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Quantum state

In quantum mechanics, a quantum state is a mathematical entity that fully describes the condition of a physical system, encoding the probabilities of all possible measurement outcomes for observables such as position, momentum, and energy. For isolated systems with complete knowledge, pure quantum states are represented by normalized vectors in a complex separable Hilbert space, often expressed as a wave function \psi in the position representation, satisfying the normalization condition \int |\psi(x)|^2 dx = 1.^[1] These states evolve deterministically according to the time-dependent Schrödinger equation, i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, where \hat{H} is the Hamiltonian operator.^[2] When dealing with ensembles of systems, partial information, or interactions with the environment, quantum states are generalized to mixed states, described by density operators \hat{\rho} that are Hermitian, positive semi-definite, and trace-normalized (\operatorname{Tr}(\hat{\rho}) = 1).^[3] A pure state corresponds to a density operator of the form \hat{\rho} = |\psi\rangle\langle\psi|, while mixed states arise as statistical mixtures, \hat{\rho} = \sum_i p_i |\psi_i\rangle\langle\psi_i|, with probabilities p_i \geq 0 summing to 1.^[4] The expectation value of an observable \hat{A} is given by \langle \hat{A} \rangle = \operatorname{Tr}(\hat{\rho} \hat{A}), unifying the formalism for both pure and mixed cases.^[5] Central to quantum states are properties like superposition, allowing a system to exist as a linear combination of multiple basis states simultaneously, such as \psi = \alpha |0\rangle + \beta |1\rangle for a qubit, until measurement collapses it probabilistically according to Born's rule: the probability of outcome |i\rangle is |\langle i | \psi \rangle|^2.^[6] For composite systems, entanglement emerges when the state cannot be factored into individual subsystems, as in the Bell state \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle), leading to correlations defying classical intuition.^[7] These features underpin applications in quantum computing, where states in high-dimensional Hilbert spaces enable exponential computational advantages, and quantum information theory, which quantifies resources like coherence and entanglement via measures such as von Neumann entropy S(\hat{\rho}) = -\operatorname{Tr}(\hat{\rho} \log \hat{\rho}).^[8]^[9] The formalism originated in the 1920s with contributions from Schrödinger, Heisenberg, and Dirac, resolving classical failures at atomic scales through probabilistic wave mechanics.^[2]

Classical and Historical Foundations

Classical states in mechanics

In classical mechanics, the state of a physical system is defined as a complete specification of the positions and momenta of all its constituent particles at a given instant, represented as a point in phase space.^[10] Phase space is a multidimensional space where each axis corresponds to a coordinate or momentum component, allowing the system's configuration to be plotted holistically.^[11] This representation encapsulates all information necessary to determine the system's future evolution under deterministic laws.^[12] The foundational framework for classical states emerged in the 18th and 19th centuries through contributions from key figures such as Joseph-Louis Lagrange, who developed Lagrangian mechanics in his 1788 treatise Mécanique Analytique, emphasizing variational principles over Newtonian forces; William Rowan Hamilton, who in 1834 introduced Hamiltonian mechanics via his reformulation in terms of action integrals; and Siméon Denis Poisson, whose work on Poisson brackets in the 1800s facilitated the transition to canonical coordinates.^[13]^[14] These developments built upon Isaac Newton's Principia Mathematica (1687), shifting focus from force-based descriptions to energy and coordinate transformations for broader applicability.^[13] Classical mechanics exhibits determinism, meaning that given precise initial conditions in phase space, the system's trajectory evolves uniquely over time according to Newton's second law or the equivalent Hamilton's equations of motion.^[15] Hamilton's equations describe this evolution as a flow in phase space: for a system with Hamiltonian H(q, p), the time derivatives satisfy \dot{q}_i = \frac{\partial H}{\partial p_i} and \dot{p}_i = -\frac{\partial H}{\partial q_i}, ensuring a single, predictable path from any starting point.^[16] This predictability underpins the predictive power of classical physics for macroscopic systems. A key example is the trajectory in Hamiltonian mechanics, where the phase space point traces a deterministic curve governed by conserved quantities like energy, illustrating how initial states dictate all future configurations without ambiguity.^[16] Complementing this, Liouville's theorem states that the evolution of an ensemble of systems preserves the volume occupied in phase space, implying incompressible flow and constant phase space density along trajectories, as derived from the divergence-free nature of the Hamiltonian vector field.^[17]^[18] In contrast, quantum states generalize this by incorporating probabilistic distributions over phase space-like structures.^[12]

Transition to quantum states

The transition from classical to quantum states was driven by empirical anomalies that classical mechanics and electromagnetism could not explain, necessitating a fundamental rethinking of how physical systems are described.^[19] In classical physics, the state of a system was fully specified by continuous variables in phase space, such as position and momentum, allowing deterministic predictions of trajectories. However, experiments on blackbody radiation revealed a critical failure: the Rayleigh-Jeans law predicted an infinite energy density at high frequencies, known as the ultraviolet catastrophe, which contradicted observations of finite radiation spectra.^[20] To resolve this, Max Planck introduced the concept of energy quantization in 1900, proposing that oscillators in the blackbody emit and absorb energy only in discrete multiples of h\nu, where h is Planck's constant and \nu is frequency, leading to his famous radiation formula that matched experimental data.^[21] Further challenges arose with the photoelectric effect, where classical wave theory failed to account for the instantaneous ejection of electrons from metals upon light absorption, independent of light intensity but dependent on frequency.^[19] In 1905, Albert Einstein extended Planck's quantization to light itself, positing that electromagnetic radiation consists of discrete energy packets called light quanta (later photons), each with energy h\nu, explaining the threshold frequency below which no emission occurs and the linear dependence of electron kinetic energy on frequency.^[22] Similarly, the discrete line spectra of atoms defied classical predictions of continuous emission from accelerating electrons in orbits, which should lead to rapid energy loss and atomic collapse.^[19] Niels Bohr addressed this in 1913 by introducing stationary orbits with quantized angular momentum n\hbar (where n is an integer and \hbar = h/2\pi), allowing electrons to jump between discrete energy levels and emit photons of specific frequencies matching observed spectral lines.^[23] These developments culminated in Louis de Broglie's 1924 hypothesis of wave-particle duality, proposing that all matter possesses wave-like properties with wavelength \lambda = h/p (where p is momentum), unifying the particle behavior of matter with the wave nature of light suggested by earlier experiments.^[24] This idea inspired Erwin Schrödinger in 1926 to formulate wave mechanics, describing quantum states via wave functions that evolve according to a differential equation, replacing classical point particles with distributed probability amplitudes.^[25] Philosophically, this marked a profound shift from deterministic classical paths to inherently probabilistic descriptions, where quantum states encode possibilities rather than certainties, as the evolution of the wave function yields outcome probabilities upon measurement.^[26]

Core Concepts in Quantum Mechanics

Role and definition

In quantum mechanics, the quantum state serves as the axiomatic foundation for describing the condition of a physical system, encapsulating all observable information that can be predicted about its behavior. This complete specification distinguishes it as the central mathematical entity from which all probabilistic outcomes of measurements are derived, assuming the foundational concept of wave-particle duality where particles exhibit both wave-like and particle-like properties.^[5] The evolution of an isolated quantum system is governed by unitary transformations, preserving the norm of the state and ensuring deterministic dynamics in the absence of measurement, as formalized in the Schrödinger picture of quantum mechanics. Upon measurement, the quantum state provides the basis for applying the Born rule, which determines the probabilities of possible outcomes: for a pure state represented by the vector |\psi\rangle in Hilbert space, the probability of projecting onto an eigenspace is |\langle \phi | \psi \rangle|^2, where |\phi\rangle is the corresponding eigenvector. For systems involving statistical ensembles or partial knowledge, the state is instead described by a density matrix \rho, a positive semi-definite operator with trace unity, from which probabilities are obtained via \operatorname{Tr}(\rho P), with P the projector onto the outcome subspace.^[27]^[5] In contrast to classical mechanics, where a state fully specifies deterministic trajectories via exact positions and momenta, the quantum state inherently embodies uncertainty, as articulated by the Heisenberg uncertainty principle, which imposes fundamental limits on the simultaneous knowledge of non-commuting observables such as position and momentum (\Delta x \Delta p \geq \hbar/2). This probabilistic nature arises not from ignorance but from the intrinsic structure of quantum theory, precluding the classical ideal of complete predictability.

Measurements and collapse

In quantum mechanics, the von Neumann measurement scheme formalizes the process of measuring an observable A by associating it with a Hermitian operator \hat{A} acting on the system's Hilbert space, where the possible measurement outcomes are the eigenvalues of \hat{A}. This scheme posits that the measurement apparatus interacts with the quantum system, leading to a definite outcome from the spectrum of \hat{A}, which corresponds to a projection onto the eigenspace of that eigenvalue. The probability of obtaining a specific outcome a_i, associated with the normalized eigenstate |\phi_i\rangle, for a system prepared in a pure state |\psi\rangle is given by the Born rule: P(a_i) = |\langle \phi_i | \psi \rangle|^2. This probabilistic interpretation, introduced by Max Born, connects the squared modulus of the wave function overlap to the likelihood of each possible result, ensuring that the outcomes are statistically predictable yet inherently indeterministic for individual measurements. Following the measurement, the wave function collapse postulate states that the system's quantum state instantaneously projects onto the eigenstate |\phi_i\rangle corresponding to the observed outcome a_i, renormalized if necessary, thereby reducing the superposition to a definite state. This projection is a non-unitary process that disrupts the prior evolution under the Schrödinger equation, marking a fundamental departure from reversible dynamics during observation. In contemporary interpretations, quantum decoherence provides an alternative explanation for the appearance of collapse, where interactions with the environment entangle the system with many degrees of freedom, suppressing interference between different outcome branches without invoking a fundamental projection postulate. Pioneered by Wojciech Zurek, this mechanism arises from the rapid spread of quantum correlations into the surroundings, effectively selecting preferred states (pointer states) that mimic classical behavior on macroscopic scales. Pure states thus serve as idealized initial conditions prior to such environmental decoherence in realistic scenarios.^[28]

Pure States

Eigenstates and eigenvalues

In quantum mechanics, eigenstates and eigenvalues arise in the context of observables modeled by Hermitian operators. An eigenstate |\psi\rangle of an operator \hat{A} satisfies the eigenvalue equation

\hat{A} |\psi\rangle = a |\psi\rangle,

where a is the eigenvalue, a real scalar representing the precise outcome of a measurement of the observable corresponding to \hat{A}. Since observables are represented by Hermitian operators, their eigenvalues are always real, ensuring physically meaningful measurement results.^[29]^[30] For Hermitian operators with discrete, non-degenerate spectra, the eigenstates form an orthonormal basis, satisfying \langle \psi_m | \psi_n \rangle = \delta_{mn}. These bases are complete, obeying the resolution of the identity

\sum_n |\psi_n\rangle \langle \psi_n | = \hat{I},

which allows any state in the Hilbert space to be expanded in terms of the eigenstates. This completeness underpins the spectral decomposition of operators and facilitates calculations of expectation values and probabilities. In cases of degeneracy, eigenstates within the same eigenspace can be chosen orthonormal but are not unique. For continuous spectra, the sum becomes an integral over the eigenbasis.^[30]^[31]^[32] Prominent examples include position and momentum eigenstates, which exhibit continuous spectra. The position eigenstate |x\rangle obeys \hat{x} |x\rangle = x |x\rangle, while the momentum eigenstate |p\rangle, often a plane wave \psi(p, x) = \frac{1}{\sqrt{2\pi \hbar}} e^{i p x / \hbar}, satisfies \hat{p} |p\rangle = p |p\rangle. Another key example is the energy eigenstate from the time-independent Schrödinger equation,

\hat{H} |\psi\rangle = E |\psi\rangle,

where \hat{H} is the Hamiltonian operator and E is the energy eigenvalue, typically discrete for bound systems.^[33]^[34] Eigenstates hold fundamental significance as stationary states, particularly energy eigenstates, where the probability densities and expectation values of observables remain time-independent under unitary evolution. The time-dependent form of such a state is |\psi(t)\rangle = e^{-i E t / \hbar} |\psi\rangle, acquiring only a global phase that does not affect measurable quantities. This stability makes eigenstates ideal for describing equilibrium configurations and simplifying time-evolution analyses in quantum systems.^[35]^[36]

Superposition principle

The superposition principle is a cornerstone of quantum mechanics, asserting that any pure quantum state can be represented as a linear combination of basis states, typically the eigenstates of a given observable. Formally, a normalized pure state |\psi\rangle is expressed as

|\psi\rangle = \sum_n c_n |n\rangle,

where the |n\rangle form a complete orthonormal basis, the complex coefficients c_n are the probability amplitudes, and the normalization condition \sum_n |c_n|^2 = 1 ensures the total probability is unity. This linearity follows directly from the structure of the Schrödinger equation, allowing solutions to be superposed without altering their validity under the theory's postulates.^[37] The amplitudes c_n govern measurement outcomes: the probability of obtaining the eigenvalue associated with |n\rangle upon measuring the corresponding observable is |c_n|^2, as dictated by the Born rule. A hallmark of superposition is quantum interference, manifested in the expectation value of an operator \hat{O}, given by

\langle \hat{O} \rangle = \langle \psi | \hat{O} | \psi \rangle = \sum_{n,m} c_m^* c_n \langle m | \hat{O} | n \rangle.

The off-diagonal "cross terms" for m \neq n introduce non-additive contributions that can constructively or destructively interfere, yielding results impossible for classical probabilistic mixtures, where such coherence is absent.^[38] Illustrative examples highlight these features. In the double-slit experiment, a particle's state is a superposition of paths through each slit, such as |\psi\rangle = \frac{1}{\sqrt{2}} (|1\rangle + |2\rangle), producing an interference pattern on the detection screen due to the cross terms in the probability distribution. Similarly, for a single qubit, the state |\psi\rangle = \alpha |0\rangle + \beta |1\rangle with |\alpha|^2 + |\beta|^2 = 1 embodies superposition, enabling computational advantages in quantum information processing through coherent interference.^[39] Superposition also underpins key no-go theorems, such as the no-cloning theorem, which prohibits perfect copying of an arbitrary unknown quantum state. If cloning were possible via a unitary operation mapping |\psi\rangle |0\rangle to |\psi\rangle |\psi\rangle, linearity would imply \frac{1}{\sqrt{2}} (|0\rangle + |1\rangle) |0\rangle maps to \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle), but it must also map to \frac{1}{\sqrt{2}} (|0\rangle |0\rangle + |1\rangle |0\rangle ) = \frac{1}{\sqrt{2}} (|00\rangle + |10\rangle), a contradiction unless the states are orthogonal. This impossibility arises inherently from the coherent nature of superpositions.^[40]

Mixed States

Density operators

In quantum mechanics, the density operator provides a general mathematical framework for describing the state of a quantum system, particularly when dealing with statistical ensembles or mixed states that cannot be represented by a single pure state vector. Introduced by John von Neumann in his foundational work on quantum statistical mechanics, the density operator \rho for a system prepared in an ensemble of pure states |\psi_i\rangle with probabilities p_i (where \sum_i p_i = 1 and p_i \geq 0) is defined as

\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|.

This operator is Hermitian (\rho^\dagger = \rho), positive semi-definite (all eigenvalues \geq 0), and normalized such that its trace equals unity: \operatorname{Tr}(\rho) = 1. These properties ensure that expectation values of observables A can be computed as \langle A \rangle = \operatorname{Tr}(\rho A), generalizing the pure state formula \langle \psi | A | \psi \rangle. A key distinction arises between pure and mixed states in this formalism. A state is pure if it corresponds to a single ket vector (up to a global phase), in which case \rho = |\psi\rangle\langle\psi| satisfies \rho^2 = \rho and has rank 1; the purity measure \operatorname{Tr}(\rho^2) = 1. For mixed states, where the system is in a probabilistic mixture, \operatorname{Tr}(\rho^2) < 1 and the rank exceeds 1, reflecting incomplete knowledge or environmental decoherence. This criterion quantifies the degree of coherence, with pure states maximizing it. The time evolution of the density operator under a Hamiltonian H follows the von Neumann equation,

i\hbar \frac{d\rho}{dt} = [H, \rho],

which is the quantum analog of the Liouville equation in classical statistical mechanics and preserves the operator's trace and positivity. Von Neumann derived this equation to describe the dynamics of ensembles consistently with unitary evolution of individual pure states. Density operators offer significant advantages over state vectors, including basis independence—allowing representation in any complete orthonormal basis without loss of information—and facilitating the description of subsystems through partial tracing. For a composite system with density operator \rho_{AB}, the reduced density operator for subsystem A is \rho_A = \operatorname{Tr}_B(\rho_{AB}), enabling the study of open quantum systems and entanglement without specifying the full joint state.

Statistical mixtures

Statistical mixtures in quantum mechanics describe situations where the quantum state cannot be represented by a single pure state vector due to inherent uncertainties or interactions with unobserved degrees of freedom. These mixtures typically originate from classical incomplete knowledge about the system's preparation, such as in thermal ensembles where the system is in contact with a heat bath at a given temperature, leading to a probabilistic distribution over energy eigenstates.^[41] Alternatively, they arise in open quantum systems, where the reduced description of the system is obtained by tracing over the environmental degrees of freedom, effectively averaging out the influence of the larger composite system.^[42] A representative example of a statistical mixture from incomplete knowledge is the thermal equilibrium state of a system, where the density operator \rho is diagonal in the energy eigenbasis with probabilities given by the Boltzmann distribution p_i = e^{-\beta E_i}/Z, with \beta = 1/(k_B T) the inverse temperature and Z the partition function.^[43] For distinguishable particles in the classical limit, such as in dilute gases, Maxwell-Boltzmann statistics apply, yielding a mixed state that reflects the classical probabilistic assignment without quantum indistinguishability effects.^[44] In contrast to coherent superpositions, the density operator for these statistical mixtures is diagonal in the basis of the mixture components, lacking coherences between the different pure states, resulting in no observable quantum interference patterns between those components upon measurement. The degree of mixedness in such states is quantified by the von Neumann entropy, defined as

S(\rho) = -\operatorname{Tr}(\rho \ln \rho),

which measures the uncertainty or information deficit relative to a pure state, where S = 0, and achieves maximum for completely random mixtures.^[45] This entropy extends the classical Shannon entropy to quantum systems and plays a key role in assessing the loss of coherence in statistical descriptions.^[45]

Mathematical Formalism

Hilbert space representation

In quantum mechanics, the mathematical framework for describing quantum states is provided by a complex Hilbert space \mathcal{H}, defined as a complete vector space equipped with an inner product that induces a norm, ensuring convergence of Cauchy sequences. This structure was formalized by John von Neumann as the foundational setting for quantum theory, where the space \mathcal{H} is typically infinite-dimensional to accommodate continuous spectra of observables like position and momentum.^[46] The completeness property guarantees that limits of sequences of states remain within the space, which is essential for the rigorous treatment of wave functions and operators in quantum dynamics.^[47] Quantum states correspond to rays in \mathcal{H}, meaning equivalence classes of vectors |\psi\rangle identified up to a global phase factor e^{i\theta}, as physical predictions depend only on probabilities derived from inner products and are invariant under such phase transformations. This ray representation ensures that distinct states are physically distinguishable only if their overlap is less than unity in magnitude. The Dirac notation, introduced by Paul Dirac, facilitates abstract manipulations: the inner product between two state vectors is denoted \langle \phi | \psi \rangle, a complex scalar satisfying \langle \phi | \phi \rangle \geq 0 with equality only for the zero vector, and linear operators A act via the bilinear form \langle \phi | A | \psi \rangle. This notation abstracts away from specific coordinate representations, emphasizing the operator algebra central to quantum mechanics.^[46] For composite systems, the total Hilbert space is constructed as the tensor product \mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B of the individual subsystem spaces, allowing for the description of correlations and entanglement; this structure is separable, possessing a countable orthonormal basis, as assumed in von Neumann's axiomatization for systems with finite or countably infinite degrees of freedom.^[46] In contrast, inseparable Hilbert spaces, lacking a countable basis, arise in contexts like quantum field theory with uncountably many modes, though standard non-relativistic quantum mechanics restricts to separable spaces to maintain mathematical tractability and physical interpretability. The Riesz representation theorem underpins the vector representation of states by asserting that every continuous linear functional \phi on \mathcal{H} is uniquely given by \phi(\psi) = \langle \xi | \psi \rangle for some fixed \xi \in \mathcal{H}, with the norm \|\phi\| = \|\xi\|. In quantum mechanics, this theorem justifies modeling pure states as vectors, as expectation values of observables—bounded self-adjoint operators—are continuous functionals on the space of states.^[47] This identification bridges the abstract duality between states and observables, ensuring that all physically realizable measurements correspond to inner products within \mathcal{H}.^[46]

Basis states and expansions

In quantum mechanics, the representation of quantum states relies on expanding them in a suitable basis within the Hilbert space. An orthonormal basis { |n\rangle } consists of states satisfying \langle n | m \rangle = \delta_{nm} and the completeness relation \sum_n |n\rangle \langle n | = \hat{I}, where \hat{I} is the identity operator, ensuring that any state can be uniquely expressed as a linear combination of basis elements. For representations involving continuous variables, such as position eigenstates |x\rangle, the completeness relation manifests as \sum_n \langle x | n \rangle \langle n | y \rangle = \delta(x - y), where \langle x | n \rangle are the basis wave functions, guaranteeing the basis spans the entire space. This completeness underpins Parseval's theorem, which preserves the norm of the state vector under basis expansion: \langle \psi | \psi \rangle = \sum_n | \langle n | \psi \rangle |^2, relating the total probability to the sum of squared projection coefficients and affirming the probabilistic interpretation of quantum states. The general expansion of a state | \psi \rangle in such a basis is

| \psi \rangle = \sum_n \langle n | \psi \rangle \, | n \rangle,

where the coefficients c_n = \langle n | \psi \rangle represent the projection of | \psi \rangle onto | n \rangle, providing the amplitude for finding the system in basis state | n \rangle upon measurement. The selection of a basis is guided by the problem's physical demands; for example, the energy eigenbasis { | E_n \rangle } facilitates solving time-dependent Schrödinger equations via stationary states, the position basis { | x \rangle } is ideal for describing spatial wave functions, and the momentum basis { | p \rangle } proves useful for analyzing free-particle propagation through Fourier transforms. In certain applications, particularly in quantum optics, overcomplete bases are advantageous despite lacking strict orthogonality; coherent states | \alpha \rangle, introduced by Glauber, form such a set, allowing efficient expansions for fields like laser light where classical-like behavior is prominent. Changing bases preserves the orthonormal structure through unitary operators \hat{U}, with \hat{U}^\dagger \hat{U} = \hat{I}, transforming the basis as | n' \rangle = \hat{U} | n \rangle and the coefficients as \langle n' | \psi \rangle = \langle n | \hat{U}^\dagger | \psi \rangle, enabling flexible computational frameworks without altering physical predictions.

Representations and Pictures

Wave function formalism

In the wave function formalism, the quantum state of a non-relativistic particle is represented by a complex-valued function \psi(\mathbf{r}, t) in position space, which encodes the probability distribution and phase information of the particle's location and dynamics. This representation emerged as a cornerstone of quantum mechanics through Erwin Schrödinger's development of wave mechanics in 1926. The wave function in the position basis is formally defined as \psi(x) = \langle x | \psi \rangle, where | \psi \rangle denotes the abstract state vector in Hilbert space and | x \rangle are the eigenstates of the position operator.^[48] For a normalized state, the wave function satisfies the condition

\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1,

which ensures that the total probability associated with the particle's position is conserved.^[48] The probabilistic interpretation of the wave function, proposed by Max Born in 1926, states that |\psi(x)|^2 \, dx represents the probability of measuring the particle's position within the infinitesimal interval dx at x.^[49] This interpretation shifts the view of \psi(x) from a physical wave to a mathematical tool for predicting measurement outcomes, resolving earlier debates about its ontological status.^[49] The temporal evolution of the wave function is governed by the time-dependent Schrödinger equation,

i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi,

where \hat{H} is the Hamiltonian operator, \hbar is the reduced Planck's constant, and i is the imaginary unit; this equation was introduced by Schrödinger as the fundamental dynamical law for wave mechanics. Solutions to this equation describe how the state propagates deterministically, while measurements yield probabilistic results via Born's rule. An equivalent representation exists in momentum space, where the momentum wave function \phi(p) is obtained via the Fourier transform of \psi(x):

\phi(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty} \psi(x) e^{-i p x / \hbar} \, dx.

This transformation, rooted in the canonical commutation relations of position and momentum, allows the state to be expressed in the basis of momentum eigenstates | p \rangle, with |\phi(p)|^2 \, dp giving the probability density in momentum.^[48] The position and momentum representations are interconvertible and complementary, illustrating the flexibility of basis expansions for quantum states.^[48]

Schrödinger vs. Heisenberg pictures

In the Schrödinger picture, the quantum state evolves in time while operators remain fixed. The time evolution of a pure state |\psi(t)\rangle is governed by the unitary operator U(t) = e^{-i H t / \hbar}, where H is the time-independent Hamiltonian, such that |\psi(t)\rangle = U(t) |\psi(0)\rangle. This formulation aligns with the time-dependent Schrödinger equation i \hbar \frac{\partial}{\partial t} |\psi(t)\rangle = H |\psi(t)\rangle, emphasizing the wave function's dynamical behavior. In contrast, the Heisenberg picture keeps the states time-independent, with operators evolving instead. Here, the Heisenberg operator A_H(t) transforms as A_H(t) = U^\dagger(t) A U(t), where A is the time-independent Schrödinger operator, preserving the commutation relations among operators. The expectation value of an operator evolves according to the Ehrenfest theorem: \frac{d}{dt} \langle A \rangle = \left\langle \frac{\partial A}{\partial t} \right\rangle + \frac{i}{\hbar} \langle [H, A] \rangle, linking quantum dynamics to classical equations of motion for expectation values in the classical limit. The two pictures are equivalent, as the expectation value \langle A \rangle_t = \langle \psi(t) | A | \psi(t) \rangle in the Schrödinger picture equals \langle \psi(0) | A_H(t) | \psi(0) \rangle in the Heisenberg picture, ensuring identical physical predictions. The choice between them depends on convenience: the Schrödinger picture suits time-independent Hamiltonians and wave function analysis, while the Heisenberg picture facilitates operator algebra and time-dependent perturbations. This framework extends to mixed states via the density operator \rho. In the Schrödinger picture, \rho(t) = U(t) \rho(0) U^\dagger(t), while in the Heisenberg picture, \rho remains fixed and operators evolve as before, yielding the same trace \operatorname{Tr}(\rho(t) A) = \operatorname{Tr}(\rho(0) A_H(t)).

Applications to Systems

Single-particle states

In non-relativistic quantum mechanics, the state of a single particle is described by a wave function that encodes its position and, if applicable, spin degrees of freedom. For a spinless particle, the state is represented by a scalar wave function \psi(\mathbf{r}, t) in three-dimensional position space, satisfying the time-dependent Schrödinger equation i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V(\mathbf{r}) \psi.^[25] This formulation, introduced by Schrödinger in 1926, captures the probabilistic nature of the particle's position, with |\psi(\mathbf{r}, t)|^2 giving the probability density.^[25] For particles with intrinsic spin, such as electrons, the wave function becomes \psi(\mathbf{r}, s, t), where s labels the discrete spin components, reflecting the internal angular momentum associated with the particle.^[50] For spin-1/2 particles like the electron, the state is a two-component spinor \psi(\mathbf{r}, t) = \begin{pmatrix} \psi_\uparrow(\mathbf{r}, t) \\ \psi_\downarrow(\mathbf{r}, t) \end{pmatrix}, or more generally \psi(\mathbf{r}, t) \chi, where \chi = \begin{pmatrix} \alpha \\ \beta \end{pmatrix}^T is the spinor with complex coefficients satisfying |\alpha|^2 + |\beta|^2 = 1.^[51] The basis states for the spin angular momentum operator S_z are the eigenstates |\uparrow\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix} with eigenvalue \frac{\hbar}{2} and |\downarrow\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix} with eigenvalue -\frac{\hbar}{2}.^[51] These spin states, proposed by Uhlenbeck and Goudsmit in 1925 to explain atomic spectral fine structure, are independent of the orbital motion and transform under rotations according to the SU(2) group representation for spin 1/2.^[50] The dynamics of such particles, including spin-orbit coupling, are governed by the Pauli equation:

i\hbar \frac{\partial \psi}{\partial t} = \left[ \frac{1}{2m} \left( \mathbf{p} - \frac{e}{c} \mathbf{A} \right)^2 - \frac{e\hbar}{2mc} \boldsymbol{\sigma} \cdot \mathbf{B} + e\phi \right] \psi,

where \boldsymbol{\sigma} are the Pauli matrices, \mathbf{A} and \phi are the vector and scalar electromagnetic potentials, and \mathbf{B} = \nabla \times \mathbf{A}. This equation, derived by Pauli in 1927 as a non-relativistic limit of the Dirac equation, incorporates the spin-magnetic field interaction essential for phenomena like the Zeeman effect.^[51] Orbital angular momentum contributes to the single-particle state through eigenfunctions that separate in spherical coordinates, with the angular part given by spherical harmonics Y_{lm}(\theta, \phi), which are simultaneous eigenfunctions of L^2 and L_z with eigenvalues \hbar^2 l(l+1) and \hbar m, respectively, where l = 0, 1, 2, \dots and m = -l, \dots, l.^[52] These functions, utilized by Schrödinger in solving the hydrogen atom problem in 1926, form a complete orthogonal basis for expanding the angular dependence of \psi(\mathbf{r}).^[52] For particles with both orbital and spin angular momentum, the total angular momentum \mathbf{J} = \mathbf{L} + \mathbf{S} is considered, leading to states classified by the quantum number j ranging from |l - s| to l + s in integer steps, with the fine-structure splitting arising from the \mathbf{L} \cdot \mathbf{S} interaction term in the Hamiltonian.^[51] This addition of angular momenta, formalized through Clebsch-Gordan coefficients, ensures proper coupling under rotations.

Many-body and entangled states

In quantum mechanics, the state of a system composed of multiple distinguishable particles is described by the tensor product of the individual Hilbert spaces, forming a composite Hilbert space \mathcal{H} = \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \cdots \otimes \mathcal{H}_N, where each \mathcal{H}_i is the Hilbert space for the i-th particle. This construction allows the total wave function to be expressed as a product or superposition of single-particle states, capturing independent behaviors while enabling correlations through linear combinations. For identical particles, however, the Hilbert space must be restricted to symmetrized or antisymmetrized subspaces to account for indistinguishability, as required by the symmetrization postulate.^[53] For identical bosons, the wave function must be fully symmetric under particle exchange, meaning \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_N) = \Psi(P\mathbf{x}_1, P\mathbf{x}_2, \dots, P\mathbf{x}_N) for any permutation P, allowing multiple particles to occupy the same quantum state.^[53] In contrast, for identical fermions, the wave function is fully antisymmetric, \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_N) = -\Psi(P\mathbf{x}_1, P\mathbf{x}_2, \dots, P\mathbf{x}_N), which enforces the Pauli exclusion principle and prevents identical fermions from sharing the same state.^[53] A practical representation for fermionic many-body states is the Slater determinant, constructed from single-particle orbitals \phi_j(\mathbf{x}_i) as

\Psi(\mathbf{x}_1, \dots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \det \begin{pmatrix} \phi_1(\mathbf{x}_1) & \phi_1(\mathbf{x}_2) & \cdots & \phi_1(\mathbf{x}_N) \\ \phi_2(\mathbf{x}_1) & \phi_2(\mathbf{x}_2) & \cdots & \phi_2(\mathbf{x}_N) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_N(\mathbf{x}_1) & \phi_N(\mathbf{x}_2) & \cdots & \phi_N(\mathbf{x}_N) \end{pmatrix},

which inherently ensures antisymmetry and is widely used in approximations like Hartree-Fock theory. Entanglement arises in many-body quantum states when the total state cannot be factored into a product of individual particle states, leading to non-local correlations that defy classical intuition. A canonical example is the Bell state for two qubits, \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle), which is entangled and exhibits perfect correlation in measurements of spin or polarization. This phenomenon was first highlighted in the Einstein-Podolsky-Rosen (EPR) thought experiment, which questioned the completeness of quantum mechanics by proposing that entangled particles imply "spooky action at a distance" through instantaneous influences violating locality.^[54] Particle statistics profoundly influence many-body phenomena; for instance, bosonic systems can form a Bose-Einstein condensate (BEC) at low temperatures, where a macroscopic number of particles occupy the lowest quantum state, exhibiting coherent matter-wave behavior akin to a laser for atoms. The first experimental realization of a BEC in a dilute rubidium-87 vapor occurred in 1995, confirming macroscopic quantum effects predicted by Bose and Einstein nearly 70 years earlier.^[55]

Advanced Generalizations

Relativistic quantum states

In relativistic quantum mechanics, the concept of quantum states must incorporate the principles of special relativity to describe particles moving at speeds comparable to the speed of light, ensuring Lorentz invariance. Unlike non-relativistic quantum mechanics, where states are represented by wave functions in a fixed Hilbert space, relativistic formulations reveal inherent challenges such as negative energy solutions and the lack of a positive-definite probability density, necessitating a transition to quantum field theory for consistency. The Klein-Gordon equation provides the foundational relativistic wave equation for spin-0 particles, derived by quantizing the relativistic energy-momentum relation E^2 = p^2 c^2 + m^2 c^4. Independently proposed by Oskar Klein and Walter Gordon in 1926, the equation is given by

\left( \Box + \frac{m^2 c^2}{\hbar^2} \right) \phi = 0,

where \Box = \partial^\mu \partial_\mu is the d'Alembertian operator in Minkowski space, \phi is a complex scalar field, m is the particle mass, c is the speed of light, and \hbar is the reduced Planck's constant. Solutions to this equation include plane waves with both positive and negative energy eigenvalues, E = \pm \sqrt{p^2 c^2 + m^2 c^4}, leading to a continuous spectrum that interprets particles and antiparticles on equal footing, though the associated probability density \rho = \frac{i}{2m} (\phi^* \partial_t \phi - \phi \partial_t \phi^*) can become negative, violating the positivity required for single-particle wave functions. For spin-1/2 particles like electrons, Paul Dirac introduced the Dirac equation in 1928 to reconcile quantum mechanics with relativity while incorporating spin naturally. The equation reads

i \hbar \frac{\partial \psi}{\partial t} = \left( c \boldsymbol{\alpha} \cdot \mathbf{p} + \beta m c^2 \right) \psi,

where \psi is a four-component spinor wave function, \mathbf{p} = -i \hbar \nabla is the momentum operator, and \boldsymbol{\alpha}, \beta are 4×4 matrices satisfying the Dirac algebra \{\alpha_i, \alpha_j\} = 2\delta_{ij}, \{\alpha_i, \beta\} = 0, \beta^2 = 1.^[56] The spinor states \psi describe relativistic electrons with positive energy solutions corresponding to particles and negative energy ones reinterpreted as positrons (antiparticles) via Dirac's hole theory, though this single-particle interpretation still suffers from issues like negative probabilities and infinite vacuum energy. These challenges in single-particle relativistic wave functions—arising from the unbounded spectrum, lack of conserved particle number, and causality violations—cannot be resolved within a fixed-particle-number framework, as relativity allows for particle creation and annihilation. The solution lies in second quantization, where states are promoted to operators in quantum field theory, constructing the Hilbert space as a Fock space of multi-particle configurations. Seminal work by Vladimir Fock in 1932 formalized this, representing relativistic quantum states as superpositions of occupation number vectors | \{ n_{\mathbf{p}, s} \} \rangle, where n_{\mathbf{p}, s} denotes the number of particles (or antiparticles) with momentum \mathbf{p} and spin s, built from fermionic or bosonic creation and annihilation operators satisfying canonical anticommutation or commutation relations. This Fock space approach ensures Lorentz-invariant states accommodating particles and antiparticles dynamically, resolving the pathologies of single-particle theories.

Quantum information perspectives

In quantum information theory, quantum states serve as the fundamental carriers of information, enabling protocols for computation, communication, and cryptography that exploit uniquely quantum phenomena such as superposition and entanglement. Unlike classical bits, which are strictly 0 or 1, quantum states allow for probabilistic outcomes and correlations that defy classical intuition, forming the basis for devices like quantum computers and secure networks. This perspective emphasizes the informational content and manipulability of states, often using density operators to describe both pure and mixed configurations, where mixedness arises from classical uncertainty or environmental interactions.^[57] The qubit, or quantum bit, represents the simplest non-trivial quantum information unit, with its state residing in a two-dimensional Hilbert space spanned by basis states |0⟩ and |1⟩. Pure qubit states can be visualized geometrically on the Bloch sphere, a unit sphere in three-dimensional real space where the state |ψ⟩ = α|0⟩ + β|1⟩ (with |α|^2 + |β|^2 = 1) corresponds to a point with coordinates (x, y, z) = (2 Re(αβ), 2 Im(αβ), |α|^2 - |β|^2); the north pole represents |0⟩, the south pole |1⟩, and equatorial points pure superpositions. This representation facilitates understanding of single-qubit operations as rotations on the sphere. Qudits extend this to higher dimensions, defining states in a d-level Hilbert space (d > 2), which for a system of multiple qubits spans a 2^n-dimensional space for n qubits, allowing denser information encoding and more efficient quantum algorithms. For instance, qudits have been demonstrated in photonic and ion-trap platforms to enhance quantum communication capacity.^[57]^[58] Entanglement in quantum states quantifies non-classical correlations essential for tasks like quantum teleportation and dense coding. For pure bipartite states, entanglement entropy, defined as the von Neumann entropy S(ρ_A) = -Tr(ρ_A log ρ_A) of the reduced density operator ρ_A on one subsystem, measures the entanglement degree, with maximum value log d for d-dimensional equally entangled states. For mixed states, which may include separable mixtures, concurrence provides a computable entanglement monotone; for two qubits, it ranges from 0 (separable) to 1 (maximally entangled Bell state) and is calculated from the eigenvalues of a specific matrix involving the state's spin-flip. These measures highlight how entanglement resources degrade under local operations, guiding resource theories in quantum information.^[59]^[60] Quantum channels model the evolution of quantum states under noisy or imperfect processes, formalized as completely positive trace-preserving maps that preserve the positivity and trace of density operators. Any such channel admits a Kraus representation ρ → ∑_i K_i ρ K_i^†, where the Kraus operators {K_i} satisfy the completeness relation ∑_i K_i^† K_i = I, capturing decoherence mechanisms like amplitude damping or phase flips in quantum devices. This framework underpins noise characterization in experiments, enabling simulations of real-world quantum hardware.^[61] Post-2000 advancements in quantum state tomography have improved state reconstruction efficiency; for example, adaptive and compressed sensing techniques reduce measurement overhead from O(d^2) to near-linear in system dimension d, achieving high fidelity with fewer copies for verification of quantum gates.^[62] In quantum error correction, states are protected by encoding logical qubits into multi-physical-qubit subspaces using codes that detect and correct errors without collapsing the superposition. Developments since 2000, including decoder optimizations via machine learning, have enabled fault tolerance with physical error rates below the ~1% threshold typical for surface code implementations under standard noise models, where logical states span a 2^k-dimensional code space for k logical qubits amid noise. Recent transformer-based decoders process syndrome data to achieve exponential error suppression with scale.^[63] In December 2024, Google demonstrated quantum error correction below the surface code threshold using distance-7 and distance-5 codes on superconducting processors, achieving further exponential suppression of logical errors.^[64] These codes, combined with channel models, ensure scalable quantum information processing.

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