PHYSICS DEPARTMENT, PRINCETON UNIVERSITYPHYSICS 301 MIDTERM EXAMINATIONOctober 26, 2005, 10:00–10:50 am, Jadwin A06This exam contains two problems. Work both problems. The problems count equallyalthough one might be harder than the other. Do all t he work you want graded in theseparate exam books.Write legibly. If I can’t read it, it doesn’t count!Put your name on all exam books that you hand in. (Only one should be necessary!!!)On the first exam boo k, rewrite and sign the honor pledge: I pledge my honor that I havenot violated the Honor Code during this examination.Copyrightc 2005, Princeton University Physics Department, Edward J. GrothPhysics 301 Midterm Exam 26-Oct-2005 Page 21. Consider atomic hydrogen in equilibrium at temperature τ. The temperature is lowenough that the el ectron is in its ground state a s far as its orbital motion is concerned. Be-cause the proton ( the nucleus of the hydrogen atom) and electron each have spin 1/2, thereare four possible states. There are very weak interactions (called hyperfine interactions)between the magnetic dipole moments of the proton and t he electron. This produces anenergy difference between the singlet and triplet states. This i s called hyperfine splitting.The singlet state occurs when the proton and electron spins anti-align to form a spin0 a tom.The three triplet stat es occur when the proton and electron spins are aligned forminga spin 1 atom.Our model is a four state system with one state (the singlet) with energy 0 and theother three (the triplet states) with energy ǫ > 0.We will be concerned with the modifications this internal hyperfine interaction makesto the statistical mechanics of a gas of N atoms. (In o ther words, we’re goi ng to ignore thestates of the atoms that correspond to the center of mass mo tion and focus on the internalstates resulting from the hyperfine splitting.)(a) Without elaborate calculation, what are the internal entropy and energy as τ → 0(that is, τ ≪ ǫ)?(b) Again without elaborate calculation, what are the internal entropy and energy ifτ →≫ ǫ?(c) What is the partition function for these i nternal states at arbitrary temperature?(Don’t forget that there are N independent atoms).(d) What are the free energy, entropy, and energy of the internal states of the N atomsat arbitrary temperature?Copyrightc 2 005, Princeton University Physics Department, Edward J. GrothPhysics 301 Midterm Exam 26-Oct-2005 Page 32. You may have read Abbott’s boo k Flatland which besides having some political andcultural themes is also an introduction to the “physics of other dimensions.” An omissionin the book is that cavity radiation in Flatland is not discussed. It turns out physics inFlatland is very simila r to physics in our own world. The speed of light is still c. Quan-tum mechanics still applies, so there is still an ¯h, and harmonic oscillators are quantumoscillators. The laws of thermodynamics are obeyed! There is a small difference: there isonly one polari zation of light.(a) In Flatland, a square box has area (volume) L × L = A. In such a box, how manymodes of electromagnetic radiation with frequency ω are there in the frequency intervaldω?(b) What is the average energy of a mode of frequency ω when it is in equilibrium attemperature τ? Ignore the zero po int energy of the oscillators.(c) What is the total energy in a Flatland cavity of area A in equilibrium at temperat ureτ? How does this total energy depend on temperature? (That is, what power oftemperature?) If you come across an integral you can’t do, be sure to put it indimensionless form so that it’s clearly just a number.Copyrightc 2 005, Princeton University Physics Department, Edward J.
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