# Pre-Term — From Classical to Quantum Mechanics

> Quantum mechanics has two front doors. The main program enters through the axioms — Hilbert
> spaces, postulates, matrices ([Term 0](../term-0-foundations/README.md) →
> [Term 1](../term-1-quantum-mechanics/README.md)). This term takes the other door: the historical,
> wave-mechanical route physicists actually stumbled through — classical mechanics at its most
> elegant, the experiments that broke it, and the differential equation Schrödinger wrote to patch
> reality. Optional for the spine, invaluable for physical intuition — and no self-respecting cat
> settles for a house with only one door open.

**Estimated time:** ~75 hours · **Lessons:** 18 across 6 courses
**Prerequisites:** multivariable calculus and ordinary differential equations (program background).
Fourier methods are strongly recommended — do
[Course 0.3 — Complex Analysis & Fourier Methods](../term-0-foundations/c03-fourier-methods/01-complex-numbers-functions.md)
first or in parallel.
**Position in the program:** optional relative to the Term 0 → Term 1 spine; can be taken **in
parallel with Term 0**. It feeds physical intuition directly into
[Term 1](../term-1-quantum-mechanics/README.md) and hardware understanding into Term 4.4
(the harmonic oscillator is the beating heart of superconducting qubits).

> **Convention note.** Unlike the rest of the program, which sets $\hbar = 1$
> ([Appendix C](../appendix/C-notation-glossary.md)), this term keeps **$\hbar$ explicit**: we deal
> with real experiments, real units, and real electron-volts here.

---

## Why this term exists

There are two honest ways into quantum mechanics, and they teach different things.

The **axiomatic route** (the program's spine) postulates finite-dimensional Hilbert spaces and
works out the consequences with linear algebra. It is clean, modern, and exactly right for quantum
*computing*, where every system is a finite register of qubits. But it hands you the postulates
fully formed, like a cat presenting a dead mouse: impressive, and you have no idea where it came from.

The **wave-mechanical route** (this term) follows history: classical mechanics is recast in
Lagrangian and Hamiltonian form, a handful of experiments (blackbody radiation, the photoelectric
effect, atomic spectra, electron diffraction) demolish the classical picture, and the Schrödinger
equation emerges as the repair. Along the way you get what the axiomatic route cannot give you:
*continuous* systems — wave packets, tunneling, the harmonic oscillator, the hydrogen atom — and
the experimental discovery of spin, the original qubit.

The two routes meet in the middle. The wavefunction $\psi(x,t)$ of this term **is** the
position-representation of the abstract ket $\lvert\psi\rangle$ of Term 1; the Poisson bracket of
Course P.1 is the classical shadow of the commutator; the ladder operators of Course P.5 return in
Term 4's superconducting circuits. If you only ever compute with qubits you can skip this term —
but when someone asks *why* observables are Hermitian operators, or *what* a transmon actually is,
this is where the answers live.

If you have taken a quantum mechanics course before, skim each lesson and jump to its
**Checkpoint**; if you can answer those, move on.

## Course Map

```mermaid
flowchart TD
    C1["P.1 Theoretical Mechanics\n(4 lessons)"] --> C2["P.2 Radiation & the Old Quantum Theory\n(3 lessons)"]
    C2 --> C3["P.3 Wave–Particle Duality\n(2 lessons)"]
    C3 --> C4["P.4 The Schrödinger Equation\n(3 lessons)"]
    C4 --> C5["P.5 Bound States & 1D Potentials\n(3 lessons)"]
    C4 --> C6["P.6 Angular Momentum & Central Potentials\n(3 lessons)"]
    C5 --> C6

    C6 --> T1["→ Term 1: Quantum Mechanics"]
    C1 -. "Poisson brackets → commutators" .-> T1
    C5 -.-> T44["→ Term 4.4: Superconducting Qubits"]
```

**Dependency notes.** The courses form a single narrative and are best taken in order. Course P.1
(especially Hamiltonian mechanics) is referenced throughout; Courses P.2–P.3 are the experimental
motivation and are lighter mathematically; Courses P.4–P.6 are the technical core. The whole term
lands exactly where [Term 1.1.1 — The State Postulate](../term-1-quantum-mechanics/c01-postulates/01-state-postulate.md)
begins.

---

## Courses & Lessons

### Course P.1 — Theoretical Mechanics  `c01-theoretical-mechanics/`
Classical waves, and classical mechanics in the formulation quantum mechanics will quantize.
1. [Waves & the Wave Equation](c01-theoretical-mechanics/01-waves-and-the-wave-equation.md) — deriving and solving the classical wave equation, the same equation in strings, rods and gases; superposition, interference, standing waves.
2. [Wave Propagation, Diffraction & Gratings](c01-theoretical-mechanics/02-wave-propagation-diffraction.md) — the 3D wave equation and plane waves; Huygens, the diffraction angle $\theta\sim\lambda/d$, and the grating equation.
3. [Lagrangian Mechanics](c01-theoretical-mechanics/03-lagrangian-mechanics.md) — from Newton to generalized coordinates, the action, and Euler–Lagrange; constraints and d'Alembert's principle; conservation laws from the symmetries of space and time.
4. [Hamiltonian Mechanics](c01-theoretical-mechanics/04-hamiltonian-mechanics.md) — Legendre transform, Hamilton's equations, non-conservative forces, Liouville, Poisson brackets: the classical shadow of the commutator.

### Course P.2 — Radiation & the Old Quantum Theory  `c02-radiation-old-quantum/`
The experiments that broke classical physics, and the first desperate patches.
1. [Blackbody Radiation & Planck's Law](c02-radiation-old-quantum/01-blackbody-radiation.md) — Kirchhoff's law, Stefan–Boltzmann, Wien, the Rayleigh–Jeans ultraviolet catastrophe, and Planck's quantum of action.
2. [Atomic Models & Spectral Series](c02-radiation-old-quantum/02-atomic-models-and-spectra.md) — Thomson, Rutherford, Bohr; the hydrogen series a formula finally explained.
3. [Correspondence & the Limits of the Old Quantum Theory](c02-radiation-old-quantum/03-correspondence-and-limits.md) — Bohr's correspondence principle, Wilson–Sommerfeld quantization, Sommerfeld's fine structure, why the old theory had to die, and the matrix and wave mechanics that replaced it.

### Course P.3 — Wave–Particle Duality  `c03-wave-particle-duality/`
Light behaves like particles; matter behaves like waves. Both, always, everywhere.
1. [Light as Particles](c03-wave-particle-duality/01-light-as-particles.md) — photoelectric effect, X-rays, Compton scattering, pair production and annihilation.
2. [Matter as Waves](c03-wave-particle-duality/02-matter-as-waves.md) — Young's double slit, the de Broglie hypothesis, and single-particle interference.

### Course P.4 — The Schrödinger Equation  `c04-schrodinger-equation/`
The equation itself, its interpretation, and its first consequences.
1. [The Schrödinger Equation & the Born Rule](c04-schrodinger-equation/01-schrodinger-equation-born-rule.md) — the equation, the probabilistic interpretation, and the conservation of probability.
2. [Stationary States & Superposition](c04-schrodinger-equation/02-stationary-states-superposition.md) — the time-independent equation; dynamics from superposed energy eigenstates.
3. [Wave Packets, Uncertainty & the Momentum Operator](c04-schrodinger-equation/03-matter-waves-uncertainty-momentum.md) — group velocity, the Heisenberg uncertainty principle, $\hat p = -i\hbar\,\partial_x$, and the energy–time relation.

### Course P.5 — Bound States & 1D Potentials  `c05-bound-states/`
The solvable one-dimensional problems every quantum mechanic must have in their paws.
1. [The Free Particle & Constant Potentials](c05-bound-states/01-free-particle-constant-potentials.md) — momentum eigenstates, allowed vs forbidden regions, the matching-conditions toolkit.
2. [Step, Well & Barrier](c05-bound-states/02-step-well-barrier.md) — reflection and transmission, bound states of the square well, tunneling.
3. [The Harmonic Oscillator](c05-bound-states/03-harmonic-oscillator.md) — analytic (Hermite) and algebraic (ladder-operator) solutions; zero-point energy; the road to superconducting qubits.

### Course P.6 — Angular Momentum & Central Potentials  `c06-angular-momentum/`
Quantum mechanics in three dimensions — and the experiment that revealed the qubit.
1. [Central Potentials & Orbital Angular Momentum](c06-angular-momentum/01-central-potentials-angular-momentum.md) — separation of variables; $L^2$, $L_z$, Legendre functions and the spherical harmonics; the cubic box and the density of states.
2. [The Radial Equation & the Hydrogen Atom](c06-angular-momentum/02-radial-equation-hydrogen.md) — the effective potential, spherical wells, Coulomb bound states, the reduced mass, and the quantum numbers $n,\ell,m$.
3. [Magnetic Moments, Stern–Gerlach & Spin](c06-angular-momentum/03-magnetic-fields-stern-gerlach-spin.md) — magnetic interactions, the two-spot experiment, and the door to the qubit.

---

## Term Learning Outcomes

By the end of the Pre-Term you will be able to:

1. Derive and solve the **classical wave equation** in one and three dimensions, predict
   **interference**, **diffraction** ($\theta \sim \lambda/d$) and **grating** orders
   ($\sin\theta_m = m\lambda/s$), and reformulate Newtonian mechanics in **Lagrangian** and
   **Hamiltonian** form, classifying constraints, deriving the conservation laws from symmetry
   (**Noether**), computing **Poisson brackets** and recognizing them as the classical counterpart
   of commutators.
2. Explain, with derivations, how **blackbody radiation**, **atomic spectra**, the
   **photoelectric effect**, and **Compton scattering** forced the quantization of energy and of
   light, how far the **old quantum theory** (Bohr, Sommerfeld) got, and how matrix and wave
   mechanics replaced it.
3. State the **de Broglie hypothesis** and analyze **matter-wave interference**, connecting it to
   the superposition principle of the main program.
4. Solve the **time-dependent and time-independent Schrödinger equations**, interpret
   $\lvert\psi\rvert^2$ via the **Born rule** in the Hilbert space $L^2$, derive the **Heisenberg
   uncertainty principle** from wave-packet Fourier analysis, and state the **energy–time relation**
   correctly.
5. Solve the standard **1D potentials** — step, well, barrier, harmonic oscillator — analytically
   *and* numerically (shooting method, transfer matrices), including **tunneling** and
   **ladder operators**.
6. Separate the **3D Schrödinger equation** in Cartesian and spherical coordinates, derive the
   spectra of $L^2$ and $L_z$, the spherical wells and the **hydrogen atom** (with the reduced mass),
   and explain how the **Stern–Gerlach experiment** reveals spin — the physical prototype of the
   qubit.

---

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