10.8: Variation Method for the Quartic Oscillator
- Page ID
- 135915
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Approximate Methods: The Quartic Oscillator
For unit mass the quartic oscillator has the following energy operator in atomic units.
\( H = - \frac{1}{2} \frac{d^{2}}{dx^{2}} \blacksquare + kx^{4} \blacksquare\) \( \int_{- \infty}^{ \infty} \blacksquare dx\)
Suggested trial wavefunction: \( \psi (x; \beta ) := ( \frac{2 \beta}{ \pi})^{ \frac{1}{4}} exp( - \beta x^{2})\)
Demonstrate that the wavefunction is normalized.
\[ \int_{- \infty}^{ \infty} \psi (x; \beta )^{2} dx~assume,~ \beta > 0 \rightarrow 1 \nonumber \]
Evaluate the variational energy integral.
\[ E( \beta ) := \int_{- \infty}^{ \infty} \psi (x, \beta ) - \frac{1}{2} \frac{d^{2}}{dx^{2}} \psi (x, \beta ) dx + \int_{- \infty}^{ \infty} \psi (x, \beta ) x^{4} \psi (x, \beta ) dx |_{simplify}^{assume,~ \beta > 0} \rightarrow \frac{1}{16} \frac{8 \beta ^{3}}{\beta ^{2}} \nonumber \]
Minimize the energy with respect to the variational parameter \( \beta\) and report its optimum value and the ground-state energy.
β := 1 β := Minimize(E, β) β = 0.90856 E(β) = 0.68142
Plot the optimum wavefunction and the potential energy on the same graph.
Calculate the classical turning point and the probability that tunneling is occurring.
\[ \begin{align} x_{ctp} &= 0.68142^{ \frac{1}{4}} \\[4pt] &= 0.90856 \end{align} \nonumber \]
\[ 2 \int_{x_{ctp}}^{ \infty} \psi (x, \beta )^{2} dx \approx 0.083265 \nonumber \]
Compare the variational result to energy obtained by numerically integrating Schrödinger's equation for the quartic oscillator using the numerical integration algorithm provided below.
Numerical Solutions for Schrödinger's Equation
Integration limit: xmax := 3 Effective mass: μ := 1 Force constant: k := 1
Potential energy: \( V(x) := kx^{4}\)
Numerical integration of Schrödinger's equation:
Given
\( \frac{-1}{2 \mu} \frac{d^{2}}{dx^{2}} \Phi (x) + V(x) \Phi (x) = energy \Phi (x)\)
\( \Phi (-x_{max} = 0\)
\( \Phi '(-x_{max} = 0.1\)
\( \Phi := Odesolve (x, x_{max}\)
Normalize wavefunction: \( \Phi (x) := \frac{ \Phi (x)}{ \sqrt{ \int_{-x_{max}}^{x_{max}} \Phi (x) ^{2} dx}}\)
Enter energy guess: Energy = 0.6679864
Compare the variational and numerical solutions for the quartic oscillator by putting them on the same graph.
