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State vector

A state vector is a mathematical representation of the state of a physical system, used in both quantum mechanics and classical dynamical systems. In quantum mechanics, it is often denoted as |\psi\rangle in Dirac notation and serves as an element of a complex Hilbert space that encodes all probabilistic information about the system's observables.^[1] This vector fully specifies the system's properties, including the probabilities of measurement outcomes for any observable, as per the Born rule, where the probability of obtaining a particular eigenvalue is given by the squared modulus of the projection onto the corresponding eigenstate.^[2] State vectors are typically normalized such that their inner product with themselves equals 1, ensuring they represent physically realizable pure states, and they can be superposed to describe more complex quantum behaviors like interference.^[3] In classical dynamical systems, the state vector captures the minimal set of variables, such as positions and velocities, needed to describe the system's evolution in phase space.^[4] The Hilbert space framework underpinning quantum state vectors provides a rigorous structure for quantum theory, where the space is separable and complete, allowing for the representation of states as infinite-dimensional vectors when necessary for continuous systems.^[1] In practice, quantum state vectors evolve deterministically according to the time-dependent Schrödinger equation, i\hbar \frac{\partial}{\partial t} |\psi\rangle = \hat{H} |\psi\rangle, where \hat{H} is the Hamiltonian operator describing the system's total energy.^[1] Upon measurement, the state vector collapses to one of the eigenstates of the measured observable, a process central to the probabilistic nature of quantum mechanics.^[2] State vectors are foundational to various applications, from atomic physics to quantum information science and classical control theory, where they enable descriptions like entanglement in multi-particle quantum systems as vectors in tensor product spaces.^[3] Different representations, such as the position-space wavefunction \psi(\mathbf{r}) = \langle \mathbf{r} | \psi \rangle, offer practical ways to compute expectations and transitions, bridging abstract formalism with experimental predictions.^[1] The quantum mechanical formulation of state vectors, introduced in the early 20th century by pioneers like Dirac and von Neumann, emphasizes features such as superposition and non-commutativity of observables that distinguish it from classical descriptions.^[3]

Mathematical background

Vector spaces

A vector space over a field F (such as the real numbers \mathbb{R} or complex numbers \mathbb{C}) is a nonempty set V of objects called vectors, equipped with two operations: vector addition +: V \times V \to V and scalar multiplication \cdot: F \times V \to V, satisfying the following axioms for all vectors \mathbf{u}, \mathbf{v}, \mathbf{w} \in V and scalars a, b \in F:

Associativity of addition: (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}).
Commutativity of addition: \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}.
Existence of zero vector: There exists \mathbf{0} \in V such that \mathbf{u} + \mathbf{0} = \mathbf{u}.
Existence of additive inverses: For each \mathbf{u}, there exists -\mathbf{u} \in V such that \mathbf{u} + (-\mathbf{u}) = \mathbf{0}.
Distributivity of scalar multiplication over vector addition: a \cdot (\mathbf{u} + \mathbf{v}) = a \cdot \mathbf{u} + a \cdot \mathbf{v}.
Distributivity of scalar addition over scalar multiplication: (a + b) \cdot \mathbf{u} = a \cdot \mathbf{u} + b \cdot \mathbf{u}.
Compatibility of scalar multiplication: a \cdot (b \cdot \mathbf{u}) = (a b) \cdot \mathbf{u}.
Identity element of scalars: $1 \cdot \mathbf{u} = \mathbf{u}, where 1 is the multiplicative identity in F.

These axioms ensure closure under the operations and provide the algebraic structure necessary for linear combinations.^[5] Common examples include the Euclidean space \mathbb{R}^n, where vectors are n-tuples of real numbers, addition is componentwise, and scalar multiplication scales each component; this forms a vector space of dimension n.^[6] Another example is the space of continuous functions C[0,1] on the interval [0,1], where addition and scalar multiplication are defined pointwise: (f + g)(x) = f(x) + g(x) and (a \cdot f)(x) = a f(x) for x \in [0,1].^[7] A subset \{\mathbf{v}_1, \dots, \mathbf{v}_k\} \subseteq V is linearly independent if the only solution to a_1 \mathbf{v}_1 + \dots + a_k \mathbf{v}_k = \mathbf{0} is a_1 = \dots = a_k = 0; otherwise, it is linearly dependent. The set spans V if every vector in V can be expressed as a linear combination \sum a_i \mathbf{v}_i. A basis for V is a linearly independent set that spans V; all bases for a given vector space have the same cardinality, called the dimension of V, denoted \dim V.^[8]^[9] Given a basis \{\mathbf{e}_1, \dots, \mathbf{e}_n\} for a vector space of dimension n, any vector \mathbf{v} \in V can be uniquely represented as \mathbf{v} = x_1 \mathbf{e}_1 + \dots + x_n \mathbf{e}_n, where the coefficients x_1, \dots, x_n form the coordinate representation of \mathbf{v} with respect to this basis, often written as the column vector \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}.^[10]^[11] The foundational ideas of vector spaces emerged in the 19th century through the development of linear algebra, notably in the works of William Rowan Hamilton on quaternions and Hermann Grassmann on extension theory.^[12]^[13]

Hilbert spaces

An inner product space, also referred to as a pre-Hilbert space, is a complex vector space equipped with a sesquilinear form denoted as the inner product \langle \phi | \psi \rangle, which satisfies three key axioms: positivity, where \langle \phi | \phi \rangle \geq 0 and equals zero if and only if \phi = 0; linearity in the second argument, meaning \langle \phi | a\psi + b\chi \rangle = a \langle \phi | \psi \rangle + b \langle \phi | \chi \rangle for scalars a, b; and conjugate symmetry, \langle \phi | \psi \rangle = \overline{\langle \psi | \phi \rangle}.^[14] These properties induce a norm \|\phi\| = \sqrt{\langle \phi | \phi \rangle}, enabling the definition of distances and angles within the space, extending the structure of general vector spaces by adding a metric compatible with the vector operations. A Hilbert space is defined as a complete inner product space, meaning every Cauchy sequence with respect to the norm converges to an element within the space; this completeness turns it into a Banach space where the norm arises from the inner product.^[15]^[16] In such spaces, orthonormal bases play a central role: an orthonormal set \{e_n\} satisfies \langle e_m | e_n \rangle = \delta_{mn}, and for a complete orthonormal basis (also called a Hilbert basis), Parseval's identity holds, stating that for any \psi in the space,

\|\psi\|^2 = \sum_n |\langle e_n | \psi \rangle|^2.

This identity equates the squared norm of a vector to the sum of the squared magnitudes of its coefficients in the basis expansion.^[17]^[18] Prominent examples of Hilbert spaces include the sequence space \ell^2, consisting of square-summable complex sequences \{x_n\} with inner product \langle x | y \rangle = \sum_n x_n^* y_n, which is complete under the induced norm.^[14] Another key example is the function space L^2(\mathbb{R}), the set of square-integrable functions f on the real line, equipped with \langle f | g \rangle = \int_{-\infty}^{\infty} f^*(x) g(x) \, dx, providing a foundational setting for representing continuous systems such as wave functions in mathematical analysis.^[19]^[20] The inner product in a Hilbert space ensures separation of distinct states: for two vectors \phi \neq \psi, the positive definiteness implies \|\phi - \psi\|^2 = \langle \phi - \psi | \phi - \psi \rangle > 0, distinguishing them via their inner product differences.^[14]

Applications in quantum mechanics

Definition of quantum state vector

In quantum mechanics, the state of a physical system in a pure state is represented by a state vector |ψ⟩, which belongs to a separable Hilbert space H equipped with an inner product.^[21] This vector encodes all observable information about the system, such as probabilities of measurement outcomes for physical observables represented as self-adjoint operators on H.^[22] Pure states correspond directly to these vectors, distinguishing them from mixed states, which require density operators to describe statistical ensembles of pure states.^[23] The Dirac notation provides a compact and basis-independent way to express these state vectors.^[24] Here, |ψ⟩ denotes the ket vector in H, while its dual ⟨ψ| is the bra vector in the dual space, and the inner product between two state vectors is given by ⟨ψ|φ⟩, a complex number satisfying ⟨ψ|ψ⟩ ≥ 0 with equality only for the zero vector.^[21] This notation facilitates operations like projections and transformations without explicit coordinate representations. State vectors are defined up to a global phase factor, meaning that |ψ⟩ and e^{iθ} |ψ⟩, where θ is a real number, describe the identical physical state, as phase differences do not affect measurement probabilities.^[22] The mathematical framework of separable Hilbert spaces for quantum states was formalized by John von Neumann in his 1932 treatise.^[22] Paul Dirac introduced the ket-bra notation and the abstract vector representation in the 1930s, building on earlier wave and matrix formulations.^[25]

Superposition and normalization

In quantum mechanics, the superposition principle asserts that a quantum state vector |\psi\rangle can be expressed as a linear combination of basis states |\phi_i\rangle, written as |\psi\rangle = \sum_i c_i |\phi_i\rangle, where the coefficients c_i are complex numbers satisfying the normalization condition \sum_i |c_i|^2 = 1. This principle, foundational to the linear structure of quantum theory, allows quantum systems to inhabit multiple states simultaneously until measurement, distinguishing them from classical systems. The normalization condition ensures the state vector has unit norm, defined as \langle \psi | \psi \rangle = 1, where \langle \psi | is the bra vector corresponding to |\psi\rangle, and the norm is \|\psi\| = \sqrt{\langle \psi | \psi \rangle}. This requirement guarantees that the total probability of all possible measurement outcomes sums to unity, providing physical interpretability to the state vector. For distinct eigenstates |\phi_i\rangle and |\phi_j\rangle (with i \neq j) of a Hermitian operator, such as the Hamiltonian, orthogonality holds: \langle \phi_i | \phi_j \rangle = 0. This property arises from the spectral theorem for Hermitian operators and facilitates the expansion of any state in an orthonormal basis of eigenstates.^[26] The probability interpretation of the state vector, known as the Born rule, states that if a system is in state |\psi\rangle, the probability of measuring it to be in eigenstate |\phi_i\rangle is |\langle \phi_i | \psi \rangle|^2. This rule, postulated by Max Born, links the mathematical formalism to observable probabilities without deriving from deeper principles in the standard formulation.^[27] A representative example is the state of a two-level quantum bit, or qubit, given by |\psi\rangle = \alpha |0\rangle + \beta |1\rangle, where |0\rangle and |1\rangle form an orthonormal basis with \langle 0 | 1 \rangle = 0, and the complex coefficients satisfy |\alpha|^2 + |\beta|^2 = 1 for normalization. Here, |\alpha|^2 is the probability of measuring the qubit in |0\rangle, illustrating superposition in quantum information processing.^[28]

Time evolution

The time evolution of a quantum state vector |\psi(t)\rangle in a closed system is described by the time-dependent Schrödinger equation,

i \hbar \frac{\partial}{\partial t} |\psi(t)\rangle = \hat{H} |\psi(t)\rangle,

where \hat{H} is the Hamiltonian operator representing the total energy of the system, \hbar is the reduced Planck's constant, and i is the imaginary unit. This equation, introduced by Erwin Schrödinger in 1926, governs the deterministic dynamics of the wave function in non-relativistic quantum mechanics.^[29] For time-independent Hamiltonians, the formal solution is given by unitary evolution,

|\psi(t)\rangle = \hat{U}(t) |\psi(0)\rangle, \quad \hat{U}(t) = e^{-i \hat{H} t / \hbar},

where \hat{U}(t) is the time-evolution operator, which is unitary (\hat{U}^\dagger \hat{U} = \hat{1}) and preserves the norm of the state vector, ensuring that probabilities remain normalized over time. This unitary nature stems from the hermiticity of \hat{H}, guaranteeing reversible dynamics without information loss in isolated systems.^[30] Two equivalent formulations describe this evolution: the Schrödinger picture, where state vectors evolve in time while operators remain fixed, and the Heisenberg picture, where state vectors are time-independent and operators evolve according to the Heisenberg equation d\hat{A}/dt = (i/\hbar) [\hat{H}, \hat{A}] + \partial \hat{A}/\partial t. The pictures are related by a unitary transformation and yield identical observable predictions, with the choice depending on convenience for calculations involving time-dependent perturbations or symmetries./03%3A_Schrodinger_and_Heisenberg_Pictures) A representative example is the quantum harmonic oscillator, with Hamiltonian \hat{H} = \hbar \omega (\hat{a}^\dagger \hat{a} + 1/2), where \omega is the angular frequency, \hat{a}^\dagger and \hat{a} are the creation and annihilation operators. The energy eigenstates |n\rangle evolve as |n(t)\rangle = e^{-i E_n t / \hbar} |n\rangle, with E_n = \hbar \omega (n + 1/2), acquiring only a global phase that does not affect expectation values. Superpositions of these states, such as coherent states, exhibit classical-like oscillatory behavior in position and momentum expectation values while maintaining quantum coherence.^[31] For time-dependent Hamiltonians that vary slowly compared to the inverse energy gaps between eigenstates, the adiabatic theorem states that if the system begins in an instantaneous eigenstate |n(0)\rangle of \hat{H}(0), it will remain in the corresponding instantaneous eigenstate |n(t)\rangle of \hat{H}(t) up to a dynamic and geometric phase, provided the change is sufficiently gradual. This result, proven by Max Born and Vladimir Fock in 1928, underpins applications like adiabatic quantum computing and molecular dynamics in slowly varying potentials.^[32]

Applications in classical dynamical systems

State-space representation

In classical dynamical systems, the state vector \mathbf{x}(t) \in \mathbb{R}^n (or occasionally \mathbb{C}^n for systems with complex dynamics) encapsulates the minimal set of variables required to uniquely determine the future evolution of the system given any inputs and initial conditions.^[33]^[34] These variables represent the complete internal configuration of the system at time t, allowing prediction of all subsequent behavior without reference to prior history beyond the initial state.^[35] The collection of all possible state vectors forms the phase space, a manifold that geometrically represents the set of all achievable states of the system.^[35] This space provides a coordinate framework for visualizing trajectories, where each point corresponds to a unique state, and the system's dynamics trace paths along the manifold.^[36] The evolution of the state vector is governed by the initial value problem

\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t), \mathbf{u}(t), t), \quad \mathbf{x}(0) = \mathbf{x}_0,

where \mathbf{u}(t) denotes the input vector influencing the system, \mathbf{f} is a vector-valued function describing the dynamics, and \mathbf{x}_0 specifies the starting state.^[37]^[35] Solutions to this problem yield the state trajectory \mathbf{x}(t) for t \geq 0, assuming local existence and uniqueness under suitable conditions on \mathbf{f}.^[37] The conceptual foundations of state-space representation trace back to Henri Poincaré's work in the 1890s, where he introduced phase space analysis to study celestial mechanics, particularly the three-body problem, revealing non-integrable dynamics.^[38] This approach was later formalized in modern control theory by Rudolf E. Kálmán during the 1960s, who developed state-space methods for linear systems, enabling systematic analysis and design.^[39] A representative example is a simple mechanical oscillator, such as a mass-spring system without damping, where the state vector comprises the position q(t) and velocity \dot{q}(t) of the mass.^[36]^[40] These two variables fully capture the system's energy and motion, allowing the future position and speed to be predicted from the initial conditions and any applied forces.^[41]

Linear systems

In linear time-invariant (LTI) dynamical systems, the state vector \mathbf{x}(t) \in \mathbb{R}^n captures the system's internal dynamics, evolving according to the state-space equations \dot{\mathbf{x}}(t) = A \mathbf{x}(t) + B \mathbf{u}(t) and \mathbf{y}(t) = C \mathbf{x}(t) + D \mathbf{u}(t), where A \in \mathbb{R}^{n \times n} is the system matrix, B \in \mathbb{R}^{n \times m} is the input matrix, C \in \mathbb{R}^{p \times n} is the output matrix, D \in \mathbb{R}^{p \times m} is the feedthrough matrix, \mathbf{u}(t) \in \mathbb{R}^m is the input vector, and \mathbf{y}(t) \in \mathbb{R}^p is the output vector.^[42] This matrix formulation, introduced by Kalman, enables the analysis of multi-input multi-output systems through linear algebra, assuming constant coefficients and superposition of solutions.^[43] The explicit solution to the state equation for zero initial conditions is given by the convolution integral:

\mathbf{x}(t) = \int_0^t e^{A(t - \tau)} B \mathbf{u}(\tau) \, d\tau,

while the full solution incorporating initial state \mathbf{x}(0) is

\mathbf{x}(t) = e^{A t} \mathbf{x}(0) + \int_0^t e^{A(t - \tau)} B \mathbf{u}(\tau) \, d\tau,

where e^{A t} is the state transition matrix, computable via matrix exponential or modal decomposition.^[44] This form highlights the system's memory through the initial state and the influence of past inputs weighted by the transition matrix. Key properties for control design are controllability and observability. A system is controllable if the controllability matrix \mathcal{C} = [B \, AB \, \cdots \, A^{n-1}B] has full row rank n, ensuring any initial state can be driven to the origin (or any target state) in finite time using admissible inputs.^[42] Similarly, the system is observable if the observability matrix \mathcal{O} = \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1} \end{bmatrix} has full column rank n, allowing reconstruction of the state from output measurements over a finite interval.^[43] These rank conditions, central to modern control theory, facilitate state feedback and observer design. Stability of the unforced system \dot{\mathbf{x}} = A \mathbf{x} is determined by the eigenvalues \lambda_i of A: the origin is asymptotically stable if all \operatorname{Re}(\lambda_i) < 0, marginally stable if \operatorname{Re}(\lambda_i) \leq 0 with simple imaginary eigenvalues, and unstable otherwise.^[44] This eigenvalue criterion provides a direct link between system matrix properties and long-term behavior, independent of input details. A representative example is the mass-spring-damper system, modeled with states x_1 = x (position) and x_2 = \dot{x} (velocity), yielding \dot{\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{c}{m} \end{bmatrix} \mathbf{x} + \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} u, where m is mass, k is spring constant, c is damping coefficient, and u is applied force; the output might be position y = x_1.^[40] Here, controllability holds as the matrix [B \, AB] is full rank, and stability requires c > 0 for positive damping, ensuring \operatorname{Re}(\lambda_i) < 0.^[42]

Nonlinear systems

In nonlinear dynamical systems, the state vector \mathbf{x}(t) \in \mathbb{R}^n encapsulates the minimal set of variables required to describe the system's evolution, where the dynamics are governed by nonlinear differential equations of the form \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u}, t), with \mathbf{u}(t) denoting the input vector and \mathbf{f} a nonlinear function.^[34]^[33] This representation contrasts with linear systems by lacking the superposition principle, leading to phenomena such as bifurcations, limit cycles, and chaos. The state space X \subseteq \mathbb{R}^n is typically a manifold, and the system's trajectory \mathbf{x}(t) = \phi(t; \mathbf{x}_0, \mathbf{u}) is determined by an initial state \mathbf{x}_0 and the transition map \phi, ensuring causality where future states depend only on prior history.^[34]^[45] For continuous-time systems, the state vector often includes position and velocity components to convert higher-order nonlinear ODEs into first-order form. A classic example is the pendulum equation \ddot{\theta} + r \dot{\theta} + \sin \theta = 0, where the state vector is \mathbf{x} = [\theta, \dot{\theta}]^T and \mathbf{f}(\mathbf{x}) = [\dot{\theta}, -r \dot{\theta} - \sin \theta]^T, exhibiting periodic motion for small angles but chaotic behavior for large swings.^[33] Another illustrative case is the Duffing oscillator, modeling nonlinear stiffness in mechanical systems: \ddot{x} + \delta \dot{x} - \omega^2 x + \epsilon x^3 = \beta \cos(\Omega t), with state vector \mathbf{x} = [x, \dot{x}]^T and dynamics \dot{\mathbf{x}} = [ \dot{x}, -\delta \dot{x} + \omega^2 x - \epsilon x^3 + \beta \cos(\Omega t) ]^T, capable of producing chaotic attractors depending on parameters like \epsilon > 0.^[34] In discrete-time settings, the logistic map x_{k+1} = a x_k (1 - x_k) for x_k \in [0,1] and a \in (0,4] serves as a one-dimensional state vector example, demonstrating period-doubling bifurcations leading to chaos as a approaches 4.^[34] Analysis of nonlinear systems often involves linearization around equilibrium points \mathbf{x}_e where \mathbf{f}(\mathbf{x}_e, \mathbf{u}_e) = 0, yielding \dot{\delta \mathbf{x}} = A \delta \mathbf{x} + B \delta \mathbf{u} with Jacobian A = \frac{\partial \mathbf{f}}{\partial \mathbf{x}} \big|_{\mathbf{x}_e, \mathbf{u}_e}, which approximates local stability via eigenvalues of A.^[33] For stochastic nonlinear systems under excitation \mathbf{w}(t), the state-space form \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{w}, t) enables emulation techniques like State Space Kriging, which learns the MIMO mapping from state and input to derivatives, mitigating high-dimensional challenges in prediction with relative errors below $10^{-3} using sparse training data.^[45] In modal analysis, the state vector \mathbf{u}(x,t) decomposes into nonlinear modes via \mathbf{u}(x,t) = \sum \phi_n(x) a_n(t), revealing energy transfers and connections between modes in conservative systems.^[46] Learning such systems from data treats the state vector as hidden, evolving via \mathbf{x}_{t} = \mathbf{f}(\mathbf{x}_{t-1}, \mathbf{u}_{t-1}) + \mathbf{w}_t with output \mathbf{y}_t = \mathbf{g}(\mathbf{x}_t, \mathbf{u}_t) + \mathbf{v}_t, often parameterized by radial basis functions and inferred using expectation-maximization with Kalman smoothing for parameter estimation in under 10 iterations.^[47] These representations underpin applications in control, prediction, and simulation, emphasizing the state vector's role in capturing complex, non-superposable behaviors.^[34]

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[PDF] Time-Domain Solution of LTI State Equations - MIT
A linear system, described by state equations ˙x = AX + Bu, is asymptotically stable if and only if all eigenvalues of the matrix A have negative real parts.
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State Space Kriging model for emulating complex nonlinear ...
The state space representation of a stochastic dynamical system can be considered as a multi-input multi-output (MIMO) function, in which the input is the state ...
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Connections between the modes of a nonlinear dynamical system ...
Jun 9, 2021 · In modal analysis, the state vector u of a dynamical system, depending on position x and time t , is assumed to be decomposed in the following ...
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### Summary of Key Concepts on State Vectors in Nonlinear Dynamical Systems