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Advanced level

Question no. 1

The Bohr model and the Schroedinger equation of hydrogen atom ( let us call them theories ) were worked out in order to account for the structure of hydrogen atom spectrum. Which of the following comments is correct?

Both the theories differ only in that the Schroedinger theory predicts the existence of spin, the intrinsic angular momentum of the electron.

Both the theories give the same results except that in order to describe the motion of the electron, the Bohr model uses the notion of classical trajectory, while the Schroedinger theory the notion of wave function.

For in principle classical theory the Bohr model brings relatively good results but does not predict that the value of the electron angular momentum can be equal zero what results from the Schroedinger theory. It also does not predict the existence of spin which predicts Dirac theory and not predict "radiative corrections" arising from the quantization of the electromagnetic field.

In fact, it is impossible to explain in detail the structure of hydrogen atom spectrum since the Bohr model is very rough one, whereas the Schroedinger theory uses the notion of wave function which is a vague concept.

Question no. 2

Originally defined in 1963, the Jaynes-Cummings model comprises a single two-state atom interacting with a single near-resonant quantized cavity mode of the electromagnetic field. Supposing that the atom is initially excited ( t = t0 ), what do you say about the occupation of the upper level of the atom at time t, if t > t0 ?

The excited level of the atom decays to its ground state according to the exponential Weisskopf-Wigner decay and does not reexcitate.
The occupation of the upper level exhibits collapses and revivals.
The upper level of the atom decays only when there is a radiation in the cavity.
The upper level of the atom decays only when the atom is in the vacuum.

Question no. 3

One knows that the solutions of the Dirac equation become singular when the nuclear charge is greater than 137. It is found with the expression for the 1s energy E1s = m √(1-Z2α2). ( Units are such that h/2π = c = 1. ) However, the Coulomb potential is not a realistic one when is considered nucleus with high value of Z. More realistic is the assumption that the nucleus is a homogeneously charged sphere and its radius is equal 1.2 A1/3 fm. What is the Z dependence of the energy of the state 1s when Z is greater than 137 and an electron is bound to a uniformly charged sphere?

Behaviour of the 1s state energy is still singular.
The value of the 1s state energy is real and always positive.
The value of the 1s state energy is real, at first positive but next negative and for Z ≥ 169 joins the lower continuum ( E ≤ -mc2 ).
The value of the 1s state energy is real and always negative.

Question no. 4

The most classical of single-mode quantum states is a coherent state. It can be expanded in terms of Fock states. Their probabilities follow a distribution. What kind of distribution is that?

A Gaussian distribution.
A Maxwell distribution.
A Rayleigh distribution.
A Poisson distribution.

Question no. 5

A muon is an elementary particle, one of the leptons. It can form muonic atom where the muon behaves exactly like heavy electron. Even in helium, the muon velocity near the nucleus is ≈ 0.2 c, while it can be several times this in heavy nuclei.

The interaction between the lepton field and the quantized electromagnetic field results in additional contributions to the energy levels of leptonic atoms. They are known as "radiative corrections" ( although perhaps improperly so for the case of vacuum polarization ). Which radiative correction is the most important for muonic hydrogen?

The contribution due to anomalous magnetic moment.
The contribution due to the self-energy.
The contribution due to vacuum polarization.
They all are similar in magnitude.

Question no. 6

Positronium is the atom consisting of an electron and a positron. The ground state of positronium is composed of three triplet states of total spin one ( 1 3S1 - called orthopositronium ) and a spin-zero singlet state ( 1 1S0 - called parapositronium ). The triplet states anihilate with lifetime of about 140 nsec and the singlet state of about 0.1 nsec. Into how many photons does primarily positronium anihilate?

1 3S1 state anihilates into three photons and 1 1S0 state into two.
1 3S1 state anihilates into two photons and 1 1S0 state into three.
All states of positronium anihilate into three photons.
All states of positronium anihilate into two photons.