kets

In quantum mechanics, a ket |ψ⟩ is a vector in a complex Hilbert space that represents a system’s state. Its dual bra ⟨φ| is a linear functional. The inner product ⟨φ|ψ⟩ is a probability amplitude. Kets obey linear superposition: a|ψ1⟩ + b|ψ2⟩ is a valid state. In a chosen orthonormal basis {|i⟩}, |ψ⟩ = ∑i c_i|i⟩ with c_i = ⟨i|ψ⟩. Normalization requires ⟨ψ|ψ⟩ = 1, and states differing by a global phase e^{iθ} are physically equivalent.

Measurement outcomes for an observable A, with eigenstates |a_j⟩, occur with probability p_j = |⟨a_j|ψ⟩|^2. After a measurement yielding the eigenvalue a_j, the state collapses to |a_j⟩ (projection postulate). The eigenbasis satisfies the completeness relation ∑_j |a_j⟩⟨a_j| = I, which ensures probabilities sum to 1.

Time evolution is unitary: iħ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩, so |ψ(t)⟩ = U(t,t0)|ψ(t0)⟩ with U†U = I. Operators act on kets from the left; bras act from the right via ⟨φ|A|ψ⟩. Outer products, such as |φ⟩⟨χ|, form operators and are used in projectors and density operators.

A ket describes a pure state; mixed states require density operators ρ = ∑i p_i |ψ_i⟩⟨ψ_i|, with expectation ⟨A⟩ = Tr(ρA). A common example is a qubit, a two-dimensional system with basis |0⟩ and |1⟩, where |ψ⟩ = α|0⟩ + β|1⟩ and |α|^2 + |β|^2 = 1.