Find the ground state energy of the He atom using the variational method.
Useful information:
ψ1s(r) = (1/πa03)½exp(-r/a0)
for the hydrogen atom.
<ψ1s|(1/r)|ψ1s> = 1/a0.
<ψ1s|∇2|ψ1s> = -1/a02.
∫∫d3r d3r'|ψ1s(r)|2|ψ1s(r')|2(1/|r
- r'|) = 5/(8a0).
In one dimension, the potential energy of a particle of mass m as a function of x is
given by
U(x) = (b2|x|)½. Here b is a positive constant
with units energy/length½.
(a)
Use the WKB method to estimate the energy of the particle in the ground state.
(b)
Use the variational method to estimate the energy of the particle in the ground
state.
(c) Which estimate is closer to the true ground-state energy?
∫0∞exp(-x2) √x dx = Γ(3/4)/2 = 0.612708
Consider a particle of mass m placed in an infinite two-dimensional potential
well of width a.
U(x,y) = 0 if 0 < x < a and 0 < y < a, U(x,y) = ∞ everywhere else.
The particle is also subject to a perturbation W described by
W(x,y) = W0 for 0 < x < a/2 and 0 < y < a/2, W(x,y) = 0 everywhere else.
(a) Calculate, to first order in W0, the perturbed energy of the ground state.
(b) Calculate, to first order in W0, the perturbed energy of the first excited
state.
Give the corresponding wave functions to 0th order in W0.
The spin Hamiltonian for an electron (a spin-½ particle), in an external
magnetic field is given by
H = -γS∙B,
where the gyromagnetic ratio γ = -qe/m, qe being the
magnitude of the charge of the electron.
Suppose that the magnetic field consists of two components along the z and y
axes, respectively, i.e. let B = B0k + B2j.
Let us consider the case that B2 << B0. Using
perturbation theory, evaluate the possible energies of the electron to second
order in the ratio B2/B0.
Suppose the potential energy between an electron and a proton had a term U0(a0
/r)2 in addition to usual electrostatic potential energy -e2/r,
where e2 = qe2/(4πε0).
To the first order in U0, where U0 = 0.01 eV, by how much
would the ground state energy of the hydrogen atom be changed?