10.30: The Variation Method in Momentum Space
- Page ID
- 136979
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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 following normalized trial wavefunction is proposed for a variational calculation on the harmonic oscillator.
\[ \psi (x, a) := \sqrt{ \frac{1}{a}} exp( \frac{-|x|}{a}) \nonumber \]
\[ \int_{- \infty}^{ \infty} \psi (x, a)^2 dx~~~assume,~a > 0 \rightarrow 1 \nonumber \]
However, the graph below shows a cusp at x = 0, indicating that the wavefunction is not well‐behaved and therefore cannot be used for quantum mechanical calculations.
Therefore, the wavefunction is Fourier transformed into the momentum representation.
\[ \Phi (p, a) := \int_{- \infty}^{ \infty} \frac{exp(-ipx)}{ \sqrt{2 \pi}} \sqrt{ \frac{1}{a}} exp( \frac{-|x|}{a}) dx~|_{simplify}^{assume,~a>0} \rightarrow (-a^{ \frac{1}{2}}) \frac{2^{ \frac{1}{2}}}{(ipa-1) \pi ^{ \frac{1}{2}} (ipa +1)} \nonumber \]
Normalization is checked and the function is graphed.
\[ \int_{- \infty}^{ \infty} \Phi (p, a)^2 dp~~~assume,~a > 0 \rightarrow 1 \nonumber \]
The momentum wavefunction appears to be well‐behaved, so a variational calculation will be carried out in momentum space.
Assuming a m = k =1 and h = 2π, we have for the harmonic oscillator in momentum space.
- Momentum space integral: \( \int_{- \infty}^{ \infty} \blacksquare dp\)
- Momentum operator: \( p \blacksquare\)
- Kinetic energy operator: \( \frac{p^2}{2}\)
- Position operator: \( i \frac{d}{dp} \blacksquare\)
- Potential energy operator: \( \frac{-1}{2} \frac{d^2}{dp^2} \blacksquare\)
Evaluate the energy integral in the momentum representation:
\[ E(a) := \int_{- \infty}^{ \infty} \Phi (p, a) \frac{p^2}{2} \Phi (p, a) dp... + \int_{- \infty}^{ \infty} \Phi (p, a) \frac{-1}{2} \frac{d^2}{dp^2} \Phi (p, a) dp~|_{assume,~a >0}^{simplify} \rightarrow \frac{1}{4} \frac{2 + a^4}{a^2} \nonumber \]
Minimize energy with respect to the variational parameter:
a := 1 a := Minimize (E, a) a = 1.189 E(a) = 0.707
Display optimum wavefunction along with exact wavefunction:
\[ Exact(p) := \frac{1}{ \pi ^{ \frac{1}{4}}} e^{ \frac{-1}{2} p^2} \nonumber \]
Naturally the agreement with the exact solution is not favorable because of the poor quality of the original coordinate space wavefunction.
\[ \frac{E(a) - 0.5}{0.5} = 41.421 \nonumber \]