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Physics · Glossary

What is State space, kets and brackets?

Definition 8.1 University Physics — Year 3 · Chapter 8 — The Formalism of Quantum Mechanics

The states of a quantum system form a complex vector space with an inner product — a Hilbert space H\mathcal H (the mathematics is developed in the Year 3 mathematics volume). A state is a vector, written as a ket ∣ψ⟩\ket\psi, normalised: ⟨ψ | ψ⟩=1\braket\psi\psi = 1. The inner product of two states is the complex number ⟨φ | ψ⟩\braket\varphi\psi, conjugate-linear in the first slot, with ⟨φ | ψ⟩=⟨ψ | φ⟩∗\braket\varphi\psi = \braket\psi\varphi^*. In an orthonormal basis {∣ei⟩}\{\ket{e_i}\},

∣ψ⟩=∑ici∣ei⟩,ci=⟨ei | ψ⟩,∑i∣ci∣2=1.\ket\psi = \sum_i c_i\ket{e_i} , \qquad c_i = \braket{e_i}\psi , \qquad \sum_i|c_i|^2 = 1 .

The wave function of the previous chapters is the family of components of ∣ψ⟩\ket\psi along position: ψ(x)=⟨x | ψ⟩\psi(x) = \braket{x}{\psi}; nothing is lost, and systems with finite state spaces — undreamable as waves — become describable.

A polarization state as a vector: its squared components on an analyser’s basis are the outcome probabilities — geometry become probability.
A polarization state as a vector: its squared components on an analyser’s basis are the outcome probabilities — geometry become probability.

Examples

Example 8.2 (The photon’s polarization: a two-dimensional world)

A photon heading down the zz axis carries a polarization state in a two-dimensional Hilbert space, with basis ∣H⟩\ket H (horizontal) and ∣V⟩\ket V (vertical). Light polarized at angle θ\theta is the superposition

∣θ⟩=cos⁡θ ∣H⟩+sin⁡θ ∣V⟩,\ket\theta = \cos\theta\,\ket H + \sin\theta\,\ket V ,

and circular polarization is the complex combination (∣H⟩±i∣V⟩)/2(\ket H \pm \iu\ket V)/\sqrt2: the coefficients being complex is not decoration but physics. Every quantum two-level system — spin, the ammonia molecule, a superconducting qubit — is this same vector space in different clothing.

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