Operators, Observables & Dirac Notation

By the end you'll be able to

1. Write quantum states as kets and bras, compute ⟨φ|ψ⟩ as an overlap, and turn it into a probability with the Born rule.
2. Explain why every observable is represented by a Hermitian operator and why its eigenvalues, the only possible measurement results, are always real.
3. Expand a state in an observable's eigenbasis, read off the probability of each result, and calculate the expectation value ⟨ψ|Â|ψ⟩.
4. Use the Pauli matrices to find the eigenvalues, eigenstates and expectation values of an electron's spin or a qubit.
5. Calculate commutators, decide whether two observables are compatible, and explain how the commutator of x̂ and p̂, which equals iħ, produces the uncertainty principle.
6. Describe in words what the momentum operator, p̂ = −iħ d/dx, does to a wavefunction, and why a plane wave has a definite momentum ħk.