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Particle in a One Dimensional Box
Feb 11, 2023
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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).
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Deriving Stirlings Formula
Feb 11, 2023
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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.
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Mandelstam Variables With Identical Particles
Feb 11, 2023
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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.
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Fixed Target vs Collider Experiments With Discussion
Feb 10, 2023
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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}
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Physics Notes, Solutions, and Related Stuff
Feb 10, 2023
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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.