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Suggestions Wanted — How to create a Physics Problem Database?

Physics
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Physics graduate students at UCSD (and many other schools) have to pass a PhD qualifying exam. At UCSD, this exam is taken across 2 days. Taking and preparing for that exam (called The Qual) is often a stressful time of life.

One of the most useful ways to study for the Qual is to work through old qualifying exams. Graduate students have had access to the tests and the solutions stretching back to 1987. The tests are offered twice a year, so there are a fair number of questions to be solved out there. I have organized PDFs of each of the exams and what solutions can be found from the past Quals, but I was hoping to make the information more accessible and more useful.

The problems on the Qual are organized into several different categories. The first category is Undergraduate or Graduate problems. (The first day of the exam is Undergraduate, the second day is Graduate). Within those categories there are several subjects that are tested: Mechanics, Electricity & Magnetism, Statistical Mechanics, Quantum Mechanics, Mathematical Methods, and Miscellaneous. Other identifying features: Year, Season, and Problem number. So, for example.

2006 Spring #13

Graduate
Electricity & Magnetism

Problem: An electron is incident with impact parameter \rho and speed v_0 on a proton at rest. Calculate the energy radiated during the collision assuming the ordering

\displaystyle  \frac{e^2}{\rho} \ll mv^2_o \ll mc^2

Hint: this simplifies the orbit approximation.

Solution:

\displaystyle  F=ma=\frac{e^2}{r^2}
\displaystyle  \left | \vec{a}\right |=\frac{e^2}{mr^2}\\
\displaystyle  \left | \vec{a}\right |^2=\frac{e^4}{m^2r^4}\\
\displaystyle  \ddot{\vec{p}}=e\ddot{\vec{r}}
\displaystyle  \vec{a}=\ddot{\vec{r}}
\displaystyle  P=\frac{2}{3}\frac{\left(\ddot{\vec{p}}\right)^2}{c^3}

\displaystyle  P=\frac{2}{3}\frac{e^2}{c^3}\left | \vec{a}\right |^2=\frac{2}{3}\frac{e^6}{m^2r^4c^3}

Energy radiated is \displaystyle \int P \,dt

\displaystyle  r^2=\rho^2+\left(v_o t\right)^2

Since
\displaystyle  mv_o^2 \gg \frac{e^2}{\rho^2}
\displaystyle  E=\frac{2}{3}\frac{e^6}{m^2c^3}\int_{- \infty  }^{  \infty }  \frac{1}{\left (\rho^2+\left (v_o t \right )^2\right)^2}\,dt
\displaystyle  E=\frac{2}{3}\frac{e^6}{m^2c^3}\frac{1}{v_o\rho^4}\underbrace{\int_{- \infty  }^{  \infty }  \frac{1}{\left (1+x^2\right)^2}\,dx}_{\frac{\pi}{2}}

Now, I think it would be very useful to be able to store the problem and solution and all the identifying pieces of information in a database in some way so that the following types of actions could be done.

  • Search and return all Electricity & Magnetism problems.
  • Search and return all graduate Statistical Mechanics problems.
  • Create a random qualifying exam — meaning a new 20 question test randomly picking 2 Undergrad mechanics problems, 2 Undergrade E&M problems, etc. from past qualifying exams (over some restricted date range).
  • Have the option to hide or display the solutions on results.

Any suggestions or comments on how best to do this project would be greatly appreciated!

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