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Bound state

A bound state is a configuration in which two or more particles, atoms, or bodies interact via attractive forces or potentials, confining them to a finite region of space such that they cannot escape to infinity without external energy input.^[1] In classical mechanics, bound states correspond to trajectories where the total energy is less than the potential energy at infinity, exemplified by planets orbiting stars in gravitational fields.^[2] In quantum mechanics, a bound state is a stable quantum configuration of particles confined within a potential well, characterized by a normalizable wave function that decays exponentially at large distances and discrete energy eigenvalues below the continuum threshold, typically with negative energy relative to the potential at infinity.^[3]^[4] Bound states differ fundamentally from scattering or free states, where wave functions extend over all space and form a continuous energy spectrum with non-negative energies; in contrast, bound state wave functions are localized, ensuring the particle has a finite probability of being found only within the confining region.^[4] Key properties include the quantization of energy levels, which arise from the boundary conditions imposed by the potential, and the absence of degeneracy in one-dimensional systems unless specified otherwise.^[5]^[6] In quantum mechanics, any attractive potential in one dimension supports at least one bound state.^[4] Examples of bound states abound in nature and are central to atomic, molecular, and nuclear physics, as well as celestial mechanics; for instance, electrons bound to protons in atoms form stable configurations with quantized energy levels, as seen in the hydrogen atom.^[7] In a finite square well potential, multiple bound states emerge depending on the well's depth and width, with the number of states increasing as the potential deepens.^[4] These states underpin phenomena such as molecular bonding and nuclear stability, where particles like neutrons and protons are trapped by strong forces, and transitions between bound states occur via absorption or emission of quanta like photons.^[7]

Classical Bound States

Definition and Characteristics

In classical mechanics, a bound state describes the confined motion of a particle within a finite region of space, occurring when the total energy E of the system is less than the potential energy V at infinity, preventing escape to unbounded distances.^[8] For typical attractive central potentials where V(r) \to 0 as r \to \infty, this condition simplifies to E < 0, ensuring the particle remains trapped by the potential well. This framework originated in Newtonian mechanics during the 17th century, with foundational examples drawn from celestial mechanics, such as the elliptical orbits of planets around the Sun as articulated in Kepler's laws of planetary motion.^[9] Key characteristics of classical bound states include periodic or quasi-periodic trajectories arising from the conservation of total energy and angular momentum, which restrict the particle to closed or nearly closed paths without dissipation./11%3A_Conservative_two-body_Central_Forces/11.07%3A_General_Features_of_the_Orbit_Solutions) In central force problems, these orbits are often elliptical for inverse-square forces like gravity but form rosette patterns—precessing loops—for other potentials, reflecting the deterministic evolution under conserved quantities./11%3A_Conservative_two-body_Central_Forces/11.07%3A_General_Features_of_the_Orbit_Solutions) This contrasts sharply with scattering states, where E > 0 permits hyperbolic trajectories that extend to infinity, representing unbound motion. The dynamics of such bound states are governed by Lagrangian or Hamiltonian formulations. For a particle of mass m in a central potential V(r), the Lagrangian is given by

L = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) - V(r),

from which Hamilton's equations derive the equations of motion, yielding conserved quantities like energy E = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\theta}^2) + V(r) and angular momentum l = m r^2 \dot{\theta}./04%3A_Hamilton%27s_Principle_and_Noether%27s_Theorem/4.09%3A_Example_2-__Lagrangian_Formulation_of_the_Central_Force_Problem)

Examples in Mechanics

One prominent example of a classical bound state is the simple harmonic oscillator, where a particle of mass m moves in a quadratic potential V(x) = \frac{1}{2} k x^2, with k > 0 being the spring constant.^[10] For any finite total energy E > 0, the motion is confined between turning points x = \pm \sqrt{2E/k}, resulting in periodic oscillations.^[11] In phase space, the trajectory forms a closed ellipse, parameterized by position x and momentum p = m \dot{x}, illustrating the bounded, periodic nature of the orbit.^[12] Another key illustration arises in gravitational systems, such as planetary motion around a central star, governed by the attractive inverse-square potential V(r) = -GMm/r, where G is the gravitational constant, M and m are the masses, and r is the separation.^[13] Kepler's first law dictates that bound orbits are ellipses with the star at one focus, ensuring the planet remains confined without escaping to infinity.^[13] For such elliptical orbits, the total energy is negative, specifically E = -GMm/(2a), where a is the semi-major axis, confirming the bound state. In broader central force problems, bound states occur for attractive $1/r potentials when E < 0, yielding closed elliptical orbits analogous to gravitational cases. This contrasts with repulsive $1/r potentials, as in Rutherford scattering, where positive energies lead to hyperbolic trajectories and unbound scattering rather than confinement.^[14] For two-dimensional anisotropic harmonic oscillators, with unequal frequencies \omega_x \neq \omega_y in potentials V(x,y) = \frac{1}{2} m \omega_x^2 x^2 + \frac{1}{2} m \omega_y^2 y^2, the trajectories trace Lissajous figures—closed curves that fill the bounded region without escaping, provided the frequencies are commensurate (rational ratio).^[15] In classical mechanics, these bound states persist indefinitely, with no mechanism for escape. This setup shares conceptual similarities with the quantum harmonic oscillator but lacks wave-like uncertainty.^[15]

Quantum Bound States

Formal Definition

In quantum mechanics, a bound state is defined as an eigenstate of the Hamiltonian operator \hat{H} satisfying the eigenvalue equation \hat{H} \psi = E \psi, where the energy E satisfies E < 0 for potentials V(\mathbf{r}) that vanish at infinity, and the corresponding wave function \psi(\mathbf{r}) is square-integrable over all space, ensuring \int |\psi(\mathbf{r})|^2 d^3\mathbf{r} = 1 and \psi(\mathbf{r}) \to 0 as |\mathbf{r}| \to \infty.^[16] This square-integrability reflects the spatial localization of the state, confining the particle to a finite region with probability approaching zero far from the potential well.^[16] Bound states are distinguished from scattering states, which correspond to continuum eigenvalues E \geq 0 above the potential threshold, yielding non-normalizable wave functions that extend indefinitely and represent free or asymptotically plane-wave behaviors. The discrete nature of bound state energies arises because only specific eigenvalues below the continuum threshold allow for normalizable solutions to the time-independent Schrödinger equation, ensuring the state's stability against dissociation.^[17] The formal framework is encapsulated in the three-dimensional time-independent Schrödinger equation for a single particle of mass m:

-\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r}) \psi(\mathbf{r}) = E \psi(\mathbf{r}),

where binding requires E < \lim_{|\mathbf{r}| \to \infty} V(\mathbf{r}), typically E < 0 when V(\infty) = 0, preventing the particle from escaping to infinity.^[18]^[17] This applies primarily to non-relativistic quantum mechanics. For multi-particle systems, such as two-body bound states like the hydrogen atom, the problem reduces to an effective one-body equation using the reduced mass \mu = \frac{m_1 m_2}{m_1 + m_2} in place of m, separating center-of-mass motion from relative coordinates.^[19]^[20]

Wave Function Properties

In quantum mechanics, the wave function \psi(\mathbf{r}) of a bound state must be square-integrable over all space to ensure it represents a physically realizable state with finite total probability. This normalizability condition is expressed as \int |\psi(\mathbf{r})|^2 d^3\mathbf{r} = 1, which conserves probability and distinguishes bound states from scattering states, whose wave functions are not normalizable.^[5]^[4] A key feature of bound state wave functions is their asymptotic behavior at large distances from the confining potential, where the potential V(\mathbf{r}) approaches zero faster than $1/r (short-range case). In this regime, for s-states (l=0), the wave function decays exponentially as \psi(\mathbf{r}) \sim \frac{e^{-\kappa r}}{r}, with \kappa = \sqrt{2m|E|}/\hbar, where E < 0 is the binding energy, m is the particle mass, and \hbar is the reduced Planck's constant. This decay ensures the normalizability and reflects the evanescent nature of the solution to the time-independent Schrödinger equation in regions where the kinetic energy would be imaginary. For long-range potentials like Coulomb, the form differs, lacking the 1/r prefactor in the leading term.^[21]^[22]^[23] In the semiclassical WKB approximation (illustrated here in one dimension for clarity), this exponential tail extends into classically forbidden regions beyond the turning points where E = V(\mathbf{r}), providing a more detailed description of the decay. There, the wave function takes the form \psi(x) \approx \frac{C}{\sqrt{\kappa(x)}} \exp\left( -\int^x \kappa(x') dx' \right), with \kappa(x) = \sqrt{2m(V(x) - E)}/\hbar > 0, ensuring the growing exponential solution is discarded to maintain normalizability. This tail arises from quantum tunneling but diminishes rapidly, confining the particle overall.^[22]^[23] In three dimensions, the asymptotic also depends on the orbital angular momentum l, generally involving a prefactor like r^l. Bound state wave functions are spatially localized, meaning the expectation value of position \langle \mathbf{r} \rangle is finite, and the probability density |\psi|^2 is concentrated in a finite region around the potential minimum. This localization contrasts with extended states in periodic potentials, such as Bloch waves, which spread indefinitely across the lattice without exponential decay, leading to delocalized probability. The penetration into forbidden regions via the exponential tail contributes to this confinement, with the probability density peaking in the classically allowed region where E > V(\mathbf{r}).^[4]^[5]

Key Properties

Discrete Energy Spectrum

In quantum mechanics, bound states exhibit a discrete energy spectrum, consisting of isolated eigenvalues lying below the continuous spectrum associated with scattering states. This discreteness arises from the requirement that the wave function must satisfy boundary conditions of vanishing at spatial infinity (or decay exponentially there) to ensure square-integrability and normalizability, which restricts the possible energy values to a countable set.^[24]^[25] Wave function normalization further enforces this discreteness by excluding continuum solutions that would otherwise be permissible for unbound states.^[26] The spacing between these discrete energy levels depends on the form of the confining potential. For the quantum harmonic oscillator, where V(x) = \frac{1}{2} m \omega^2 x^2, the levels are equally spaced with energies given by

E_n = \hbar \omega \left( n + \frac{1}{2} \right), \quad n = 0, 1, 2, \dots

yielding a constant separation of \hbar \omega.^[26] In contrast, for the Coulomb potential of the hydrogen atom, V(r) = -\frac{e^2}{4\pi \epsilon_0 r}, the bound-state energies follow a Rydberg-like formula

E_n = -\frac{13.6 \, \mathrm{eV}}{n^2}, \quad n = 1, 2, 3, \dots

where the spacing decreases as n increases, becoming denser near the ionization threshold.^[27] As a toy model illustrating this general discreteness, consider the infinite square well potential of width L, where V(x) = 0 for $0 < x < L and infinite elsewhere. The energy eigenvalues are

E_n = \frac{n^2 \pi^2 \hbar^2}{2 m L^2}, \quad n = 1, 2, 3, \dots

demonstrating quadratic dependence on the quantum number n; even for finite potentials, similar isolated levels emerge below the continuum onset due to the same asymptotic boundary conditions.^[28] For smoother potentials, the semiclassical Bohr-Sommerfeld quantization rule provides an approximate method to predict these discrete levels without solving the full Schrödinger equation, stating that the action integral over the classical turning points equals (n + \frac{1}{2}) h:

\oint p \, dx = (n + \frac{1}{2}) h, \quad n = 0, 1, 2, \dots

where p = \sqrt{2m(E - V(x))}. This approach yields accurate estimates for high-lying states in potentials like the harmonic oscillator or Coulomb, bridging classical and quantum descriptions.^[29]^[30]

Spatial Localization

In quantum mechanics, the spatial localization of bound states refers to the confinement of the particle's probability density to a finite region in position space, distinguishing them from scattering or free states. This localization is a direct consequence of the square-integrable nature of the bound state wave functions, which decay exponentially at large distances from the potential well. A key measure of this localization is the finiteness of the expectation value ⟨r²⟩, defined as ∫ r² |ψ(r)|² d³r, which quantifies the second moment of the position distribution and remains finite for all bound states due to the rapid decay of |ψ(r)|².^[31] The position variance, Δr = √(⟨r²⟩ - ⟨r⟩²), provides a standard deviation of this spread, with the ground state typically exhibiting the smallest Δr, rendering it the most localized among the energy eigenstates; for example, in the hydrogen atom, the 1s ground state has ⟨r²⟩ = 3 a₀², where a₀ is the Bohr radius, while higher-n states show larger values scaling as n⁴.^[32] This confinement has significant implications for physical observables. In true bound states, the exponential decay ensures stability with infinite lifetimes, as the particle has negligible probability of escaping to infinity. However, in unstable or quasi-bound systems—such as narrow resonances above the potential threshold—spatial localization leads to finite lifetimes via quantum tunneling, where the wave function penetrates the barrier, allowing decay with a rate proportional to the imaginary part of the complex energy.^[33] In stark contrast to free particles, whose delocalized plane-wave descriptions yield uniform probability densities over infinite space and divergent ⟨r²⟩ (due to non-normalizability), bound orbitals are compact and peaked within the attractive potential, enabling phenomena like atomic shells in multi-electron systems.^[31] A fundamental aspect of this localization is that bound state wave functions in position space are not eigenstates of the momentum operator, implying inherent spreads in both position (Δx bounded and finite) and momentum (Δp finite but non-zero). The Heisenberg uncertainty principle then requires Δx Δp ≥ ħ/2, with the bounded Δx enforcing a minimal Δp that reflects the superposition of momentum components needed to maintain confinement./University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/07%3A_Quantum_Mechanics/7.03%3A_The_Heisenberg_Uncertainty_Principle) This uncertainty product often achieves near-minimal values in ground states, correlating with their tightest spatial localization and discrete energy spacing.

Theorems and Constraints

Non-Degeneracy in One Dimension

In one-dimensional quantum mechanics, for a time-independent potential V(x) that is real-valued and supports bound states, each bound energy eigenvalue E is non-degenerate, meaning there is exactly one eigenfunction \psi(x) (up to an overall phase factor) corresponding to that energy.^[34]^[17] This non-degeneracy follows from the structure of the one-dimensional time-independent Schrödinger equation:

-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + V(x) \psi = E \psi,

where the eigenfunctions \psi(x) can be chosen real, ensuring they are uniquely determined for each bound state energy.^[34]^[17] The proof relies on the Sturm-Liouville formulation of the Schrödinger equation, which guarantees orthogonal eigenfunctions for distinct eigenvalues; for the same eigenvalue, assume two independent real solutions \psi_1 and \psi_2. Their Wronskian W = \psi_1 \frac{d\psi_2}{dx} - \psi_2 \frac{d\psi_1}{dx} satisfies W' = 0, implying W is constant. For bound states vanishing at \pm \infty, W = 0 everywhere, so \psi_1 and \psi_2 are linearly dependent, contradicting independence and proving non-degeneracy.^[34]^[17] This property simplifies the solution of one-dimensional quantum problems by eliminating the need to consider multiple states per energy level, in contrast to higher-dimensional systems like the three-dimensional hydrogen atom where accidental degeneracies arise due to symmetry.^[34]^[17]

Node Theorem

The node theorem, also known as the oscillation theorem in the context of Sturm-Liouville problems, states that in one-dimensional quantum mechanics, the wave function \psi_n(x) of the nth bound state—ordered by increasing energy E_n—possesses exactly n-1 nodes (points where \psi_n(x) = 0) within the classically allowed region between the turning points, excluding the boundaries at infinity where the wave function decays to zero.^[35] This holds for real-valued, normalizable eigenfunctions of the time-independent Schrödinger equation -\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi, assuming a potential V(x) that supports bound states and tends to infinity or a positive constant at |x| \to \infty.^[36] A sketch of the proof relies on the separation and comparison theorems from Sturm-Liouville theory. The separation theorem implies that zeros of consecutive eigenfunctions interlace, with no two eigenfunctions sharing a common zero without coinciding entirely. The comparison theorem, applied to potentials with infinite walls at \pm a (approximating the bound state problem as a \to \infty), shows that higher-energy states oscillate more rapidly, introducing an additional node between those of lower states; nodes persist under adiabatic expansion of the well, as adding or removing a node would require \psi = \psi' = 0 at some interior point, leading to the trivial solution \psi \equiv 0, which contradicts normalization.^[35] Physically, higher energies allow the wave function to "wiggle" more times in the allowed region before exponential decay in the forbidden regions dominates.^[37] This theorem facilitates labeling bound states in solvable potentials, such as the finite square well or harmonic oscillator, where the ground state (n=1) has no nodes and is nodeless (always positive or negative), while excited states gain nodes sequentially.^[36] Nodes signify phase changes in the wave function, corresponding to regions of constructive and destructive interference in the probability density. In one dimension, non-degeneracy ensures a unique state per nodal count.^[36] The node theorem extends to three-dimensional central potentials by reducing the radial Schrödinger equation to an effective one-dimensional problem for u(r) = r R(r), where R(r) is the radial wave function; the effective potential includes the centrifugal term \frac{\hbar^2 l(l+1)}{2m r^2}, and u(r) satisfies u(0) = 0 with exactly n-1 nodes in (0, \infty) for the nth bound state, yielding n_r = n - l - 1 radial nodes in R(r).

Existence Conditions

Mathematical Requirements

In quantum mechanics, bound states are elements of the Hilbert space \mathcal{H} = L^2(\mathbb{R}^d), the space of square-integrable functions where the wave function \psi satisfies \int |\psi|^2 d\mathbf{r} < \infty, ensuring normalizability and physical interpretability as probability densities.^[38] This L^2 condition implies that \psi \to 0 as |\mathbf{r}| \to \infty, distinguishing bound states from scattering states that extend to infinity.^[33] The Hamiltonian operator H = -\frac{\hbar^2}{2m} \nabla^2 + V(\mathbf{r}) must be self-adjoint on a suitable dense domain in \mathcal{H} to guarantee real eigenvalues and unitary time evolution, as required for observables in quantum mechanics. Self-adjointness ensures the spectrum is real and that bound state energies correspond to isolated eigenvalues.^[39] For the time-independent Schrödinger equation H\psi = E\psi, the potential V(\mathbf{r}) is typically assumed to approach 0 as |\mathbf{r}| \to \infty, with V bounded below and locally integrable, allowing solutions with E < 0 that are confined by the attractive regions of V.^[33] Finite or infinite potential wells satisfy these assumptions, provided there exists at least one E < 0 eigenvalue solution.^[40] Boundary conditions require \psi and its first derivative \psi' (or the radial derivative in 3D) to be continuous across finite potential discontinuities to ensure the probability current is well-defined and the wave function remains single-valued.^[41] In one dimension, \psi(x) \to 0 and \psi'(x) remains finite as x \to \pm \infty; in three dimensions, the radial wave function u(r) = r \psi(r) satisfies u(0) = 0 and u(r) \to 0 as r \to \infty.^[42] Bound states correspond to the point spectrum of H lying below the essential spectrum, which begins at 0 for potentials vanishing at infinity, ensuring discrete, negative energies separated from the continuous spectrum of free-particle states.^[43] This spectral condition, rooted in the decomposition of the spectrum into point and essential parts, confirms the existence of localized eigenstates.^[44]

Stability Considerations

The stability of bound states under perturbations is a central concern in quantum mechanics, as small changes to the Hamiltonian can shift energies and potentially disrupt localization. In time-independent perturbation theory, the first-order correction to the energy of a bound state is given by \delta E = \langle \psi | \delta V | \psi \rangle, where \psi is the unperturbed wave function and \delta V is the perturbing potential.^[45] This shift can unbind the state if it pushes the energy above the continuum threshold, transforming it from a stable bound state to a resonance embedded in the scattering continuum.^[46] For instance, in the Stark effect, an external electric field applied to atomic bound states induces such shifts, leading to ionization if the field strength exceeds a critical value.^[46] Quasi-bound states, often arising from these perturbations, exhibit temporary spatial localization but possess finite lifetimes due to tunneling or decay into the continuum. These states are characterized by complex energies of the form E = E_r - i \Gamma / 2, where E_r is the real part corresponding to the resonance position and \Gamma is the decay width inversely proportional to the lifetime \tau = \hbar / \Gamma.^[47] Unlike true bound states with real, negative energies below the continuum, quasi-bound states represent metastable configurations that broaden the otherwise discrete spectrum.^[48] In time-dependent scenarios, dynamical stability depends on the rate of potential variation; rapid changes can excite bound states to the continuum, while slow evolutions preserve them via the adiabatic theorem. This theorem states that if the Hamiltonian varies sufficiently slowly compared to the inverse energy gaps, the system remains in the instantaneous eigenstate, maintaining bound character. For example, in the adiabatic application of an electric field inducing the Stark effect, atomic bound states follow the evolving potential without dissociation.^[46] The discrete spectrum of bound states renders them particularly susceptible to broadening under such perturbations.^[45] In open quantum systems, environmental interactions introduce decoherence, which erodes quantum coherence and can destabilize bound states by facilitating energy dissipation or entanglement with the bath.^[49] However, certain dissipationless localized bound states demonstrate robustness, with decoherence rates minimized when the state aligns with dark subspaces decoupled from the environment.^[49] Studies of time-dependent potentials further reveal that bound state formation and persistence follow the pulse profile, with stability enhanced for gradual modulations that avoid non-adiabatic transitions.^[50]

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Nov 20, 2007 · We report the prediction of quasibound states (resonant states with very long lifetimes) that occur in the eigenvalue continuum of ...
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[2410.20840] Two-mode Open Quantum Systems: Decoherence ...
Oct 28, 2024 · In this work, the decoherence dynamics and dissipationless localized bound states of a two-mode open quantum system are investigated.
[47]
Bound-state formation in time-dependent potentials | Phys. Rev. C
This study examines bound state formation in a one-dimensional potential using time-dependent pulses, showing that formation follows the pulse shape, not the ...