# Term 1 — Quantum Mechanics

> Now we do physics. Term 0 gave us the mathematics of complex Hilbert spaces; here we adopt the
> **four postulates** that turn that mathematics into a theory of nature — and into the operating
> system of a quantum computer. By the end you'll know precisely what a quantum *state* is, how it
> *evolves*, what *measurement* does to it, and how systems *combine* to produce entanglement. This
> is also where the curriculum becomes hands-on: **your first AWS Braket code** appears in Course 1.2.

**Estimated time:** ~90 hours · **Lessons:** 19 across 6 courses
**Prerequisites:** [Term 0 — Mathematical & Computational Foundations](../term-0-foundations/README.md)
(especially [Course 0.1 — Linear Algebra](../term-0-foundations/c01-linear-algebra/01-complex-vector-spaces.md)).
**Lab setup:** complete [Appendix A — Braket Setup](../appendix/A-braket-setup.md) before Course 1.2.

---

## The Four Postulates (the spine of this term)

All of quantum mechanics hangs on four statements. Course 1.1 introduces them; the rest of the term
unpacks each one with the full machinery.

| # | Postulate | Mathematical content | Developed in |
|---|---|---|---|
| 1 | **State** | A system's state is a unit vector $\lvert\psi\rangle$ in a Hilbert space. | 1.1.1, Course 1.2 |
| 2 | **Observables & Measurement** | Observables are Hermitian operators; outcomes are eigenvalues; probabilities obey the Born rule; the state collapses. | 1.1.2, Course 1.3 |
| 3 | **Evolution** | Closed-system dynamics is unitary: $\lvert\psi(t)\rangle = e^{-iHt}\lvert\psi(0)\rangle$. | 1.1.3, Course 1.6 |
| 4 | **Composite systems** | Joint systems combine via the tensor product — the origin of entanglement. | 1.1.4, Courses 1.4–1.5 |

## Course Map

```mermaid
flowchart TD
    C1["1.1 Postulates of QM\n(4 lessons)"] --> C2["1.2 Qubits & Bloch Sphere\n(3 lessons)"]
    C2 --> C3["1.3 Measurement\n(3 lessons)"]
    C2 --> C4["1.4 Entanglement\n(4 lessons)"]
    C3 --> C4
    C4 --> C5["1.5 Density Matrices\n(3 lessons)"]
    C2 --> C6["1.6 Dynamics\n(2 lessons)"]
    C1 --> C6
    C5 --> T2["→ Term 2: QC Core"]
    C6 --> T2
    LAB["Braket setup (App. A)"] -.-> C2
```

---

## Courses & Lessons

### Course 1.1 — The Postulates of Quantum Mechanics  `c01-postulates/`
The axioms, stated once, cleanly, with their immediate consequences.
1. [The State Postulate](c01-postulates/01-state-postulate.md) — states as rays; normalization; global vs relative phase.
2. [Observables & the Measurement Postulate](c01-postulates/02-observables-measurement-postulate.md) — Hermitian observables, the Born rule, collapse.
3. [The Evolution Postulate](c01-postulates/03-evolution-postulate.md) — unitarity and the Schrödinger equation.
4. [The Composite Systems Postulate](c01-postulates/04-composite-systems-postulate.md) — tensor products and the seed of entanglement.

### Course 1.2 — Qubits, Superposition & the Bloch Sphere  `c02-qubits-bloch/`
The qubit, made concrete — and your first runnable Braket code.
1. [The Qubit](c02-qubits-bloch/01-the-qubit.md) — two-level systems, superposition; **first Braket circuit**.
2. [The Bloch Sphere](c02-qubits-bloch/02-bloch-sphere.md) — the geometry of one qubit.
3. [Bases & Expectation Values](c02-qubits-bloch/03-bases-and-expectation.md) — X/Y/Z bases; measuring on Braket.

### Course 1.3 — Measurement  `c03-measurement/`
What measurement really is, from projectors to POVMs.
1. [Projective Measurement](c03-measurement/01-projective-measurement.md) — Born rule, collapse, measuring in any basis.
2. [Expectation & Uncertainty](c03-measurement/02-expectation-uncertainty.md) — variance and the Robertson uncertainty relation.
3. [POVMs & Generalized Measurement](c03-measurement/03-povm-generalized.md) — the most general measurements.

### Course 1.4 — Entanglement & Composite Systems  `c04-entanglement/`
The defining quantum resource.
1. [Multi-Qubit States](c04-entanglement/01-multi-qubit-states.md) — product vs entangled.
2. [Bell States](c04-entanglement/02-bell-states.md) — maximal entanglement.
3. [Nonlocality & the CHSH Inequality](c04-entanglement/03-nonlocality-chsh.md) — Bell's theorem, $2\sqrt2$.
4. [Schmidt Decomposition](c04-entanglement/04-schmidt-decomposition.md) — quantifying entanglement.

### Course 1.5 — Density Matrices & Mixed States  `c05-density-matrices/`
The formalism for noise, ensembles, and subsystems.
1. [The Density Operator](c05-density-matrices/01-density-operator.md) — pure vs mixed states.
2. [Partial Trace & Reduced States](c05-density-matrices/02-partial-trace-reduced-states.md) — the state of a subsystem.
3. [The Bloch Ball & Purity](c05-density-matrices/03-bloch-ball-purity.md) — mixed states geometrically.

### Course 1.6 — Dynamics  `c06-dynamics/`
How states change in time — and where gates come from.
1. [Time Evolution](c06-dynamics/01-time-evolution.md) — the Schrödinger equation and the propagator.
2. [Two-Level Dynamics & Rabi Oscillations](c06-dynamics/02-two-level-rabi.md) — the physical origin of single-qubit gates.

---

## Term Learning Outcomes

By the end of Term 1 you will be able to:

1. State and apply the **four postulates** of quantum mechanics in Dirac notation.
2. Represent a qubit as a state vector and a **Bloch-sphere** point, and compute measurement
   probabilities and **expectation values**.
3. Analyze **projective** and **POVM** measurements and derive the **uncertainty relation**.
4. Identify and quantify **entanglement**, and prove the **CHSH** violation of local realism.
5. Use the **density-matrix** formalism for mixed states, partial traces, and reduced subsystems.
6. Solve **time evolution** under a Hamiltonian and explain how **Rabi oscillations** become gates.
7. Run and interpret your first **AWS Braket** circuits on the local simulator.

---

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