Dirac notation: kets, bras and projectors

In ℂ² the machinery is bare: |ψ⟩ = (3/5, 4i/5)ᵀ is a column, and its bra ⟨ψ| = (3/5, −4i/5) is the conjugate transpose row. The pairing ⟨ψ|ψ⟩ is then a 1×2 by 2×1 matrix product: 9/25 + (−4i/5)(4i/5) = 9/25 + 16/25 = 1. A bra is not decoration and not a second copy of the state: it is a linear functional, a machine that eats one ket and returns one complex number. The Riesz representation theorem is what licenses the notation — on a complete inner-product space every continuous linear functional is ⟨φ|⋅⟩ for exactly one vector |φ⟩, so kets and bras pair off one-to-one. The pairing is conjugate-linear on the bra side: c|ψ⟩ corresponds to c*⟨ψ|, which is why ⟨ψ|φ⟩ = ⟨φ|ψ⟩* and why the norm ⟨ψ|ψ⟩ comes out real and non-negative.

Written side by side, a bra then a ket is a number; a ket then a bra is an operator. In ℂ² the second is a 2×1 column times a 1×2 row — a full 2×2 matrix. Take |+⟩ = (|0⟩ + |1⟩)/√2: the outer product |+⟩⟨+| has every entry ½. Square that matrix and you get it back, P² = P, because ⟨+|+⟩ = 1 collapses the middle. Idempotency is the defining property of a projector — projecting a second time finds the vector already in the subspace — and P† = P makes it an observable whose only eigenvalues are 1 (inside) and 0 (outside). Its expectation ⟨ψ|P|ψ⟩ = |⟨+|ψ⟩|² is a squared amplitude, which is exactly how Units 2 and 3 will write the Born rule. Every operator this course builds — projectors, ladder operators, density matrices, the raising operator |1⟩⟨0| that lifts |0⟩ to |1⟩ — is a sum of outer products, so read |m⟩⟨n| as an instruction: take the |n⟩ component, deliver it along |m⟩.

Completeness of an orthonormal basis is an operator statement: Σₙ|n⟩⟨n| = 1. Multiplying by 1 changes nothing, so you may insert it into any product, and each insertion converts one abstract object into components. Between a bra and a ket: ⟨φ|ψ⟩ = Σₙ⟨φ|n⟩⟨n|ψ⟩ = Σₙ φₙ*ψₙ — the abstract overlap becomes arithmetic on coefficients. Around an operator, twice: ⟨φ|Â|ψ⟩ = Σₘₙ ⟨φ|m⟩⟨m|Â|n⟩⟨n|ψ⟩, a row times a matrix times a column, and the operator has become the matrix Aₘₙ = ⟨m|Â|n⟩. Try it with |ψ⟩ = (3|0⟩ + 4i|1⟩)/5 and the analyser |+⟩: ⟨+|ψ⟩ = (3 + 4i)/(5√2), whose squared modulus is (9 + 16)/50 = 1/2. The insertion move is not a trick; it is the definition of 'representation'. ψ(x) = ⟨x|ψ⟩ and φ(p) = ⟨p|ψ⟩ are one state's components along two different complete sets, and the Fourier transform of topic 1.8, which Unit 3 will read as a unitary change of basis, is nothing but the identity ∫dx|x⟩⟨x| = 1 inserted inside ⟨p|ψ⟩.

Position and momentum have no eigenvectors in the space. A plane wave e^(ipx/ħ) has constant |ψ|², so ∫|ψ|²dx diverges: |p⟩ is not square-integrable, and 'the state of definite momentum' is not a state. The notation keeps working because the normalisation is changed, not abandoned: demand ⟨p|p′⟩ = δ(p − p′) in place of a Kronecker δ, which fixes the prefactor in ⟨x|p⟩ = e^(ipx/ħ)/√(2πħ) and makes ∫dp|p⟩⟨p| = 1 exact — check it by inserting the p-identity into ⟨x|x′⟩ and recovering δ(x − x′) as a Fourier integral. Dimensions confirm the object is strange: normalising ∫|⟨x|ψ⟩|²dx = 1 forces ψ(x) to carry L⁻¹⁄², so ⟨x|ψ⟩ is a density amplitude, not a probability, and |x⟩ itself carries L⁻¹⁄². The honest statement is that |x⟩ and |p⟩ live in a rigged Hilbert space of distributions, and every physical state is a normalisable wave packet smeared over them.

Everything above is executable. A ket is a complex array, ψ = np.array([0.6, 0.8j]); the bra is its conjugate, and np.vdot(φ, ψ) conjugates its first argument for you — it computes ⟨φ|ψ⟩ in the physics convention, where phi.T @ psi would silently drop the conjugate. The outer product np.outer(n, n.conj()) builds |n⟩⟨n|, and the two projector laws become one-line assertions: np.allclose(P @ P, P) and np.allclose(P, P.conj().T). The resolution of the identity is the test that a claimed basis really is one: take the eigenvector columns returned by numpy.linalg.eigh, sum the outer products, and compare against np.eye(N). On the eigenvectors of any Hermitian matrix the check passes to machine precision; drop one eigenvector and the sum falls short of the identity by exactly the missing |n⟩⟨n| — completeness failing in code the same silent way an incomplete basis loses probability on paper.