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Series

Quantum Mechanics I Studied

20 posts

  1. #1

    Introduction to Quantum Mechanics: On Probability

    Two weeks into quantum mechanics and already in full hair-loss mode — here's my honest attempt to make sense of the Schrödinger equation and what a wave function even is.

    · 9 min read
  2. #2

    Normalization

    Why normalizing your wavefunction once is enough — turns out the Schrödinger equation secretly guarantees it stays normalized for all time.

    · 3 min read
  3. #3

    The Momentum Operator

    Cranking through the math to differentiate ⟨x⟩ with respect to time — and stumbling our way into the momentum operator.

    · 6 min read
  4. #4

    The Time-Independent Schrödinger Equation

    A breezy play-by-play of how Planck, Einstein, de Broglie, and Schrödinger loaded the bases and derived the time-independent Schrödinger equation — no mysticism required!

    · 4 min read
  5. #5

    The Time-Independent Schrödinger Equation, Part 2

    We assume the wave function splits into space and time parts, plug it into the Schrödinger equation, and crank out two way simpler ODEs — same trick as E&M!

    · 6 min read
  6. #6

    The Infinite Square Well

    We dive into the infinite square well, crank through boundary conditions on the Schrödinger equation, and land on quantized wave functions — same moves as E&M, surprisingly!

    · 8 min read
  7. #7

    The Harmonic Oscillator and Ladder Operators

    We tackle case 2 of V(x) — the harmonic oscillator — unpack why springs are *everywhere* in physics, then get into ladder operators to solve it.

    · 11 min read
  8. #8

    Ladder Operators

    We pick up the ladder-operator trick to climb from the ground state through every ψ_n, then chase down the normalization constants c_n and d_n purely in terms of n.

    · 5 min read
  9. #9

    Algebraic Solution to the Harmonic Oscillator

    Picking up right where we left off — muscling through the harmonic oscillator's Schrödinger equation with a slick substitution and some algebraic tricks!

    · 8 min read
  10. #10

    The Free Particle

    We crack open the free particle case (V(x) = 0 everywhere) — it looks deceptively easy until two existential problems crash the party almost immediately.

    · 6 min read
  11. #11

    The Delta-Function Potential

    Time for the delta-function potential — we sort out bound vs. scattering states, meet the Dirac delta, and motivate why V(x) = −αδ(x) is a solid model for electrons in a material.

    · 5 min read
  12. #12

    Delta-Function Potential: Scattering States and Quantum Tunneling

    Tackling the E>0 case of the delta-function potential — scattering states, complex exponentials that refuse to die, and how tunneling actually works.

    · 8 min read
  13. #13

    The Finite Square Well

    Tackling the finite square well — the sneaky middle ground between the infinite well and the delta function — through bound states, boundary conditions, and all that fun stuff.

    · 8 min read
  14. #14

    Formalism: Hilbert Space

    We finally crack open the math behind quantum mechanics — why it works, where it came from, and how Heisenberg's beach epiphany kicked the whole thing off.

    · 7 min read
  15. #15

    Formalism: Observables and Hermitian Operators

    Why measurement in QM always spits out real numbers — and how that forces every observable to be a Hermitian operator. (Daggers are literally knives, lol.)

    · 3 min read
  16. #16

    Formalism: Determinate States

    Transcribing Griffiths on determinate states — turns out demanding zero standard deviation just pops out the eigenvalue equation, and it all chains together beautifully.

    · 3 min read
  17. #17

    The Uncertainty Principle

    We derive the generalized uncertainty principle and chase down exactly when it hits its minimum — turns out the answer is when ψ is Gaussian, and here's why!

    · 7 min read
  18. #18

    Two-Level Systems

    We set up a two-state quantum system with a 2×2 Hermitian Hamiltonian and crank out the eigenvalues and eigenvectors using the Schrödinger equation.

    · 3 min read
  19. #19

    The Schrödinger Equation in Spherical Coordinates (3D)

    We bump quantum mechanics up to 3D for the hydrogen atom, watch Cartesian coordinates immediately implode, and brace ourselves for the spherical coordinate nightmare.

    · 16 min read
  20. #20

    The Hydrogen Atom

    We finally plug the real hydrogen electric potential into the radial equation, grind through the algebra, and find R(r) — normalization and all.

    · 10 min read