pi\hbar(n+1/2)</math>.

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==== General connection conditions ====

Thus, from the two cases the connection formula is obtained at a classical turning point, <math>x=a
</math>:<ref name=":2" />

<math> \frac{N}{\sqrt{|p(x)|}} \sin{\left(\frac 1 \hbar \int_{x}^{a} |p(x)| dx - \frac \pi 4\right)} \Longrightarrow - \frac{N}{\sqrt{|p(x)|}}\exp{\left(\frac 1 \hbar \int_{a}^{x} |p(x)| dx \right)} </math>

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and:

<math> \frac{N'}{\sqrt{|p(x)|}} \cos{\left(\frac 1 \hbar \int_{x}^{ba} |p(x)| dx - \frac \pi 4\right)} \Longleftarrow \frac{N'}{2\sqrt{|p(x)|}}\exp{\left(-\frac 1 \hbar \int_{ba}^{x} |p(x)| dx \right)} </math>

The positionWKB ofwavefunction equationsat coincidethe withclassical theturning positionpoint ofaway correspondingfrom regionsit ofis wavefunctionsapproximated onby theoscillatory xsine axisor atcosine function in the turningclassically pointsallowed region, represented in the left and growing or decaying exponentials in the forbidden region, represented in the right. The implication follows due to the dominance of growing exponential compared to decaying exponential. Thus, the solutions of oscillating or exponential part of wavefunctions can imply the form of wavefunction on the other regionsregion of potential as well as at the associated turning point.

===Probability density===