What this course is actually doing

The notes jump between models — Einstein, Debye, Drude, Sommerfeld, tight-binding — and scatter equations, and the connective tissue is easy to lose. Here's the spine everything hangs on. Once you see it, each chapter stops being a new random model and becomes the same machine run on different inputs.

The one question the whole course answers

A solid is ~10²³ atoms glued together. Given only that microscopic picture, can we predict the things you can actually measure — how much heat it stores, whether it conducts electricity, whether it's shiny, why silicon runs your computer? The surprising answer is yes, with embarrassingly simple models. The whole course is a tour of those models, each one fixing the previous one's biggest lie.

The recipe secretly in every chapter

This is the part the notes never say out loud. Every model — Einstein, Debye, Drude, Sommerfeld, tight-binding, all of them — is the same four-step pipeline:

1. Pick your "stuff" — either the atoms vibrating (→ heat) or the electrons moving (→ electricity, optics).
2. Find the allowed states / the dispersion E(k) — what energies are possible, as a function of momentum/wavevector k.
3. Count them: the density of states g(E) — how many states sit at each energy. This is the workhorse that reappears constantly.
4. Fill the states with the right statistics, then integrate to get a measurable number — bosons (phonons) use Bose–Einstein; electrons (fermions) use Fermi–Dirac/Pauli.

The narrative arc (mapping to the lectures)

Core takeaways

• It's always: states → density of states g(E) → fill with the right statistics → integrate. Same four steps, every chapter.
• Two casts of characters: vibrating atoms (phonons, bosons → heat) and electrons (fermions → electricity/optics). The course braids these two threads.
• Each model deliberately fixes the previous one's worst assumption (Einstein→Debye, Drude→Sommerfeld, free electrons→bands). Knowing what lie each model tells beats memorizing its formula.
• Bands are the climax. Metal vs. insulator vs. semiconductor = "are the bands full, and where's the Fermi level?"
• g(E) and the dispersion E(k) are everywhere. Confused by a new equation? Ask: is this defining E(k), counting states, or filling them?

Derivation: where P(n) ∝ e−En/kBT comes from

The Boltzmann factor is the bedrock — every distribution in the course sits on it. It follows from one postulate plus one Taylor expansion.

The single postulate

For an isolated system at equilibrium, every accessible microstate is equally likely. Your oscillator isn't isolated (it swaps energy with its surroundings), so apply the postulate to system + reservoir together, which is isolated.

The setup

System S in contact with a huge reservoir R, fixed total energy E_tot. If S is locked in a state of energy E_n, all remaining freedom lives in R, which must hold E_tot − E_n. So the probability is proportional to how many ways R can arrange itself:

P(n) ∝ Ω_R(E_tot − E_n)

A high-energy state of S is unlikely not because S "dislikes" energy, but because every unit S grabs is stolen from R — leaving R fewer arrangements. The suppression is bookkeeping on the bath's side.

Turning that into the exponential

Ω is astronomically huge, so work with its log — entropy, S_R = k_B ln Ω_R, i.e. Ω_R = eS_R/k_B. Then Taylor-expand the bath's entropy in the tiny energy E_n, using the definition of temperature dS/dE ≡ 1/T:

P(n) ∝ e^( S_R(E_tot − E_n)/k_B )
     ≈ e^( [S_R(E_tot) − E_n/T] / k_B )
     = e^( S_R(E_tot)/k_B ) · e^( −E_n/k_BT )
        └── constant in n ──┘   └ Boltzmann factor ┘

The constant is absorbed into normalization (that's what Z is). What remains: P(n) ∝ e−E_n/k_BT. The whole exponential is just ebath entropy, and the bath's entropy is linear in the borrowed energy with slope 1/T. Log → linear → exponential.

Derivation: the Bose–Einstein distribution ⟨n⟩ = 1/(eℏω/kBT − 1)

This is how you fill each phonon mode. It comes from one object you already know — a single quantum harmonic oscillator in thermal equilibrium.

A vibrational mode of frequency ω is an oscillator with energy ladder E_n = ℏω(n + ½), n = 0,1,2,…. Reinterpret n as "the number of phonons in this mode", so ⟨n⟩ is the average rung. Weight each rung by Boltzmann and sum. With β = 1/k_BT and x = e−βℏω (the ½ℏω cancels top and bottom):

Z    = Σ x^n = 1/(1 − x)                 (geometric series)
⟨n⟩ = (Σ n x^n)/(Σ x^n)
     = [ x/(1−x)² ] / [ 1/(1−x) ]
     = x/(1 − x)
     = e^(−βℏω)/(1 − e^(−βℏω))
     = 1/(e^(ℏω/k_BT) − 1)              ✓

The two limits are what to actually remember:

• High T (k_BT ≫ ℏω): ⟨n⟩ ≈ k_BT/ℏω — grows linearly, energy ≈ k_BT. Recovers classical equipartition → Dulong–Petit.
• Low T (k_BT ≪ ℏω): ⟨n⟩ ≈ e−ℏω/k_BT → 0 — the mode freezes out. This is exactly why heat capacity dies at low T.

Phonons are bosons (any number per mode → infinitely many rungs), and their number isn't conserved (heat the crystal, make more), so the chemical potential is μ = 0 — which is why the phonon form has no μ. Multiply ⟨n⟩ by ℏω, sum over all modes with g(ω), and you get the internal energy; differentiate by T for the heat capacity. Einstein and Debye differ only in which modes exist — they fill them with this exact distribution.

Written as an orientation to the Open Solid State Notes for TN2844. The course follows Steve Simon's Oxford Solid State Basics.