10.1: Trial Wavefunctions for Various Potentials

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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, \( \sqrt{x-2}\)]