# 417: Trial Wavefunctions for Various Potentials

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, $$\sqrt{x-2}$$]