10.32: Momentum-Space Variation Method for the Abs(x) Potential
- Page ID
- 136982
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The energy operator in atomic units in coordinate space for a unit mass particle with potential energy V = |x| is given below.
\[ H = \frac{-1}{2} \frac{d^2}{dx^2} \blacksquare + |x| \blacksquare \nonumber \]
Suggested trial wave function:
\[ \Psi (x, \beta ) := ( \frac{2 \beta}{ \pi})^{ \frac{1}{4}} exp(- \beta x^2) \nonumber \]
Demonstrate that the wave function is normalized.
\[ \int_{- \infty}^{ \infty} \Psi (x, \beta )^2 dx~~assume,~ \beta >0 \rightarrow 1 \nonumber \]
Carry out Fourier transform to get momentum wave function:
\[ \Phi (p, \beta ) := \frac{1}{ \sqrt{2 \pi}} \int_{- \infty}^{ \infty} exp(-ipx) \Psi (x, \beta ) dx |_{simplify}^{assume,~ \beta > 1} \rightarrow \frac{1}{2} \frac{2^{ \frac{3}{4}}}{ \pi ^{ \frac{1}{4}}} \frac{e^{\frac{-1}{4}} \frac{p^2}{ \beta}}{ \beta ^{ \frac{1}{4}}} \nonumber \]
Demonstrate that the momentum wave function is normalized.
\[ \int_{- \infty}^{ \infty} \overline{ \Phi (p, \beta )} \Phi (p, \beta ) dp~~~assume,~ \beta > 0 \rightarrow 1 \nonumber \]
The energy operator in momentum space is:
\[ H = \frac{p^2}{2} \blacksquare + |i + \frac{d}{dp} \blacksquare| \nonumber \]
Evaluate the variational energy integral:
\[ E( \beta ) := \int_{- \infty}^{ \infty} \overline{ \Phi (p, \beta )} \frac{p^2}{2} \Phi (p, \beta ) dp + \int_{- \infty}^{ \infty} \overline{ \Phi (p, \beta )} |i \frac{d}{dp} \Phi (p , \beta )| dp |_{simplify}^{assume,~ \beta >0} \frac{1}{2} \frac{ \pi^{ \frac{1}{2}} \beta^{ \frac{3}{2}} + 2^{ \frac{1}{2}}}{ \beta^{ \frac{1}{2}} \pi^{ \frac{1}{2}}} \nonumber \]
Minimize the energy with respect to the variational parameter β and report its optimum value and the ground-state energy.
\( \beta\) := 1 \( \beta\) := Minimize (E, \( \beta\)) \( \beta\) = 0.542 E( \( \beta\)) = 0.813
Plot the coordinate and momentum wave functions and the potential energy on the same graph.
