# A Problems from Lev Landau's "Theoretical Minimum"

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1. Mar 20, 2017

### JoePhysics

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
1. 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
1. 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.
2. A hemisphere lies on an infinite two-dimensional plane. The electron falls on the hemisphere, determine the scattering cross section in the Born approximation.
3. 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).
Does anyone know where to find more problems? These should be entertaining and good practice for the graduate qualifying exam.

2. Mar 20, 2017

### Staff: Mentor

3. Mar 22, 2017

### JoePhysics

Yes, that PSE is exactly where I got the quantum mechanics problems from. It does seem to mention his books, though I'm not sure which problems in his books showed up in the exam later on. I was hoping there was a compendium of some sort somewhere.

4. Mar 22, 2017

### Staff: Mentor

Any Lev problem is worthy of solving. They were all so difficult and I didn't have the patience to solve them or if I did I'd get stuck and couldn't prove my answer was correct. It takes a certain level of knowledge and brilliance which I have yet to attain to get through his stuff.

Look through each chapter and assume the last few problems are the toughest ones.