Posts
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Three-Input Cost Minimization with One Fixed Input [Microeconomics I Studied #35]
A worked guide to short-run cost minimization with labor and materials variable, capital fixed, and a three-input Cobb-Douglas production function.
7 min read -
Opportunity Cost vs. Sunk Cost [Microeconomics I Studied #30]
A university-choice example that separates opportunity cost from sunk cost and shows how to make a forward-looking stay-or-transfer decision.
6 min read -
Average and Marginal Product of Labor [Microeconomics I Studied #23]
We flip to the supplier side and dig into how output Q responds to labor L (with K fixed) — including why piling on more workers eventually starts hurting.
7 min read -
WKB Connection Formulas [Quantum Mechanics I Studied #36]
At an isolated simple turning point, a local Airy solution and asymptotic matching connect the allowed and forbidden WKB regions.
7 min read -
The WKB Approximation (Wentzel–Kramers–Brillouin) [Quantum Mechanics I Studied #35]
Ch8 is all about WKB — the go-to approximation when V(x) varies slowly, letting you tackle bound states and tunneling without solving Schrödinger exactly.
8 min read -
The Variational Principle [Quantum Mechanics I Studied #34]
The variational principle is basically quantum gambling — guess a trial wavefunction, and your calculated energy is always ≥ the true ground state energy.
7 min read -
Van der Waals Interaction and the Stark Effect [Quantum Mechanics I Studied #33]
A walkthrough of Griffiths Problem 6.31, deriving the weak van der Waals attraction between two polarizable atoms using perturbation theory and Taylor series.
5 min read -
Perturbation Theory Practice Problems [Quantum Mechanics I Studied #32]
Working through classic perturbation theory homework problems — delta-function bumps in infinite square wells and harmonic oscillator perturbations, step by step.
5 min read -
Spin-Orbit Coupling (LS Coupling) and the Fine Structure of Hydrogen [Quantum Mechanics I Studied #31]
We flip to the electron's frame, crank through Biot-Savart and magnetic moments, and nail down the spin-orbit Hamiltonian correction behind hydrogen's fine structure.
5 min read -
Relativistic Correction and the Fine Structure of Hydrogen [Quantum Mechanics I Studied #30]
We're back to hydrogen again?! Using perturbation theory, we dig into fine structure — relativistic corrections, spin-orbit coupling, and the Lamb shift.
3 min read -
Kramers' Relation [Quantum Mechanics I Studied #29]
Proving ⟨1/r³⟩ for the hydrogen atom using Kramers' relation — brutal-looking but apparently the least painful option, thanks Uncle Griffiths T_T
6 min read -
The Feynman-Hellmann Theorem [Quantum Mechanics I Studied #28]
Working through the virial theorem proof step by step before tackling hydrogen's fine structure — commutators, expectation values, and all.
6 min read -
Time-Independent Perturbation Theory [Quantum Mechanics I Studied #27]
A friendly derivation of nondegenerate and degenerate time-independent perturbation theory, from first- and second-order corrections to the splitting of degenerate energy levels.
6 min read -
Two-Particle Systems and the Exchange Force [Quantum Mechanics I Studied #26]
Chapter 5 kicks off with identical particles — turns out in QM you literally can't tell two electrons apart, and that changes everything about how we write wave functions!
9 min read -
Clebsch-Gordan Coefficients [Quantum Mechanics I Studied #25]
A casual walkthrough of how to read Clebsch-Gordan coefficient tables — because thankfully some genius already worked out all those messy spin-coupling combos for us.
5 min read -
Addition of Angular Momenta [Quantum Mechanics I Studied #24]
How two spin-1/2 angular momenta combine into a spin-1 triplet and a spin-0 singlet, with coupled and uncoupled bases, ladder operators, and Clebsch-Gordan coefficients.
4 min read -
Larmor Precession and the Stern-Gerlach Experiment [Quantum Mechanics I Studied #23]
How a spin-1/2 magnetic moment precesses in a uniform magnetic field, and how a field gradient separates a silver-atom beam into two Stern-Gerlach components.
5 min read -
Spin [Quantum Mechanics I Studied #22]
An intuitive introduction to intrinsic spin, spin-1/2 states, Pauli matrices, measurement probabilities, and the spin-1 representation.
7 min read -
Angular Momentum [Quantum Mechanics I Studied #21]
Kicking off 2nd semester QM by turning the classical angular momentum L = r × p into a proper quantum operator using our existing momentum operator tools.
6 min read -
Magnetic Vector Potential: What Does B = Curl A Mean? [Electromagnetism I Studied #19]
Build an intuition for B = ∇×A, then derive the Coulomb-gauge vector potential of a steady current and separate the pinwheel analogy from the physics.
5 min read