This is a microcanonical approach to the problem of the mixture of two ideal gases. It involves a somewhat tricky integral over the surface of an N-dimensional hypersphere, and as far as I can tell is an example beloved of professors for exam questions. It might be a good one to become familiar with if you’re taking a statistical physics course. Question Enclosed in a box of volume V are \(N_1\) molecules of species 1 with mass \(m_1\) and \(N_2\) molecules of species 2 with mass \(m_2\).

The reason this general problem is so useful in a wide range of areas of physics is in physics we love to deal with harmonic approximations of systems. The point of solving the problem of N harmonic oscillators in this way is that they approximate (actually, correspond to) the behaviour of the particles in an ideal gas. Here, we work out the number of states (microstates) available to the system, the entropy of the system, and derive the energy of the corresponding ideal gas as a function of temperature.

The one-dimensional random walk is a key foundational concept in statistical physics which crops up in a variety of situations, albeit normally with far more dimensions. This light-hearted question which is circling amongst statistical physics professors serves as a good introduction to random walks. If you were set this as a homework and were struggling with where to go, about now would be a good time to pat yourself on the back.

Question A point particle of mass \(m\) moves in the region \(0 \le x \le l\) and is reflected elastically at the walls at \(x=0\) and \(x=l\). Calculate the volume \(\Gamma_0(E)\) of the classical phase space with an energy smaller than \(E\). Assume that a particle initially has an energy \(E_0\). Demonstrate that the phase-space volume \(\Gamma_0(E)\) of this particle remains constant when the wall at \(x=l\) is moved slowly (adiabatic invariance).

## A little background to Stirling’s Formula ## Stirling’s approximation is vital to a manageable formulation of statistical physics and thermodynamics. It vastly simplifies calculations involving logarithms of factorials where the factorial is huge. In statistical physics, we are typically discussing systems of \(10^{22}\) particles. With numbers of such orders of magnitude, this approximation is certainly valid, and also proves incredibly useful. There are numerous ways of deriving the result, and further refinements to the approximation to be found elsewhere.

For the elastic scattering of identical particles, \(A + A → A + A\), what are the Mandelstam variables? ## Solution ## The Mandelstam variables are defined, for a process \(1 + 2 → 3 + 4\), as \begin{eqnarray} s &=& (p_1 + p_2)^2 \nonumber \\ t &=& (p_1 − p_3)^2 \nonumber \\ u &=& (p_1 − p_4)^2 \nonumber \end{eqnarray} where the \(p\)s are the four-momenta. Relevant elastic collision for the \(t\) and \(u\) Mandelstam variables.

A proton beam with a momentum of \(p=100\text{GeV}\) hits a fixed Hydrogen target (discussion beneath). a) What is the centre of mass energy \(\sqrt{s}\) for this interaction? In such an interaction, obviously only a fraction of the momentum carried by the incoming proton will be accessible to the interaction. Let’s calculate the momentum 4-vectors (in natural units) for this interaction: \begin{eqnarray} p_1^\mu &=& (E_p, \vec {p}_1) \\ p_2^\mu &=& (m_p, \vec 0) \end{eqnarray}

Physics Notes, Solutions, and Related Stuff I’ve written up a couple of foundational and a few somewhat challenging physics problems and notes which I think the web could benefit from. I’ve started with a few from Statistical Physics, and will be adding more in time that’s all, folks!. Note: I recently migrated these pages from my old site, so be sure to look out for typos.

I’ve typed up solutions to a few physics problems often set in BSc. courses, some because they aren’t easily found on the web and I’d like to make the lives of fellow labourers that little bit easier, others are here because by writing them up I was drilling them into my head. I’ve tried to make them as clear as possible in my write-ups from my often less-than-legible notes, but if you have any problems/comments or spot any errors, do get in contact.

Recently I returned from a particle physics experiment at the Paul Scherrer Institut, a nuclear research lab in Switzerland. I was one of ten students from the University of Heidelberg and ETH, Zürich who had two weeks of (nearly) free reign to carry out an experiment on the PSI’s proton beam line. To put into perspective how crazy that is, ordinarily the going rate for such a privilege is €10,000s per day!

Our goal was to measure a mysterious number called the “Panofsky ratio”. The ratio is named after Wolfgang Panofsky, first to attempt to measure it, and corresponds is the relative likelihood of two events involving particles called protons and pions occurring. It is important, because historically its value strongly contradicted the expectation of theoretical physicists.