Hydrogenic atoms

Problem:

Use the measured ionization energy of Helium (Eionization = 24.6 eV) to calculate the energy of interaction of the two electrons in the ground state of the atom.

Solution:

Problem:

Consider an "atom" of positronium made up of an electron and a positron. 
Calculate the energies of the lowest three states of the system and briefly enumerate correction terms you have neglected.

Solution:

Problem:

The ground state wave function for a hydrogen like atom is
Φ100(r) = (1/√π)(Z/a0)3/2 exp(-Zr/a0),
where  a0 = ħ2/(μe2) and μ is the reduced mass, μ ~ me = mass of the electron. 
(a)  What is the ground state wave function of tritium?
(b)  What is the ground state wave function of 3He+?
(c)  An electron is in the ground state of tritium.  A nuclear reaction instantaneously changes the nucleus to 3He+.  Assume the beta particle and the neutrino are immediately removed from the system.  Calculate the probability that the electron remains in the ground state of 3He+.

Problem:

The emission spectrum of hydrogenic Lithium ions is measured and the wavelength of a series of emission lines are recorded.  The longest line in that series has a wavelength of 450.25 nm.  What is the wavelength of the shortest line in that series?

Solution:

Problem:

A certain species of ionized atoms produces an emission line spectrum according to the Bohr model, but the number of protons in the nucleus is unknown.  A group of lines in the spectrum forms a series in which the shortest wavelength is 22.79 nm and the longest wavelength is 41.02 nm.  Find the next-to-longest wavelength in the series of lines.

Solution:

Problem:

Certain nuclei can occasionally de-excite by internal conversion, which is a process whereby the excitation energy is transferred directly to one of the atomic electrons, causing it to be ejected from the atom.  (This process competes with de-excitation by photon emission.)  It is a reasonable assumption that the probability of this occurrence on a particular electron is directly proportional to the probability of that electron being at the nucleus.  Of the n = 2 electrons, which are the most likely to undergo internal conversion and why?  Estimate the ratio of conversion of 1s electrons to that of 2s electrons.  Assume that the nuclear excitation energy is much greater than the ionization energy of the 1s electron.

Hydrogenic atoms wave functions:
R10(r) = 2 (Z/a)3/2 exp(-Zr/a), 
R20(r) = (Z/(2a))3/2 (2 - Zr/a) exp(-Zr/(2a)),
R21(r) = 3(Z/(2a))3/2(Zr/a) exp(-Zr/(2a)),
Y00 = (4π),  Y1±1 = ∓(3/8π)½sinθ exp(±iφ),  Y10 = (¾π)½cosθ.

Solution:

Problem:

For hydrogen-like atoms, such as the alkali atoms, the screening effect of the "closed-shell" electrons can be accounted for by considering the electron to move in the potential V(r) = -(e2/r)(1 + α/r), where α is a constant.  Find the energy eigenvalues for this potential.

Solution:

Problem:

Consider a quasi-stable muonic atom composed of a muon in a 1s state and 9 electrons around a Ne nucleus (Z = 10).
(a)  Calculate the approximate energy and radius of the lowest energy bound state of the muon.  Comment on the effect of the outer electrons and of the finite size of the nucleus on the wave functions of this state.
(b)  Calculate the approximate energy difference between the lowest energy electron states in the muonic atom and in ordinary neon.

Solution: