## Transcribed Text

Part 1
Consider states of the hydrogen atom with principal quantum number 71 - 2. (Neglect spin)
a) Using degenerate perturbation theory find corrections to energies of all these states due to
static uniform electric field F directed in z-direction and the respective zero order wave
functions.
b) Now imagine that you want to observe the energy splitting induced by the electric field
(Stark effect) by measuring optical absorption using monochromatic linearly polarized
light. Assuming that the atom is in its ground state and the light is polarized in z.
direction, determine frequencies of spectral lines in the absorption spectrum, which will
be observed, and their relative strength
c) Can Stark effect be observed with light polarized in x-direction?
Part 2
a. Now consider helium atom with two electrons assuming that both electrons are in the
same spin state so that you can ignore the spin, but you cannot ignore the fermion nature
of electrons. Including in your consideration only single electron states with quantum
numbers (1,0,0),(2,0,0),(2,1,0),(2,1,-1),(2,1,1) (I use standard notation (1,1,m) here),
introduce fermion creation and annibilation operators based on these states. Neglecting
electron-electron interaction, rewrite the Hamiltonian of this system in the occupation
number representation.
b. Assume that the atom is illuminated by a time-dependent quantum electromagnetic field
propagating in the x-direction, polarized in z-direction and characterized by modal
functions
(1.1)
and add to the electron Hamiltonian found above terms describing the electromagnetic
field and the field-atom interaction in the occupation number representation
(representation using fermionic and photonic creation- -annihilation operators)
c.
Carefully evaluate matrix elements in the atom-photon interaction Hamiltonian using
2m
single-particle electron functions (assume that x<< 1, so that you can set x 0 in the
2
exponential term in Eq. (1.1)), and determine which of them are different from zero.
d. Assume that the initial state of the photon-atom system is (one electron is in
the ground state, the other is in 1,0) state, and there is one photon in the given mode).
What is the total energy of the system in this state? Determine, which optical transitions
have non-zero probability to occur, describe the energies and structure of each final state.
Use energy conservation to verify possibility or impossibility of each suggested
transition.

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.