The variational method

Problem:

A particle moves non-relativistically in a three-dimensional harmonic oscillator potential.  The potential energy function is U(r) = αr2.  Use spherical coordinates.
(a)  Use the trial function ψ = A exp(-br2) in the variational method to find the ground-state energy and the normalized wave function.
(b)  Comment on the quality of the following trial functions.  Do not do the calculations, just comment on the functions' good and bad points.
ψ = A exp(-br)
ψ = A sech(-br)  [sech(x) = 1/cosh(x) = 2/(ex + e-x)]
ψ = A (1 + b2r2)-1
ψ = A sin(br)/(br)

Note:  [∫0 r4exp(-r2)dr = 3π½/8.  ∫0  r2exp(-r2)dr = π½/4.]

Solution:

Problem:

Assume that you are asked to use the variational method to estimate the ground state energy of a particle with mass m, given a one-dimensional potential energy function U(x).  Assume U(x) = -Bx, x < 0,  U(x) = Fx, x > 0, where B and F are positive constants with the appropriate units.
(a)  Assume B ≠ F.  Choose a trial wave function ψ(x) with two adjustable parameters and explain why it is an acceptable wave function.  (You do not have to find the ground-state enery.)
(b)  Assume B = F.  Choose a trial wave function ψ(x) with one adjustable parameter and explain why it is an acceptable wave function.  (You do not have to find the ground-state enery.)
(c)  Let B --> ∞.  Choose a trial wave function ψ(x) with one adjustable parameter and estimate the ground state energy

Solution:

Problem:

Assume the potential for the deuterium is given by
U(r) = -U0exp(-r2/a2), with U0 = 37 MeV and a = 2.2*10-15 m. 
Estimate the ground state energy of the deuterium.

Solution:

Problem:

Consider the half space linear potential in one dimension
U(x) = ∞, x < 0, U(x) = Fx, x > 0.
Use the variational method to estimate the ground state energy.  Choose your own variational wave function.

Solution:

Problem:

A particle is bound to a site at the origin by a linear “string” potential, U(r) = br, where b is the string tension.  In 3D the Hamiltonian is H = a∇2 + br, where a = -ħ2/2m.
(a)  The energy eigenvalues are all proportional to aqbp.  Use dimensional analysis to determine both p and q.
(b)  Assume a trial wave function of the form ψ(r) = Bexp(-d2r2/2).
(1)  Determine the normalization constant B.
(2)  Calculate the expected energy and find the optimum variational parameter d.

Solution:

Problem:

It is known that the stable H- exists (two electrons bound to a proton).  Estimate the ground state energy of H- using the variational method.  Assume that each electron moves in a 1s orbit.  Neglect spin.

Useful information:
ψ1s(r) = (1/πa03)½exp(-r/a0)
< ψ1s |(1/r)| ψ1s > = 1/a0
< ψ1s |∇2| ψ1s > = -1/a02
∫∫d3r d3r'|ψ1s(r)|21s(r')|2(1/|r - r'|) = 5/(8a0)

Solution: