 Matt Evans

An interesting blog about interesting things.

# physics-problems

• Particle in a One Dimensional Box Feb 11, 2023 - 2 min read

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).

• Deriving Stirlings Formula Feb 11, 2023 - 3 min read

## 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.

• Mandelstam Variables With Identical Particles Feb 11, 2023 - 2 min read

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.

• Fixed Target vs Collider Experiments With Discussion Feb 10, 2023 - 2 min read

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}

• Projects Feb 10, 2023 - 1 min read

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.

• Physics Notes, Solutions, and Related Stuff Feb 10, 2023 - 3 min read

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.

• Physics Problems and Related Stuff Dec 2, 2012 - 1 min read

Just a quick one here. It’s pushing a month since I last checked in, and not for want of desire. I’ve not been entirely negligent though: I’ve been busy putting together a few typed up solutions to both foundational and pretty challenging/rare problems in statistical physics. My motivation is two-fold: a) to get extremely familiar with the material, what with exams just around the corner and b) for a few of the problems, the solutions are nearly impossible to do or to find help online for if you can’t do them.