# 10.41: Variation Method Using the Wigner Function - The Quartic Oscillator

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Define potential energy:

$V(x) = x^4 \nonumber$

Display potential energy:

Choose trial wave function:

$\psi (x, \beta = \left( \frac{2 \beta}{ \pi} \right)^{ \frac{1}{4}} exp( - \beta x^2) \nonumber$

Calculate the Wigner distribution function:

$W(x, p, \beta ) = \frac{1}{2 \pi} \int_{- \infty}^{ \infty} \psi \left(x + \frac{s}{2}, \beta \right) exp(isp) \psi (\left( x- \frac{s}{2}, \beta \right) ds~ \bigg|_{assume,~ \beta > 0}^{simplify} \rightarrow \frac{1}{ \pi} e^{ \frac{-1}{2} \frac{4 \beta ^2 x^2 + p^2}{ \beta}} \nonumber$

Evaluate the variational integral:

$E( \beta ) = \int_{- \infty}^{ \infty} \int_{- \infty}^{ \infty} W(x, p, \beta ) \left( \frac{p^2}{2} + V(x) \right)dx~dp \nonumber$

Minimize the energy integral with respect to the variational parameter, $$\beta$$.

$$\beta$$ = 1 $$\beta$$ = Minimize (E, $$\beta$$) $$\beta$$ = 0.90856 E( $$\beta$$) = 0.68142

Calculate and display the coordinate distribution function:

$Px(x, \beta ) = \int_{- \infty}^{ \infty} W(x, p, \beta ) dp \nonumber$

Classical turning points: $$x_{cl} = 0.681^{ \frac{1}{4}}~~~ x_{cl} = 0.90842$$

Probability that tunneling is occurring:

$2 \int_{0.908}^{ \infty} Px (x, \beta ) dx = 0.08345 \nonumber$

Calculate and display the momentum distribution function:

$Pp(p, \beta ) = \int_{- \infty}^{ \infty} W(x, p, \beta ) dx \nonumber$

Display the Wigner distribution function:

N = 60 i = 0 .. N xi = $$-3 + \frac{6i}{N}$$ j = 0 .. N pj = $$-5 + \frac{10j}{N}$$ Wigneri, j = W( xi, pj, $$\beta$$)

This page titled 10.41: Variation Method Using the Wigner Function - The Quartic Oscillator is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Frank Rioux via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.