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Virtual state

In , a virtual state refers to an intermediate, non-stationary configuration of a quantum system that is not an eigenstate of the , characterized by a vanishingly small probability of direct observation and serving as a theoretical mediator in interactions such as and transitions. Unlike bound states, which are stable and correspond to poles on the physical sheet of the energy plane, virtual states are located on the unphysical Riemann sheet, rendering them non-normalizable and inaccessible as asymptotic states. This concept arises in , where virtual states facilitate the description of off-resonant couplings between real states, often eliminated adiabatically to derive effective s. The notion of virtual states emerged in the late 1920s amid early developments in quantum mechanics, with Friedrich Hund introducing related ideas in molecular spectroscopy and quantum tunneling in 1927. Guido Beck formalized the term in 1930, associating it with resonances in particle scattering, such as alpha-particle interactions, where it described quasi-discrete levels blending into the continuous spectrum. By the mid-1930s, following the discovery of the neutron, the concept gained prominence in nuclear physics; Enrico Fermi applied it to the deuteron's singlet state in 1936, distinguishing virtual states (with negative binding energy) from real bound states (positive binding energy). Gregory Breit and Eugene Wigner further refined its use in analyzing neutron capture cross-sections, establishing virtual states as essential for interpreting low-energy scattering phenomena. Virtual states remain fundamental in contemporary , particularly in —where the related concept of particles underpins perturbative expansions—and in open quantum systems. In theory, they manifest as poles influencing observables such as lengths, as seen in the shallow state in the ^1S_0 channel of nucleon-nucleon interactions, with a length of approximately 23.7 fm. Recent studies explore unconventional mechanisms for populating virtual states in dissipative environments, revealing non-zero occupation probabilities (e.g., up to 1/3) over extended timescales, challenging traditional views of their unobservability. These states also appear in spectroscopic processes like , where transitions enable energy shifts without populating intermediate levels directly.

Fundamental Concepts

Definition

In quantum mechanics, a virtual state refers to a transient quantum configuration that does not correspond to an eigenstate of the and thus fails to satisfy the time-independent as a stationary solution. These states arise as intermediate configurations during quantum processes, where the system's energy deviates from the eigenvalues of the , placing them "off-shell" in the formalism of and scattering. Virtual states can exist only briefly, temporarily violating as permitted by the , \Delta E \Delta t \geq \hbar/2, which allows energy fluctuations \Delta E over short timescales \Delta t. This short-lived nature stems from the rapid decay of such configurations back to real states, as their off-resonant character prevents sustained occupation. In contrast to real states, which are , eigenstates with well-defined energies that can be measured directly, virtual states are not directly and possess vanishingly small probabilities of detection due to their non- and intermediary role.

Mathematical Description

In , virtual states are mathematically described within the framework of the or resolvent operator, which encapsulates the propagation of off the energy shell. The free for a non-relativistic particle is given by G_0(E) = \frac{1}{E - H_0 + i\epsilon}, where H_0 = \mathbf{p}^2 / (2m) is the , E is the complex energy parameter, and \epsilon \to 0^+ prescribes the boundary conditions for outgoing waves. States are off-shell when E does not coincide with an eigenvalue of the full H = H_0 + V, allowing for intermediate propagations where energy is not conserved locally. This off-shell nature is fundamental to virtual states, which do not correspond to stationary states but influence and amplitudes through their role in the full . The full interacting G(E) = [E - H + i\epsilon]^{-1} exhibits a rich analytic structure in the complex plane, with a branch cut along the positive real axis representing the . Bound states appear as poles on the physical Riemann sheet at negative real energies E < 0. Virtual states, in contrast, correspond to poles located on the unphysical (second) Riemann sheet, accessed by analytic continuation around the branch cut. Specifically, these poles lie on the negative real energy axis (\operatorname{Re} E < 0, \operatorname{Im} E = 0), close to the E = 0, distinguishing them from resonances, whose poles on the unphysical sheet have \operatorname{Re} E > 0 and substantial negative imaginary part (|\operatorname{Im} E| \gg 0). The proximity to for virtual states leads to large lengths when \operatorname{Re} E \approx 0^-. In the second quantization formalism, suitable for many-body systems or field-theoretic treatments of few-body scattering, the single-particle propagator is the vacuum expectation value of the time-ordered product of field operators: i G(\mathbf{x}, t; \mathbf{x}', t') = \langle 0 | T \psi(\mathbf{x}, t) \psi^\dagger(\mathbf{x}', t') | 0 \rangle, where \psi annihilates a particle. In momentum space, the free propagator is G_0(p, E) = 1 / (E - p^2/(2m) + i\epsilon). The interacting propagator satisfies the Dyson equation G = G_0 + G_0 \Sigma G, where \Sigma is the self-energy operator encoding interactions. Iterating this yields the Dyson series expansion: G = G_0 + G_0 \Sigma G_0 + G_0 \Sigma G_0 \Sigma G_0 + \cdots, a perturbative resummation of irreducible diagrams. Virtual states emerge in this expansion as off-shell intermediate propagators in the terms G_0 \Sigma G_0 \cdots G_0, where the total energy E deviates from the on-shell condition, contributing to the analytic continuation of G(E) and the placement of poles on unphysical sheets. Solving the Dyson equation non-perturbatively shifts the free poles according to the interaction, with virtual state poles arising when the denominator E - H_0 - \Sigma(E) = 0 admits solutions on the unphysical sheet near threshold. A concrete example illustrating the emergence of a virtual state pole occurs in a simple finite square-well potential V(r) = -V_0 for r < a and $0 otherwise, with small depth V_0 such that no bound state exists. For s-wave (l=0) scattering, the radial wave function inside the well is u(r) \propto \sin(qr)/q, where q = \sqrt{2m(V_0 + E)}/\hbar, and outside u(r) \propto e^{i k r} - S(k) e^{-i k r}, with k = \sqrt{2m E}/\hbar. The pole condition is determined by the zeros of the Jost function f(k), derived from matching logarithmic derivatives at r = a. For shallow V_0, the would-be bound state pole (at imaginary k = i\kappa, \kappa > 0, E = -\hbar^2 \kappa^2 / (2m) < 0) on the physical sheet migrates to the unphysical sheet at k = i \kappa_0 (\kappa_0 > 0), corresponding to a virtual state near E \approx 0^-. The wave function norm for this pole is positive real, \int_0^\infty |\phi_0(k_0, r)|^2 dr = (\kappa_0 a - 1) / [2\kappa_0 (q_0^2 + \kappa_0^2) q_0^4 \sin^2(q_0 a)], confirming its unphysical character. This configuration yields a large negative scattering length a \approx -1/\kappa_0, linking the virtual state to low-energy scattering observables.

Role in Quantum Processes

Perturbation Theory and Intermediate States

In time-dependent , virtual states serve as non-physical intermediate states in the expansion of the operator, facilitating transitions between real eigenstates of the unperturbed . The second-order contribution to the transition amplitude from an initial |i\rangle to a final |f\rangle is expressed as a over these virtual intermediate states |m\rangle, proportional to \sum_m \frac{\langle f | V | m \rangle \langle m | V | i \rangle}{E_i - E_m + i [\epsilon](/page/Epsilon)}, where V is the , E_i and E_m are the unperturbed energies, and the infinitesimal i \epsilon ensures proper around the . This sum-over-states formulation arises from inserting a complete set of unperturbed states between successive perturbation interactions in the . Virtual states inherently involve an apparent violation of energy conservation, as the energy E_m of the intermediate state need not equal E_i + \hbar \omega for a perturbation of frequency \omega, differing from real transitions. This off-energy-shell character is permitted by the time-energy uncertainty principle, \Delta E \Delta t \gtrsim \hbar/2, where \Delta t \approx \tau is the brief duration of the virtual process, often comparable to the period of the perturbation or shorter. This allows significant energy mismatches without contradicting the exact conservation in the full time evolution. A representative example occurs in atomic multi-photon processes, such as —predicted theoretically in 1931 by Maria Göppert-Mayer— where the first excites the atom to a virtual intermediate state without real population, followed by the second driving the transition to the final bound or continuum state; this process enables selection-rule-forbidden transitions otherwise inaccessible in single- absorption. The adaptation of to include virtual contributions appears in higher-order transition s, where the effective matrix element incorporates the second-order sum over virtual states, yielding a w_{i \to f} = \frac{2\pi}{\hbar} |\sum_m \frac{\langle f | V | m \rangle \langle m | V | i \rangle}{E_i - E_m + i \epsilon}|^2 \rho(E_f) for a continuum of final states with \rho(E_f), extending the rule to processes mediated by off-resonant intermediates.

Scattering Theory

In quantum scattering theory, virtual states manifest prominently in s-wave interactions at low energies, where they correspond to poles in the on the unphysical Riemann sheet close to the . These near-threshold poles result in a large negative scattering length a_s < 0, distinguishing virtual states from bound states, which have positive scattering lengths. The effective range expansion captures this behavior through the relation k \cot \delta = -\frac{1}{a_s} + \frac{1}{2} r_0 k^2 + \cdots, where \delta is the s-wave phase shift, k is the wave number, and r_0 is the effective range; for virtual states, the dominant term -\frac{1}{a_s} is large and positive due to the negative a_s. Unlike bound states, which lie below threshold with negative energy E < 0, virtual states occur at positive energies E > 0 but on the second sheet, behaving as if weakly unbound and leading to enhanced low-energy amplitudes. This results in anomalously large low-energy cross-sections, \sigma \approx 4\pi a_s^2, where the magnitude |a_s| is much larger than typical interaction ranges, contrasting with the diminishing cross-sections expected for short-range potentials without such poles. The nature implies no real bound system forms, yet the mimics a near-miss , amplifying interactions at energies far below the . A classic example is the neutron-proton interaction in the ^1S_0 , where a virtual state exists at an energy of approximately 0.077 MeV, leading to a large negative a_s \approx -23.7 fm and correspondingly large low-energy cross-sections. Another instance occurs in formation during positron-atom , where virtual states contribute significantly to the amplitude at low energies, enhancing the total cross-section by up to 30% in cases like positron-hydrogen interactions. These examples illustrate how virtual states drive observable enhancements in processes without forming stable composites. The phase shift for virtual states exhibits a distinctive linear behavior at low energies, \delta \approx -k |a_s|, remaining negative and monotonically decreasing without crossing \pi/2, in contrast to resonances where the phase shift rises rapidly through \pi/2 near the resonance energy, producing a peak in the cross-section. This smooth, negative progression underscores the repulsive-like low-energy despite the underlying , as the virtual avoids the physical sheet.

Spectroscopic Applications

Raman Spectroscopy

In , the Raman effect arises from of light by s, where an incident excites the molecule to a virtual intermediate state before emission of a scattered with shifted , corresponding to Stokes ( ) or anti-Stokes ( ) processes. This virtual state is non-resonant and does not correspond to any real excited electronic state of the , serving as a transient superposition of multiple excited states during the two-photon process. The energy of the virtual state is determined by the incident photon's energy added to the ground state energy, expressed as E_v = E_{\text{ground}} + \hbar \omega_{\text{incident}}, where \hbar is the reduced Planck's constant and \omega_{\text{incident}} is the angular frequency of the incident light; this energy typically does not match the energies of real vibrational or electronic levels, ensuring the process remains off-resonant. In the energy diagram of Raman scattering, the virtual state lies above the ground vibrational level but below or offset from real excited electronic states, facilitating transitions between vibrational levels in the ground electronic state without direct electronic excitation. The involvement of virtual states relaxes certain selection rules compared to direct absorption-emission processes, enabling fundamental vibrational transitions (\Delta v = \pm 1) that may be dipole-forbidden in but are allowed in Raman due to changes in molecular rather than . This is captured in the Kramers-Heisenberg-Dirac dispersion formula, developed in 1925 by Kramers and Heisenberg (with Dirac's quantum mechanical refinement in 1927), which describes the scattering cross-section through summation over states and emphasizes the non-resonant nature of the process. Raman spectroscopy, leveraging these virtual state-mediated transitions, is widely applied in chemistry for vibrational analysis, such as identifying molecular bond vibrations in solid materials like polymers or crystals, where it provides complementary information to techniques for symmetric modes.

Other Techniques

In two-photon absorption (TPA), a virtual state serves as an intermediate step in the sequential absorption of two photons, enabling the transition from an initial ground state |i⟩ to a final excited state |f⟩ without populating a real intermediate energy level. This nonlinear process, predicted theoretically by Maria Göppert-Mayer in 1931, relies on the off-resonant coupling through the virtual state, where the absorption rate is proportional to the square of the transition amplitude given by second-order perturbation theory: \Gamma \propto \left| \sum_m \frac{\langle f | \hat{E} | m \rangle \langle m | \hat{E} | i \rangle}{\omega - \omega_m} \right|^2, with the sum over intermediate states |m⟩, electric field operator \hat{E}, photon frequency \omega, and intermediate state frequency \omega_m. TPA is widely applied in microscopy and photochemistry due to its ability to excite molecules at longer wavelengths, reducing photodamage compared to single-photon processes. Coherent anti-Stokes () utilizes states to generate a coherent, phase-matched anti-Stokes signal enhanced by the interaction of and Stokes fields. In this process, the excites the to a state, followed by stimulated to a real vibrational state, and subsequent of the anti-Stokes via another intermediate. The states ensure the process is non-resonant with electronic transitions, allowing selective probing of vibrational modes with high in applications. signals are particularly strong in condensed phases, enabling label-free imaging of biological samples. In ultrafast , virtual states manifest as transient populations in pump-probe experiments, where short pulses create coherent superpositions without real excitation. For instance, in coherent studies of materials like , the pump pulse drives electrons to a virtual state, and the probe monitors the ensuing vibrational dynamics, revealing lattice responses on timescales. These experiments highlight the role of virtual states in mediating non-equilibrium processes, such as generation, without violating over the ultrashort pulse duration. Hyper-Raman scattering involves higher-order nonlinear processes where virtual states facilitate multiple photon interactions, typically requiring two incident photons to excite a vibrational mode inaccessible to standard Raman selection rules. In this second-order Raman effect, the molecule absorbs two photons to reach a virtual electronic state, followed by emission of a shifted photon, allowing detection of infrared-active modes through the intermediate virtual steps. Theoretical models emphasize the contribution of excitonic virtual states in semiconductors, enhancing scattering efficiency for phonon studies. This technique is valuable for probing symmetry-forbidden transitions in complex materials.

Theoretical and Experimental Aspects

Relation to Resonances and Bound States

In quantum scattering theory, bound states correspond to poles of the on the physical Riemann sheet with real energies E < 0 below the threshold, representing stable configurations where the wave function is normalizable. Virtual states, in contrast, are characterized by poles on the real positive energy axis (E > 0) but located on the unphysical Riemann sheet, accessed by across the branch cut; these indicate unstable, non-normalizable states that influence low-energy scattering without forming true bound systems. Resonances differ by having complex energies with significant negative imaginary parts (\operatorname{Im}(E) < 0) on the unphysical sheet, leading to short-lived quasi-bound states that manifest as Breit-Wigner peaks in scattering cross-sections. The distinction in pole locations gives rise to differing scattering behaviors: resonances produce pronounced, Lorentzian-shaped enhancements due to their imaginary energy component, whereas virtual states, with poles on the real axis of the unphysical sheet, result in monotonic phase shifts without oscillatory peaks, often yielding large negative scattering lengths. Levinson's theorem, which relates the zero-energy phase shift \delta(0) to the number of bound states as \delta(0) = n\pi (where n is the number of bound states), is modified in the presence of virtual states; these contribute an additional \pi/2 phase shift, effectively counting as a "half-bound" state in the topological index of the matrix. A classic example is the nucleon-nucleon interaction: the deuteron in the ^3S_1 forms a shallow with 2.224 MeV, corresponding to a on the physical sheet, while the ^1S_0 np exhibits a virtual state with a large negative scattering length a \approx -23.7 fm, indicative of a near-threshold on the unphysical sheet. Virtual states often signal loosely bound or molecular-like structures, as quantified by the compactness parameter Z, which approaches 0 for virtual states (high compositeness X \approx 1) compared to Z \approx 1 for compact resonances or s, reflecting extended spatial distributions.

Observability and Recent Developments

Virtual states in cannot be directly observed because they are non-, off-shell configurations with zero probability density, defying conventional position or measurements. Instead, their presence is inferred indirectly through effects on observable quantities in experiments, such as modifications to the at low , where a virtual state on the negative real axis leads to characteristic enhancements or anomalies in cross-sections. Similarly, virtual states influence linewidths in resonant processes by contributing to the imaginary part of the denominator, and they manifest in time delays, quantified via the Wigner-Smith , which captures the duration of interaction influenced by nearby virtual . Quantum measurement effects offer pathways to transiently "localize" virtual occupations, bypassing their inherent ephemerality. The , arising from frequent or continuous projective measurements, can suppress evolution away from a configuration, effectively trapping the system in it for measurable durations despite its forbidden energy. Continuous monitoring, modeled as stochastic unraveling of the , further enables such localization by conditioning trajectories on measurement outcomes, increasing average occupation probabilities in virtual states without altering the underlying . Recent theoretical advances have demonstrated unconventional mechanisms for populating virtual states in open quantum systems. A 2022 study revealed that can lead to a finite long-time in virtual states, counterintuitively stabilizing them through environmental coupling rather than isolation. Building on this, a 2025 of measurements in a triple showed that repeated charge detections can capture an in a virtual state at energies below the conduction band, yielding average occupations exceeding 50% over classical expectations, even as individual trajectories fluctuate. Looking ahead, ultrafast spectroscopy holds promise for probing virtual state lifetimes directly by resolving sub-cycle electron dynamics. Experiments with extreme ultraviolet pulses have already revealed oscillations in virtual states of , hinting at femtosecond-scale interventions to map their transient contributions to real-time processes like .