The quantum state is the mathematical object used in quantum physics to encode all the information that can be predicted about a physical system. It does not itself constitute a classical description of properties but provides probabilities for the outcomes of measurements such as position, momentum, or energy. A familiar example is the state of a hydrogen atom, whose measurable properties include position and energy; see hydrogen and atom for context.
Representations
There are several equivalent ways to describe a quantum state depending on the context. In nonrelativistic quantum mechanics a state is often given by a complex wavefunction or state vector in a Hilbert space. Mixed states, which represent statistical ensembles or partial knowledge, are described by a density operator. Key mathematical features include linearity (allowing superposition) and normalization (total probability equals one).
Typical measurable quantities are associated with operators. For example, momentum is related to the momentum operator and energy to the Hamiltonian; see momentum and energy. Angular momentum and intrinsic spin are additional observables with quantized spectra; compare angular momentum and spin for further reading.
Important distinctions exist between pure and mixed states. A pure state corresponds to maximal information about preparation and can be represented by a single state vector; mixed states are statistical mixtures of pure states and require a density matrix. Measurement in quantum mechanics yields probabilistic outcomes and usually changes the state according to the measurement postulates of the theory; various interpretations disagree about whether this change is a physical collapse or an update of knowledge.
Properties and examples
- Superposition: a system can exist in a linear combination of basis states until measurement.
- Entanglement: composite systems can exhibit correlations stronger than classical statistics, with important consequences for information transfer and nonlocality.
- Unitary evolution: isolated states evolve deterministically by the Schrödinger equation; open systems require more general dynamical maps.
Concrete examples help connect abstract ideas to practice. The quantum state of a single electron in a hydrogen-like atom determines its allowed energy levels and spatial probability distribution. In quantum information, a two-level quantum state (qubit) is the basic carrier of quantum information, exploiting superposition and entanglement for tasks such as computation and secure communication.
Quantum states underpin much of modern physics and technology. They are central to spectroscopy, quantum chemistry, condensed-matter physics, and emerging technologies like quantum computing and sensing. While the formalism is precise and predictive, conceptual questions about interpretation and the nature of measurement remain topics of active discussion and research.