10.1: Trial Wavefunctions for Various Potentials
- Page ID
- 135894
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This is list of functions and the potentials for which they would be suitable trial wave functions in a variation method calculation.
\( \psi (x, \alpha) = 2 \cdot \alpha ^{ \frac{3}{2}} \cdot x \cdot exp(- \alpha \cdot x)\)
\( \psi (x, \alpha) = ( \frac{ 128 \cdot \alpha ^{3}}{ \pi})^{ \frac{1}{4}} \cdot exp(- \alpha \cdot x^{2})\)
- Particle in a gravitational field V(z) = mgz (z = 0 to ∞)
- Particle confined by a linear potential V(x) = ax (x = 0 to ∞)
- One-dimensional atoms and ions V(x) = -Z/x (x = 0 to ∞)
- Particle in semi-infinite potential well V(x) = if[ x \( \leq a, 0, b\)] (x = 0 to ∞)
- Particle in semi-harmonic potential well V(x) = kx2 (x = 0 to ∞)
\( \psi (x, \alpha) = ( \frac{ 2 \cdot \alpha}{ \pi})^{ \frac{1}{4}} \cdot exp(- \alpha \cdot x^{2})\)
- Quartic oscillator V(x) = bx4 (x = -∞ to ∞)
- Particle in the finite one-dimensional potential well V(x) = if[(x \( \geq\) -1 \( \cdot\) (x \leq 1), 0, 2] (x = -∞ to ∞)
- 1D Hydrogen atom ground state
- Harmonic oscillator ground state
- Particle in V(x) = | x | potential well
\( \psi (x, \alpha ) = \sqrt{ \alpha} \cdot exp(- \alpha \cdot |x|)\)
- This wavefunction is discontinuous at x = 0, so the following calculations must be made in momentum space
- Dirac hydrogen atom V(x) = - \( \Delta\) (x)
- Harmonic oscillator ground state
- Particle in V(x) = | x | potential well
- Quartic oscillator V(x) = bx4 (x = -∞ to ∞)
\( \psi (x) = \sqrt{30} \cdot x \cdot (1-x)\)
\( \Gamma (x) = \sqrt{105} \cdot x \cdot (1-x)^{2}\)
\( \Theta (x) = \sqrt{105} \cdot x^{2} \cdot (1-x)\)
- Particle in a one-dimensional, one-bohr box
- Particle in a slanted one-dimensional box
- Particle in a semi-infinite potential well (change 1 to variational parameter)
- Particle in a gravitational field (change 1 to variational parameter)
\( \Phi (r, a) = (a-r)\)
\( \Phi (r, a) = (a - r)^{2}\)
\( \Phi (r, a) = \frac{1}{ \sqrt{2 \cdot \pi \cdot a}} \cdot \frac{ \sin \frac{ \pi \cdot r}{a}}{r}\)
- Particle in a infinite spherical potential well of radius a
- Particle in a finite spherical potential well (treat a as a variational parameter)
\( \psi (r, \beta) = ( \frac{2 \cdot \beta}{ \pi})^{ \frac{3}{4}} \cdot exp (- \beta \cdot r^{2})\)
- Particle in a finite spherical potential well
- Hydrogen atom ground state
- Helium atom ground state
\( \psi (r, \beta) = \sqrt{ \frac{3 \cdot \beta ^{3}}{ \pi ^{3}}} \cdot sech( \beta \cdot r)\)
- Particle in a finite potential well
- Hydrogen atom ground state
- Helium atom ground state
\( \psi (x, \beta) = \sqrt{ \frac{ \beta}{2}} \cdot sech( \beta \cdot x)\)
- Harmonic oscillator
- Quartic oscillator
- Particle in a gravitational field
- Particle in a finite potential well
\( \psi ( \alpha, \beta) = \sqrt{ \frac{12 \alpha ^{3}}{ \pi}} \cdot x \cdot sech( \alpha \cdot x)\)
- Particle in a semi-infinite potential well
- Particle in a gravitational field
- Particle in a linear potential well (same as above) V(x) = ax (x = 0 to ∞)
- 1D hydrogen atom or one-electron ion
Some finite potential energy wells.
V(x) = if[(x \( \geq\) -1 \( \cdot\) (x \( \leq\) 1), 0, V0]
V(x) = if[(x \( \geq\) -1 \( \cdot\) (x \( \leq\) 1), 0, |x| - 1]
V(x) = if[(x \( \geq\) -1 \( \cdot\) (x \( \leq\) 1), 0, \( \sqrt{|x| - 1}\)]
Some semi-infinite potential energy well.
V(x) = if (x \( \leq\) a, 0, b)
V(x) = if[(x \( \leq\) 2), 0, \( \frac{5}{x}\)]
V(x) = if[(x \( \geq\) 2), 0, (x - 2)]
V(x) = if[(x \( \leq\) 2), 0, \( \sqrt{x-2}\)]