Assignment 10

Problem 1:

Find the ground state energy of the He atom using first-order perturbation theory.

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

Problem 2:

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.
1s |(1/r)|ψ1s> = 1/a0.
1s |∇2| ψ1s> = -1/a02.
∫∫d3r d3r'|ψ1s(r)|21s(r')|2(1/|r - r'|) = 5/(8a0) .

Problem 3:

Consider a quantum system with just three linearly independent states.
Suppose the Hamiltonian, in matrix form, is

H = V0  

  1 - ε    0    0   
     0    1    ε   
     0    ε   2   

.  


where V0 is a constant and ε << 1.
(a)  Write down the eigenvalues and eigenvectors of the unperturbed Hamiltonian,  H0 (ε = 0).
(b)  Solve for the exact eigenvalues of H.
Expand each of them in a power series in ε up to second order.
(c)  Use first- and second-order non-degenerate perturbation theory to find the approximate eigenvalue for the state that grows out of the non-degenerate eigenvector of H0.
(d)  Use degenerate perturbation theory to find the first-order corrections to the initially degenerate eigenvalues.

Problem 4:

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/lemgth½.
(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

Problem 5:

The hydrogen atom states 2S½ and 2P½ are still degenerate, even after including the spin-orbit interaction.  However, they are split by a small "Lamb shift" energy ħω0 = 2πħ*1.06 GHz.
The 2P½ state has the lower energy.
Consider only the subspace spanned by these two states.  With the appropriate choice of the zero of the energy scale, the Hamiltonian in the absence of an external field is

H0 = ħ 

  ω0   0   
   0   0   

.  


(a)  Find the eigenvalues of H in an applied electric field E = Efield k.
(Use the matrix element <2P½|z|2S½> = 3a0, where the Bohr radius a0 ≈ 0.053 nm.)
Write down the low-field and the high-field approximations for the eigenvalues.
Sketch these eigenenergies vs. the field strength Efield.
What are the approximate low-field and high-field eigenstates of H?

(b)  How large should the electric field be for the energy shifts (i.e. Stark shifts) to become linear?
(c)  What is the value of the linear Stark shift (in MHz/(V/cm)) at large electric fields?