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## Main Question or Discussion Point

It seems Lev Landau created an entry exam to test his students, and the exam was known to be ridiculously hard. To get an idea as to how hard the test really was, I've been scouring the Internet for problems Landau proposed... so far I've managed to find only four.

**Electrodynamics**- A dielectric sphere with the electric and magnetic susceptibilities ##\varepsilon_1## and ##\mu_1## is rotating with angular frequency ##\omega## in a constant electric field ##\mathbf{E}## in a medium, characterized by the parameters ##\varepsilon_2## and ##\mu_2##. The angle between the rotation axis and the direction of ##\mathbf{E}## is ##\alpha##. Find the electric and magnetic fields inside the sphere and in the medium.

**Quantum mechanics**- The electron enters a straight pipe of circular cross section (radius ##r##). The tube is bent at a radius ##R \gg r## by the angle ##\alpha## and then is aligned back again. Find the probability that the electron will jump out.
- A hemisphere lies on an infinite two-dimensional plane. The electron falls on the hemisphere, determine the scattering cross section in the Born approximation.
- The electron "sits" in the ground state in the cone-shaped "bag" under the influence of gravity. The lower end of the plastic bag is cut with scissors. Find the time for the electron to fall out (in the semi-classical approximation).