Stationary perturbation theory, non-degenerate states

Perturbed oscillator

Problem:

A one-dimensional harmonic oscillator has momentum p, mass m, and angular frequency ω.  It is subject to a perturbation U = bx4, where b is a suitable parameter, so that perturbation theory is applicable.
(a)  Derive an expression for a and a in terms of x and p, using the fact that they satisfy
[a, a] = 1, H = ħω (aa + ½).
(b)  Evaluate the first-order shift of the energy eigenvalues due to the above perturbation using creation and annihilation operators.
(c)  Does your result suggest that something “goes wrong” for large n even for small b-values?  Argue qualitatively why this may be the case.

Solution:

Problem:

A charged particle of mass m and charge q is sitting in a harmonic potential U0(x) = ½mω2x2.  A weak constant electric field E is applied in the +x-direction, so that the potential is perturbed by U1(x) = -qEx.
(a)  Show that there is no change in the energy levels to first order in E.
(b)  Calculate the second-order change in the energy levels.
(c)  This problem can be solved exactly by changing variables to x' = x - qE/(mω2).  Show that the exact energy levels agree with your results in parts (a) and (b).

The following may be used without proof:
x = (ħ/(2mω))½(a + a),  a|n> = n½|n-1>,  a|n> = (n+1)½|n+1>.

Solution:

Problem:

Consider a single electron moving with an anharmonic potential energy given by
U(x) = ½ kx2 + k’x4.  Let H0 be the Hamiltonian of the system when k’ = 0.
(a)  Use perturbation theory to find the energies of the ground and excited states to first order, assuming that k’ << k.
(b)  Expand the ground state to first order in terms of the eigenstates {|n>} of H0.

Solution:


Perturbed hydrogenic atoms

Problem:

Calculate the first-order shift in the ground state of the hydrogen atom caused by the finite size of the proton.  Assume the proton is a uniformly charged sphere of radius r = 10-13 cm.  The ground state wave function of the hydrogen atom is
|1,0,0> = (πa03)exp(-r/a0)
and the Bohr constant is a0 = 0.53*10-10 m.

Solution:

Problem:

The correctly normalized hydrogen ground state wave function in 3D is given by
ψ0(r) = (πa03)exp(-r/a0),
where a0 = ħ2/(mee2) is the Bohr radius, which is numerically ~0.529 Ǻ.
(a)  Confirm that this does indeed satisfy the radial Schroedinger equation for hydrogen, and that the wave function is normalized to ∫d3r|ψ(r)|2 = 1.
(b)  Two identical ions are introduced on the z-axis at locations z = +d and -d.  Assuming that the effect of each ion on the electron can be treated as a point interaction,
Ue – ion = ν0 δ(r rion),
calculate the change in the hydrogen atom's ground state energy using first order perturbation theory.

Solution:

Problem:

(a)  If the Hamiltonian for a system may be written as  H = H0 + H1 where H1 is a small perturbation and H0ψ0 = E0ψ0, then the effect of H1 gives rise to an energy correction term ΔE0.  Deduce an expression for the energy correction term ΔE0 if ψ0 is non-degenerate.
(b)  The relativistic dependence of mass on velocity introduces into the energy operator for the hydrogen atom a correction term for the state
ψ0 = (πa03)exp(-r/a0)
that may be written as   (-E02/(2mc2))[1 + 2a0/r]2.
Determine, using first order perturbation theory, the consequent level shift.

Solution:

Problem:

A muonic atom is one in which an atomic electron is replaced by a muon.  The muon is 209 times more massive than the electron.
(a)  Compute the energy of the 2p – 1s muonic transition in 208Pb (Z = 82) under the assumption that Pb is a point nucleus.   Make reasonable assumptions and explain your assumptions.   Compare your result with the observed value of 5.8 MeV.
(b)  Use the transition-energy values computed and given in part (a) and simple scaling rules for hydrogenic atoms to give an order-of-magnitude estimate of the nuclear radius of Pb (whose actual nuclear charge radius is ~ 6 fm).
(c)  Use perturbation theory to calculate the first-order shift in the ground-state energy of an electron in hydrogenic  208Pb (Z = 82) due to the finite size of this nucleus.  Assume the nucleus is a uniformly charged sphere.  Why is this not a valid approach for the muonic Pb atom?

The ground state wave function of the hydrogen atom is |1,0,0> = (πa03)exp(-r/a0).

Solution: