State VectorEdit

The state vector is the central mathematical object used to describe a pure quantum state within the standard formalism of quantum mechanics. Written most commonly in Dirac notation as |ψ⟩, it is a vector in a complex Hilbert space that contains all the information needed to predict the outcomes of measurements on the system, as well as its evolution in time under a given dynamics. The probabilities of possible measurement results are obtained from the state vector via the Born rule, which associates the squared magnitude of expansion coefficients in a chosen basis with measurement likelihoods. Because an overall phase factor e^{iθ} leaves all physical predictions unchanged, a state vector is defined only up to a global phase; what matters is the direction of the vector, not its length or phase, a notion captured by the concept of a ray in Hilbert space.

The state vector formalism is built on a few durable ideas: linear structure, superposition, and probabilistic prediction. Superposition allows a system to be described as a combination of basis states, with complex amplitudes that encode both magnitude and phase relationships. Observables correspond to linear operators, and the act of measurement connects these mathematical objects to experimental outcomes through expectation values like ⟨ψ|Â|ψ⟩, where Â is an observable. The framework is remarkably successful at predicting phenomena across atomic, molecular, optical, and condensed-mense environments, and it underpins modern technologies such as quantum information processing and quantum communication.

See what this article covers

The formal definition and the mathematical setting of the state vector
How time evolution and measurements are expressed within the framework
Common representations (bases, spin systems, qubits) and practical examples
The relationship between pure states (state vectors) and mixed states (density operators)
Historical development and the principal interpretive debates surrounding the meaning of the wavefunction

Formal definition

A state vector |ψ⟩ resides in a complex Hilbert space H associated with a physical system. When a basis {|i⟩} is fixed, the state vector can be expanded as |ψ⟩ = ∑i ψi|i⟩, where the complex coefficients ψi are the probability amplitudes for the system to be found in the corresponding basis state |i⟩. The Born rule states that the probability of obtaining the outcome associated with |i⟩ is Pi = |ψi|^2, provided the state is properly normalized: ⟨ψ|ψ⟩ = 1. The normalization condition ensures total probability sums to one. A global phase e^{iθ} multiplies the vector without changing any physical predictions, so states are more precisely identified with rays in H rather than with individual vectors.

In many contexts, the most natural setting is a finite-dimensional Hilbert space, such as a qubit spanned by {|0⟩, |1⟩}, or a d-level system with basis {|0⟩, |1⟩, ..., |d−1⟩}. More generally, |ψ⟩ can live in an infinite-dimensional space, for example for the quantum harmonic oscillator, where the basis can be the energy eigenstates {|n⟩}. See Hilbert space for the formal structure that supports these constructions.

Dynamics and measurement

Time evolution in the Schrödinger picture is governed by the Schrödinger equation iħ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩, where H is the system’s Hamiltonian, a Hermitian operator representing the total energy. If H is time-independent, the solution is |ψ(t)⟩ = U(t)|ψ(0)⟩ with the unitary operator U(t) = exp(−iHt/ħ). In more general, time-dependent cases, the evolution remains unitary when the system is closed, meaning probabilities are conserved over time.

Measurement of an observable Â, represented by a Hermitian operator, yields one of its eigenvalues a_j with probability p_j = ⟨ψ|P_j|ψ⟩, where P_j is the projector onto the eigenspace corresponding to a_j. After a measurement resulting in a_j, the system is typically described by the post-measurement state |ψ'⟩ ∝ P_j|ψ⟩, illustrating the nontrivial link between the mathematical state and experimental data. The probabilistic nature of outcomes and the role of measurement are central to the theory's interpretation and practical use.

In more applied terms, the state vector is the primary input to predicting expectation values and variances of observables, and it evolves under dynamics that may be unitary (isolated systems) or effectively non-unitary when considering interactions with environments or measurement devices. A compact way to handle mixed situations is to move to the density operator formalism, where a pure state |ψ⟩⟨ψ| corresponds to a density matrix ρ, and mixed states are convex combinations of such projectors. See Density matrix for the broader framework that encompasses both pure and mixed states.

Representations and examples

Choosing a basis is a matter of convenience and problem structure. In a two-level system (a qubit), the state can be written as |ψ⟩ = α|0⟩ + β|1⟩, with complex amplitudes α, β satisfying |α|^2 + |β|^2 = 1. Polarization states of photons and spin-1/2 particles provide concrete physical realizations of this formalism, with common measurement bases such as the computational basis {|0⟩, |1⟩} or the eigenbasis of a spin component along a chosen axis. Quantum information science frequently uses the Bloch sphere representation, where any pure qubit state corresponds to a point on the unit sphere, and unitary operations correspond to rotations of the sphere.

In practical terms, the state vector framework supports prediction and control of experimental outcomes, from interference patterns in interferometers to the probabilities of detecting specific energy levels in spectroscopy. The same mathematics underlies quantum algorithms and error-correcting schemes in Quantum information processing and Quantum computation.

State vector and mixed states

Not all physical situations are adequately described by a single pure state. When there is statistical uncertainty about the preparation or genuine classical randomness, the system may be described by a mixed state, captured most generally by a density operator ρ with Tr(ρ) = 1 and ρ ≥ 0. A pure state |ψ⟩ yields ρ = |ψ⟩⟨ψ|, which satisfies Tr(ρ^2) = 1. Mixed states satisfy Tr(ρ^2) < 1. The density formalism provides a unified way to treat both coherent superpositions and incoherent mixtures, and it is essential for describing open systems interacting with environments, decoherence, and realistic measurement processes. See Density matrix for further development.

Interpretations and debates

Beyond computation and prediction, the state vector has been the subject of philosophical and interpretive debates. A central question is whether the wavefunction represents a real physical state (ontic view) or merely information about our knowledge (epistemic view). In the ontic view, the state vector is a real object that exists independently, while in the epistemic view it reflects subjective information about a system. This debate intersects with different interpretive programs, such as the traditional Copenhagen approach, many-worlds (which treats the universal wavefunction as the primary object and denies collapse in a fundamental sense), and collapse theories that posit a genuine dynamical reduction of the state.

From a practical standpoint, many physicists emphasize operational content: the state vector provides a compact and predictive bookkeeping device, and different interpretations often yield the same experimental predictions. Critics across the spectrum argue about the necessity and ontology of the wavefunction, with discussions ranging from empirical adequacy to deeper questions about realism. In the assessment of these debates, the emphasis tends to be on what the formalism reliably explains and how it guides technological progress, such as in Quantum technologies and Quantum computing.

Historical development

The state vector formalism emerged in the 1920s and 1930s in the work of pioneers such as Schrödinger and Heisenberg, who developed complementary representations of quantum states. Dirac introduced the bra–ket notation that unifies these pictures and provides a clean language for superpositions, inner products, and operator actions. The Born rule, linking the squared amplitude to probabilities, provided the critical link between the abstract state vector and experimental outcomes. Over the ensuing decades, the framework was refined and extended to treat systems of increasing complexity, including many-body states, entanglement, and quantum field theoretical contexts.