WKB approximation: Difference between revisions

Revision as of 11:38, 9 April 2025

Revision as of 08:21, 15 April 2025

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By the requirement of continuity of wavefunction and its derivatives, the following relation can be shown:<math display="block">\frac {|ED|^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>.

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<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}|ED|^2) </math>

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

<math display="block">T = \frac {|ED|^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" />