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

Useful information:

ψ_{1s}(**r**) = (1/πa_{0}^{3})^{½}exp(-r/a_{0})
for the hydrogen atom.

<ψ_{1s} |(1/r)|ψ_{1s}> = 1/a_{0}.

<ψ_{1s} |∇^{2}|
ψ_{1s}> = -1/a_{0}^{2}.

∫∫d^{3}r d^{3}r'|ψ_{1s}(**r**)|^{2}|ψ_{1s}(**r'**)|^{2}(1/|**r
**-** r**'|) = 5/(8a_{0}) .

Find the ground state energy of the He atom using the variational method.

Useful information:

ψ_{1s}(**r**) = (1/πa_{0}^{3})^{½}exp(-r/a_{0})
for the hydrogen atom.

<ψ_{1s} |(1/r)|ψ_{1s}> = 1/a_{0}.

<ψ_{1s} |∇^{2}|
ψ_{1s}> = -1/a_{0}^{2}.

∫∫d^{3}r d^{3}r'|ψ_{1s}(**r**)|^{2}|ψ_{1s}(**r'**)|^{2}(1/|**r
**-** r**'|) = 5/(8a_{0}) .

Consider a quantum system with just three linearly independent states.

Suppose the Hamiltonian, in matrix form, is

H = V_{0}

1 - ε | 0 | 0 | ||

0 | 1 | ε | ||

0 | ε | 2 |

.

where V_{0} is a constant and ε << 1.

(a) Write down the eigenvalues and eigenvectors of the
unperturbed Hamiltonian, H_{0} (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 H_{0}.

(d) Use degenerate perturbation theory to find the first-order corrections to
the initially degenerate eigenvalues.

In one dimension, the potential energy of a particle of mass m as a function of x is
given by

U(x) = (b^{2}|x|)^{½}. Here b is a positive constant
with units energy/lemgth^{½}.

(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(-x^{2}) √x dx = Γ(3/4)/2 = 0.612708

The hydrogen atom states ^{2}S_{½} and ^{2}P_{½}
are still degenerate, even after including the spin-orbit interaction. However,
they are split by a small "Lamb shift" energy ħω_{0} = 2πħ*1.06 GHz.

The ^{2}P_{½} 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

H_{0} = ħ

ω_{0} |
0 | ||

0 | 0 |

.

(a) Find the eigenvalues of H in an applied electric field **E** = E_{field}
**k**.

(Use the matrix element <^{2}P_{½}|z|^{2}S_{½}>
= 3a_{0}, where the Bohr radius a_{0} ≈ 0.053 nm.)

Write down the low-field and the high-field approximations for the eigenvalues.

Sketch these eigenenergies vs. the field strength E_{field}.

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?