File:Wigner function for tunnelling.ogv

Summary

The Wigner function for tunneling through the potential barrier Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>8</mn><msup><mi>e</mi><mrow class="MJX-TeXAtom-ORD"><mo>−<!-- &minus; --></mo><mn>0.25</mn><msup><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></mrow></msup></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle U(x)=8e^{-0.25x^{2}}}</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4e7ecd3eb5d2cdf6cc584a719822d6e254fe35" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.595ex; height:3.509ex;" alt="{\displaystyle U(x)=8e^{-0.25x^{2}}}"> in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>H</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>p</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn><mo>+</mo><mi>U</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle H(x,p)=p^{2}/2+U(x)}</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2a76f1a7e5645d322e5f18d378e158497422bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.979ex; height:3.176ex;" alt="{\displaystyle H(x,p)=p^{2}/2+U(x)}">. The Wigner's function normalization is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): <semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo>∫<!-- &int; --></mo><mi>d</mi><mi>x</mi><mi>d</mi><mi>p</mi><mspace width="thinmathspace"></mspace><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>W</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo><msup><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \int dxdp\,|W(x,p)|^{2}=1}</annotation></semantics> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf2d685c01e3863ba1004a27e799f040af201ec6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:22.449ex; height:5.843ex;" alt="{\displaystyle \int dxdp\,|W(x,p)|^{2}=1}"> during the entire evolution.

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