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Ground state

In quantum mechanics, the ground state of a physical system is defined as the stationary state with the lowest possible energy, corresponding to the lowest eigenvalue of the system's Hamiltonian operator.^[1] This state represents the most stable configuration for the system, persisting indefinitely without external perturbations due to its time-independent nature.^[1] Unlike classical systems, quantum ground states exhibit zero-point energy, a non-zero minimum energy arising from the Heisenberg uncertainty principle, which prevents particles from being completely at rest.^[2] For atomic and molecular systems, the ground state is particularly significant as it dictates the baseline electronic configuration and stability.^[3] In atoms, electrons fill the lowest energy orbitals (starting with principal quantum number n = 1), forming the most energetically favorable arrangement that governs chemical reactivity and bonding.^[3] This configuration is the default state under normal conditions, with excitations to higher energy levels requiring energy input, such as from photons, after which systems typically relax back to the ground state by emitting radiation.^[4] In broader contexts like quantum chemistry and many-body physics, determining the ground state energy is a foundational task for predicting molecular properties, reaction pathways, and material behaviors.^[5] For instance, it is the most common computational challenge in quantum chemistry, enabling simulations of complex systems from simple diatomic molecules to biomolecules.^[5] Additionally, in interacting systems, the ground state may exhibit spontaneous symmetry breaking, leading to emergent phenomena such as ferromagnetism or superfluidity^[6], where the lowest-energy configuration lacks the symmetry of the underlying Hamiltonian.^[1]

Definition and Fundamentals

Definition in Quantum Mechanics

In quantum mechanics, the ground state of a system is defined as the stationary state corresponding to the lowest possible energy eigenvalue of the Hamiltonian operator, representing the most stable configuration under the system's dynamics.^[7] This state is typically denoted as |\psi_0\rangle, with its associated energy E_0, satisfying the eigenvalue equation \hat{H} |\psi_0\rangle = E_0 |\psi_0\rangle, where \hat{H} is the Hamiltonian and E_0 is the minimum eigenvalue among all possible energy levels.^[8] Unlike excited states, which are higher-energy eigenstates with energies E_n > E_0 for n \geq 1, the ground state has no lower-energy alternatives and thus serves as the baseline for all quantum transitions and stability analyses in the system.^[9] This distinction arises directly from the time-independent Schrödinger equation, which yields the eigenstates and eigenvalues describing stationary states.^[10] Quantum systems naturally evolve toward the ground state as temperature approaches absolute zero, in accordance with the third law of thermodynamics, which implies that the system's entropy approaches zero and the system occupies its ground state configuration in thermal equilibrium, with particles filling the lowest available energy levels according to their quantum statistics (e.g., all in the lowest state for non-interacting bosons, or up to the Fermi level for fermions).^[7] At T = [0](/page/0) K, the internal energy converges to E_0, ensuring the ground state dominates without thermal excitations.^[11]

Historical Context and Development

The concept of the ground state in quantum mechanics originated from efforts to address inconsistencies in classical physics regarding radiation and atomic stability. In December 1900, Max Planck introduced energy quantization to derive the blackbody radiation spectrum, positing that harmonic oscillators exchange energy in discrete multiples of h \nu, where h is a universal constant and \nu is frequency; this implicitly established discrete energy levels, with the lowest level representing the ground state of the oscillators. Building on this, Albert Einstein's 1905 explanation of the photoelectric effect applied quantization to electromagnetic radiation itself, proposing that light consists of discrete energy quanta (later called photons) with energy E = h \nu, thereby extending the idea of discrete states—including a minimal energy ground state—to light-matter interactions. A pivotal advancement occurred in 1913 with Niels Bohr's semiclassical model of the hydrogen atom, which explicitly defined the ground state as the lowest-energy (n=1) circular orbit where the electron remains stable without radiating energy, in accordance with angular momentum quantization L = n \hbar. This model resolved the classical instability of orbiting electrons and accounted for hydrogen's emission spectrum by transitions from excited states to this ground state, marking the first clear articulation of ground states in atomic physics. The formalization of ground states advanced significantly in 1925–1926 through the founding of modern quantum mechanics. Werner Heisenberg's matrix mechanics treated dynamical variables as non-commuting matrices, yielding discrete energy eigenvalues where the smallest eigenvalue corresponds to the ground state energy. Independently, Erwin Schrödinger's 1926 wave mechanics described quantum systems via a linear partial differential equation, identifying the ground state as the lowest-energy eigenfunction with no nodes in certain symmetric potentials, providing a continuous wave interpretation equivalent to Heisenberg's discrete formulation. The mathematical equivalence of these approaches was demonstrated shortly thereafter, unifying the field.^[12] Key extensions in the late 1920s and 1930s broadened the ground state to statistical and relativistic contexts. In 1926, Enrico Fermi developed a quantum statistical framework for identical particles, showing that for fermions at absolute zero temperature, the system occupies its ground state by filling the lowest momentum states up to the Fermi surface, introducing degeneracy and zero-point energy in many-particle ensembles. Paul Dirac's 1928 relativistic wave equation incorporated special relativity into quantum mechanics, redefining ground states for high-speed particles and predicting negative-energy solutions filled in the vacuum (Dirac sea) to ensure stability.^[13] By the 1930s, many-body theory emerged, with Dirac's antisymmetric wave functions for electrons enabling ground state calculations in interacting systems, paving the way for applications in quantum field theory where ground states represent the vacuum or lowest collective excitations.^[14]

Theoretical Foundations

Schrödinger Equation and Eigenstates

In quantum mechanics, the ground state of a system is fundamentally described through the solutions to the time-independent Schrödinger equation, which takes the form

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

where \hat{H} is the Hamiltonian operator representing the total energy of the system, \psi is the wave function, and E is the corresponding energy eigenvalue. This equation arises as the eigenvalue problem for stationary states, where the solutions \psi_n are the eigenfunctions associated with discrete energy eigenvalues E_n, ordered such that E_0 \leq E_1 \leq E_2 \leq \cdots. The ground state corresponds to the lowest-energy solution, denoted \psi_0 and E_0, which represents the most stable configuration of the quantum system under the given potential. For bound states—systems confined by a potential well—the wave functions must satisfy boundary conditions ensuring they are normalizable, meaning \int |\psi_n|^2 dV = 1 over all space, which requires the functions to decay sufficiently at infinity. These conditions impose square-integrability on the solutions, resulting in a discrete energy spectrum bounded from below, with the ground state energy E_0 forming the lowest point of this spectrum and prohibiting energies below it. This discreteness distinguishes bound states from scattering states, which have continuous spectra and non-normalizable wave functions.^[15] Since the Hamiltonian \hat{H} is a Hermitian operator, its eigenfunctions corresponding to distinct eigenvalues are orthogonal, satisfying \int \psi_0^* \psi_n \, [dV](/page/DV) = 0 for all n > 0. This orthogonality ensures that the ground state wave function \psi_0 is uniquely separated from excited states in the Hilbert space, facilitating the expansion of any wave function in terms of the complete set of eigenstates. In the presence of a small perturbation \delta \hat{H} to the Hamiltonian, the first-order correction to the ground state energy is given by the expectation value \delta E_0 = \langle \psi_0 | \delta \hat{H} | \psi_0 \rangle, providing a means to quantify how weak interactions shift the lowest energy level without altering the unperturbed eigenfunction to leading order. This Rayleigh-Schrödinger perturbation approach is foundational for analyzing real systems where exact solutions are intractable.^[16]^[17]

Variational Principle

The variational principle, also known as the Rayleigh-Ritz method in its foundational form, serves as a cornerstone for approximating the ground state energy and wave function in quantum mechanics. It asserts that for any trial wave function \phi, the Rayleigh quotient R(\phi) = \frac{\langle \phi | \hat{H} | \phi \rangle}{\langle \phi | \phi \rangle} provides an upper bound to the true ground state energy E_0, such that R(\phi) \geq E_0, with equality holding if and only if \phi is proportional to the exact ground state wave function \psi_0.^[18] This principle originates from classical variational methods for vibration problems, as introduced by Lord Rayleigh in his analysis of mechanical systems.^[19] The derivation follows from the Hermitian nature of the Hamiltonian operator \hat{H}, which ensures real eigenvalues and orthogonal eigenstates. Expanding the normalized trial function in the complete basis of exact eigenstates \{ \psi_n \} as \phi = \sum_n c_n \psi_n (with \sum_n |c_n|^2 = 1), the expectation value becomes \langle \phi | \hat{H} | \phi \rangle = \sum_n |c_n|^2 E_n. Since E_n \geq E_0 for all n and E_0 is the lowest eigenvalue, the weighted average satisfies \sum_n |c_n|^2 E_n \geq E_0 \sum_n |c_n|^2 = E_0.^[18] This inequality highlights the principle's reliance on the spectral properties of \hat{H}, providing a systematic way to bound the ground state without solving the full eigenvalue problem. Choosing an appropriate trial function is crucial for practical application, typically involving adjustable parameters that are varied to minimize the Rayleigh quotient. For the quantum harmonic oscillator, a Gaussian trial function \phi(x) = \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2 / 2} (with variational parameter \alpha > 0) yields the exact ground state energy upon minimization, demonstrating the method's potential for simple systems.^[18] In atomic systems, Slater-type orbitals of the form \chi_{nlm}(r, \theta, \phi) \propto r^{n-1} e^{-\zeta r} Y_{lm}(\theta, \phi) (with effective nuclear charge \zeta as the parameter) are employed to approximate hydrogen-like wave functions, facilitating calculations in multi-electron atoms via linear combinations. The accuracy of the approximation improves as the trial function more closely resembles \psi_0, yielding tighter upper bounds on E_0; for instance, expanding the trial space with more basis functions in the Rayleigh-Ritz procedure converges monotonically from above to the exact value.^[20] The Hellmann-Feynman theorem enhances optimization by relating derivatives of the energy with respect to parameters to expectation values: \frac{dE}{d\lambda} = \left\langle \psi \middle| \frac{\partial \hat{H}}{\partial \lambda} \middle| \psi \right\rangle, where \lambda is a variational parameter, allowing efficient parameter updates without recomputing the full wave function. This theorem, independently derived by Hellmann and Feynman, is particularly valuable in parameter-dependent Hamiltonians.^[21] A key limitation of the variational principle is its provision of an upper bound strictly for the ground state; approximations to excited states require additional constraints, such as orthogonality to lower states, and do not inherently guarantee upper bounds without modifications.^[20]

Key Properties

Node Structure in Wave Functions

In one-dimensional quantum mechanical systems governed by the time-independent Schrödinger equation, the node theorem, rooted in Sturm-Liouville theory, asserts that the ground state wave function \psi_0(x) possesses no zeros—or nodes—within the interior of the potential domain, in contrast to the nth excited state, which exhibits exactly n nodes.^[22] This result follows from the Sturm oscillation theorem, which characterizes the eigenfunctions of self-adjoint Sturm-Liouville operators by the number of sign changes in their oscillatory behavior.^[23] A straightforward derivation proceeds by contradiction: assume \psi_0(x) vanishes at some interior point x_0, dividing the domain into two subintervals where \psi_0 maintains constant sign. Integrating the Schrödinger equation over one subinterval and applying integration by parts yields \int \psi_0 (-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V) \psi_0 \, dx = E_0 \int \psi_0^2 \, dx, but the boundary terms at x_0 imply that a trial function constructed by taking the absolute value |\psi_0| (which lacks the node) would have lower kinetic energy while preserving potential energy, violating the variational principle and indicating \psi_0 cannot be the ground state.^[24] Thus, \psi_0(x) remains of definite sign throughout the domain, ensuring the probability density |\psi_0(x)|^2 > 0 everywhere, which reflects the most concentrated distribution compatible with the uncertainty principle.^[23] This theorem generalizes to higher dimensions for potentials separable into one-dimensional factors, where the ground state wave function, as a product of nodeless one-dimensional components, contains no nodal hypersurfaces.^[24] In non-separable potentials, the ground state typically lacks nodal hypersurfaces passing through the origin, as demonstrated by enclosing the system in an infinite well and applying the one-dimensional argument radially, though exact nodal structure becomes more complex due to coupling.^[24] Representative examples illustrate this in bound systems. For the one-dimensional infinite square well of width L, the ground state is \psi_0(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) for $0 < x < L, which maintains positive values without interior zeros.^[25] Similarly, in the symmetric finite square well, the even-parity ground state wave function involves a cosine-like form inside the well that does not vanish within the domain, decaying exponentially outside.^[26]

Zero-Point Energy

In quantum mechanics, the ground state energy E_0 of systems confined by potential minima is inherently positive, a direct consequence of the Heisenberg uncertainty principle, which requires a finite uncertainty in both position \Delta x and momentum \Delta p such that \Delta x \Delta p \geq \hbar/2. This principle implies that a particle cannot be precisely localized at the potential minimum without an accompanying spread in momentum, leading to a non-zero kinetic energy that offsets the potential energy and prevents the system from collapsing to zero energy. For bounded systems like atoms or molecules, this results in a lowest-energy configuration above the classical minimum, fundamentally distinguishing quantum from classical mechanics.^[27]^[28] A canonical illustration is the quantum harmonic oscillator, modeled by the time-independent Schrödinger equation:

-\frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + \frac{1}{2} m \omega^2 x^2 \psi = E \psi,

where m is the mass, \omega the angular frequency, and \psi(x) the wave function. Solving this yields quantized energy eigenvalues E_n = (n + 1/2) \hbar \omega for quantum number n = 0, 1, 2, \dots, so the ground state energy is E_0 = (1/2) \hbar \omega. Alternatively, using ladder operators a and a^\dagger that annihilate and create excitations, the Hamiltonian becomes H = \hbar \omega (a^\dagger a + 1/2), confirming the same zero-point contribution of \hbar \omega / 2 even in the absence of excitations (n=0). This derivation underscores how quantum delocalization enforces a minimal energy scale.^[29]^[30] The zero-point energy gives rise to zero-point motion, manifesting as residual oscillations that persist at absolute zero temperature and influence physical properties across scales. In molecular systems, it elongates bond lengths beyond classical predictions and contributes to vibrational zero-point energies, which must be accounted for in thermochemical calculations to match spectroscopic data. In condensed matter physics, these motions affect electron-lattice interactions, playing a role in superconductivity by altering phonon-mediated pairing energies, as seen in the Bardeen-Cooper-Schrieffer theory where shifts in lattice zero-point energies contribute to the condensation energy. The node-free structure of the ground state wave function facilitates this minimal-energy oscillation without additional nodes that would raise the energy further. Extending to quantum field theory, the vacuum state—the field's ground state—exhibits fluctuating modes with non-zero zero-point energies summing to the vacuum energy density, which is theoretically divergent but renormalized in practice; this vacuum energy is hypothesized to underpin the observed cosmological constant driving cosmic acceleration. Experimental confirmation of zero-point effects appears in low-temperature studies of crystals, where techniques like X-ray or neutron scattering reveal persistent atomic displacements and vibrations at T \to 0 K, violating classical equipartition by maintaining motion amplitudes up to 25% of interatomic spacings in quantum solids like helium-4. For instance, solid helium persists despite classical melting tendencies due to this dominant zero-point contribution.^[31]^[32]^[33]^[34]^[35]

Applications and Examples

Quantum Harmonic Oscillator

The quantum harmonic oscillator models a particle of mass m confined by a parabolic potential V(x) = \frac{1}{2} m \omega^2 x^2, where \omega denotes the classical angular frequency of oscillation. This exactly solvable system provides essential insights into quantum ground states, as the time-independent Schrödinger equation yields stationary states with discrete energies. The ground state wave function takes the form of a Gaussian:

\psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2 / 2},

where \alpha = m \omega / \hbar. This normalized function is symmetric about x = 0 and exhibits no nodes, reflecting the lowest-energy configuration.^[36] The ground state energy is E_0 = \frac{1}{2} \hbar \omega, representing the zero-point energy inherent to quantum systems. This value arises directly from solving the Schrödinger equation for the harmonic potential, establishing the evenly spaced energy ladder E_n = \hbar \omega (n + 1/2) for n = 0, 1, 2, \dots. The Gaussian profile of \psi_0(x) ensures maximum localization consistent with the Heisenberg uncertainty principle, with position and momentum uncertainties satisfying \Delta x \Delta p = \hbar / 2. Consistent with the nodal theorem, the absence of nodes in \psi_0(x) distinguishes it as the lowest eigenstate.^[36] An elegant algebraic formulation employs creation (a^\dagger) and annihilation (a) operators, defined as a = \sqrt{\frac{m \omega}{2 \hbar}} \left( x + \frac{i p}{m \omega} \right) and a^\dagger = \sqrt{\frac{m \omega}{2 \hbar}} \left( x - \frac{i p}{m \omega} \right), where p = -i \hbar d/dx. These satisfy the commutation relation [a, a^\dagger] = 1, and the Hamiltonian becomes H = \hbar \omega (a^\dagger a + 1/2). The ground state |0\rangle is the vacuum state annihilated by a |0\rangle = 0, from which excited states are generated by applying a^\dagger. This operator method simplifies computations of expectation values and transitions.^[37] Physically, the harmonic oscillator ground state captures the vibrational zero-point motion in diatomic molecules under the harmonic approximation, where bond stretching is modeled by the quadratic potential near equilibrium. In solids, it describes phonon modes as collective lattice vibrations, with the ground state embodying residual thermal fluctuations even at absolute zero. Unlike the classical oscillator, which achieves zero energy at the potential minimum with no motion, the quantum ground state enforces persistent oscillations due to E_0 > 0, manifesting as zero-point fluctuations that broaden spectral lines and influence material properties.^[38]^[39]

Atomic and Molecular Systems

In atomic systems, the ground state of the hydrogen atom is described by the 1s orbital, with the wave function given by

\psi_{100}(r) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0},

where a_0 is the Bohr radius, and the corresponding energy is E_0 = -13.6 eV.^[40] This solution arises from solving the Schrödinger equation for a single electron in a Coulomb potential, representing the lowest-energy configuration where the electron is most probable near the nucleus.^[40] For multi-electron atoms, the ground state electronic configuration follows the Aufbau principle, which dictates that electrons occupy the lowest available energy orbitals, respecting the Pauli exclusion principle and Hund's rule.^[41] For example, in helium, the ground state is 1s², with both electrons filling the 1s orbital.^[14] The Hartree-Fock approximation provides a mean-field treatment to estimate the ground state energy, solving self-consistent equations for single-particle orbitals while accounting for electron-electron repulsion in an average sense; for helium, this yields an energy of approximately -2.86 hartree, close to the exact value of -2.90 hartree. This method assumes a Slater determinant wave function, offering a practical way to approximate the many-body ground state beyond simple independent-particle models. In molecular systems, the Born-Oppenheimer approximation separates the much faster electronic motion from the slower nuclear motion, allowing the electronic ground state to be computed for fixed nuclear positions, which defines the potential energy surface for nuclear dynamics.^[42] For the simplest molecular ion, H₂⁺, the ground state corresponds to the lowest σ_g orbital formed by the linear combination of 1s atomic orbitals from each proton, resulting in a bonding configuration with a dissociation energy of about 2.65 eV and equilibrium bond length of 1.06 Å.^[43] This σ orbital concentrates electron density between the nuclei, stabilizing the ion.^[43] To improve accuracy beyond mean-field approximations like Hartree-Fock, configuration interaction (CI) methods incorporate electron correlation by expanding the wave function as a linear combination of multiple Slater determinants, capturing instantaneous electron-electron interactions.^[44] For atomic and molecular ground states, CI lowers the energy estimate toward the exact value within a basis set; for example, in helium, full CI recovers nearly 100% of the correlation energy missing from Hartree-Fock.^[44] Seminal applications, such as Hylleraas' variational CI for two-electron atoms, demonstrate how this approach refines ground state properties like ionization potentials. The ground state electronic configuration fundamentally governs atomic and molecular stability, influencing chemical reactivity and equilibrium bond lengths through the minimization of total energy.^[45] Closed-shell ground states, such as the 1s² of helium or the σ_g² of H₂, exhibit high stability and low reactivity due to filled orbitals and a large HOMO-LUMO gap, while open-shell configurations promote reactivity by facilitating electron promotion or transfer.^[45] Bond lengths in molecules are determined by the ground state potential minimum, where the balance of attractive and repulsive forces yields optimal nuclear separation, as seen in H₂⁺ at around 1.06 Å.^[43]

Condensed Matter Systems

In condensed matter systems, the ground state of electrons often emerges from the collective behavior of many interacting particles within a periodic lattice, leading to distinct phases such as metals, insulators, and ordered states. Unlike isolated systems, these ground states are shaped by the interplay of kinetic energy, Coulomb interactions, and lattice potentials, resulting in phenomena like band formation and symmetry breaking. The free electron gas model provides a foundational description, where the ground state at absolute zero temperature consists of a filled Fermi sea, with all single-particle states up to the Fermi energy E_F occupied according to the Pauli exclusion principle. This configuration minimizes the total kinetic energy of the non-interacting electron gas in a metal, as derived in the seminal work applying Fermi-Dirac statistics to metallic conduction.^[46] Building on this, band theory refines the picture by incorporating the periodic potential of the crystal lattice, leading to energy bands separated by gaps. In insulators, the ground state features a completely filled valence band and an empty conduction band, with the Fermi level lying within a large band gap that prevents charge transport at low temperatures. This state arises from the Bloch wave solutions to the Schrödinger equation in a periodic potential, ensuring no low-energy excitations across the gap. For semiconductors, the ground state is similar but with a small band gap E_g (typically 0.1–3 eV), allowing thermal activation to the conduction band while maintaining an insulating character at low temperatures; examples include silicon and germanium, where the filled valence band configuration dominates the zero-temperature properties. Superconductivity introduces a remarkable collective ground state, described by Bardeen-Cooper-Schrieffer (BCS) theory, where electrons form bound Cooper pairs due to attractive phonon-mediated interactions, condensing into a single quantum state with zero resistivity and perfect diamagnetism. The BCS ground state is a coherent superposition of paired states across the Fermi surface, breaking gauge symmetry and opening an energy gap in the excitation spectrum, as calculated from the reduced BCS Hamiltonian.^[47] In magnetic systems, the ferromagnetic ground state occurs when exchange interactions favor parallel alignment of electron spins, minimizing the total exchange energy in itinerant or localized models. This alignment, first theoretically grounded in the quantum mechanical exchange between electrons, leads to spontaneous magnetization in materials like iron, where the Stoner criterion for band ferromagnetism confirms the stability of the spin-polarized state. Strong electron correlations can drive deviations from band theory, as in Mott insulators, where the ground state exhibits charge localization and insulating behavior despite partially filled bands, due to the dominance of on-site Coulomb repulsion U over kinetic hopping t. In the Mott-Hubbard model, this results in a charge-ordered state with one electron per site, suppressing double occupancy and opening a correlation-induced gap; classic examples include transition metal oxides like NiO, where the strong U (several eV) enforces the insulating ground state at half-filling.

Experimental Determination

Spectroscopic Methods

Absorption and emission spectroscopy serve as primary experimental techniques to probe the ground state by inducing and observing electronic transitions to higher energy levels. In absorption spectroscopy, photons are absorbed by the system in its ground state, promoting electrons to excited states, with the transition frequency \nu related to the energy difference by \nu = (E_n - E_0)/h, where E_n is the excited state energy, E_0 the ground state energy, h Planck's constant, allowing indirect determination of E_0 relative to known excited levels.^[48] Emission spectroscopy complements this by detecting photons released as the system relaxes from excited states back to the ground state, confirming the same energy gaps and providing insights into ground state stability through line positions and intensities.^[49] These methods are widely applied in atomic and molecular systems, where the onset of the absorption spectrum marks the energy required to depart from the ground state.^[50] Photoelectron spectroscopy (PES) directly accesses ground state electronic structure by measuring the kinetic energies of electrons ejected upon ionization from the ground state using high-energy photons, typically in the ultraviolet or X-ray range. The ionization potential, which corresponds to the negative of the highest occupied molecular orbital energy in the ground state, is calculated as the binding energy E_b = h\nu - KE_e, where KE_e is the photoelectron kinetic energy and \nu the incident photon frequency, yielding precise orbital energies and valence electron configurations.^[51] In X-ray photoelectron spectroscopy (XPS), core-level binding energies further reveal ground state chemical environments and local charge distributions with atomic specificity.^[52] This technique has been instrumental in studying ground state properties of solids and molecules, such as in time-resolved measurements of ultrafast dynamics.^[53] Raman spectroscopy elucidates ground state vibrational properties by scattering light off the molecule, where inelastic scattering (Stokes or anti-Stokes shifts) corresponds to transitions between vibrational levels within the ground electronic state. The frequency shifts \Delta \nu equal the vibrational energy spacings \Delta E_v = h \Delta \nu, with the lowest observed Raman line originating from the zero-point vibrational level, which is elevated above the classical minimum due to quantum uncertainty, thus confirming the non-zero ground state vibrational energy.^[54] In molecules, this allows mapping of the ground state potential energy surface through overtone and combination bands, particularly enhanced in surface-enhanced Raman for single-molecule sensitivity.^[55] For instance, in diatomic molecules like N_2, Raman spectra reveal the anharmonic corrections to the ground state vibrational ladder.^[56] Laser cooling techniques achieve ultracold temperatures to populate the ground state with minimal thermal excitation, facilitating precision spectroscopy of its intrinsic properties. By using resonant laser light to reduce atomic or molecular velocities via photon recoil and optical pumping, systems like neutral atoms or diatomic molecules are cooled to microkelvin or nanokelvin regimes, suppressing Doppler broadening for high-resolution ground state measurements such as hyperfine splittings or transition frequencies.^[57] In symmetric top molecules like CaOCH_3, direct Sisyphus cooling scatters hundreds of photons to reach near-ground rovibrational states, enabling applications in quantum simulation and precise energy level determinations.^[58] This method has extended to complex species, including positrons and heavy ions, for ground state probing in precision metrology.^[59] Despite their power, spectroscopic methods face challenges in directly accessing ground state details due to broadening effects and selection rules. Spectral broadening, including homogeneous (lifetime) and inhomogeneous (environmental) contributions, can widen lines by several meV, obscuring subtle ground state features like fine structure in disordered systems or low-dimensional materials.^[60] Selection rules, governed by symmetry and angular momentum conservation, restrict observable transitions—for example, \Delta |N| = \pm 1 in conventional optics but altered by interactions like trigonal warping—preventing direct excitation of certain ground state modes and requiring indirect inference or specialized setups.^[61] These limitations often necessitate cryogenic environments or high-purity samples to resolve ground state signatures accurately.^[62]

Computational Approaches

Computational approaches to determining the ground state are essential when exact analytical solutions to the Schrödinger equation are infeasible, particularly for many-electron systems. These methods leverage numerical techniques to approximate the ground-state energy E_0 and wave function, often grounded in the variational principle, which ensures that trial wave functions yield upper bounds to the true E_0. Density Functional Theory (DFT) provides an efficient framework for ground-state calculations by reformulating the many-body problem in terms of the electron density n(\mathbf{r}) rather than the wave function. The Hohenberg-Kohn theorems establish that the ground-state density uniquely determines the external potential and total energy, allowing minimization of the energy functional E to obtain E_0. In practice, the Kohn-Sham approach maps the interacting system onto a fictitious non-interacting one with the same density, solved via self-consistent Kohn-Sham equations that include an exchange-correlation functional to capture many-body effects.^[63] This enables accurate approximations of E_0 for molecules and solids, with computational cost scaling favorably compared to wave function-based methods. Quantum Monte Carlo (QMC) methods employ stochastic sampling to evaluate multidimensional integrals in the ground-state wave function, offering high accuracy for strongly correlated systems where other approaches falter. Variational Monte Carlo (VMC) optimizes a trial wave function by minimizing the energy expectation value through random walks in configuration space, providing an upper bound to E_0. Diffusion Monte Carlo (DMC) projects the ground state from the trial function using imaginary-time evolution, effectively solving the time-independent Schrödinger equation stochastically and yielding near-exact energies for small systems after fixed-node approximations to handle the fermion sign problem.^[64] QMC excels in many-body problems like quantum liquids and solids, often achieving chemical accuracy (1 kcal/mol) with errors below 1% for binding energies. Configuration Interaction (CI) methods expand the ground-state wave function as a linear combination of Slater determinants from a chosen basis set, directly incorporating electron correlation beyond mean-field approximations. The full CI (FCI) includes all possible determinants within the basis, yielding the exact E_0 for that basis, though its exponential scaling limits it to small systems. Truncated variants, such as singles and doubles CI (CISD), selectively include excitations from a reference determinant (typically Hartree-Fock) to approximate correlation energy, with the lowest eigenvalue providing a variational upper bound to E_0.^[44] These approaches are particularly valuable for molecular ground states, where they benchmark other methods and refine properties like dissociation energies. Recent advances in machine learning potentials, particularly neural network-based interatomic potentials (NNPs) developed post-2020, enable rapid prediction of ground-state energies and geometries by learning from quantum mechanical data. These models, such as deep neural networks trained on DFT or coupled-cluster calculations, approximate the potential energy surface as a function of atomic coordinates, achieving quantum accuracy for large systems like proteins or materials. For instance, graph neural networks encode molecular topology to predict E_0 with mean absolute errors under 1 meV/atom, facilitating ground-state exploration in high-dimensional configuration spaces.^[65] Validation of these methods often involves comparing computed E_0 or derived properties, such as bond dissociation energies, to experimental benchmarks. In DFT, for example, the B3LYP functional with a 6-31G* basis predicts the equilibrium dissociation energy D_e of H_2 as 4.74 eV, closely matching the experimental value of 4.748 eV, demonstrating reliable performance for simple diatomic systems despite approximations in the exchange-correlation functional.

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We start with the exact energy eigenvalue problem, given by ˆH|n> = En|n>. But, perhaps this problem is too difficult to solve. Then, if ˆH = ˆH0 + ˆV, ...
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[PDF] 2. Approximation Methods - DAMTP
The variational method provides a simple way to place an upper bound on the ground state energy of any quantum system and is particularly useful when trying to ...
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The historical bases of the Rayleigh and Ritz methods - ScienceDirect
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[PDF] Theory of variational quantum simulation - arXiv
In this work, we first review the conventional variational principles, including the Rayleigh-Ritz method for solving static problems, and the Dirac and Frenkel ...
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Hans Hellmann of the Hellmann–Feynman Theorem - SpringerLink
Hellmann [2] formulated the famous Hellmann-Feynman theorem [3], to which the American physicist Richard Feynman came independently [4] and which has retained ...
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18.39 Applications in the Physical Sciences
Mar 27, 2024 · ... Theorem 3.3, p. 35) , such eigenfunctions satisfy the Sturm oscillation theorem. Namely the k th eigenfunction, listed in order of ...
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Nodes of wavefunctions | American Journal of Physics - AIP Publishing
Mar 1, 2007 · Two of the main theorems in Sturm-Liouville theory are the separation and the comparison theorems. For the Schrödinger equation, the separation ...
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[PDF] arXiv:quant-ph/0702260v1 27 Feb 2007
Feb 27, 2007 · The node structure of wavefunctions is important. For example, the fact that the ground state function has no nodes was one of the ingredients ...
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[PDF] 4. The Infinite Square Well - Physics
The lowest energy level is called the ground state, while the rest are called excited states. The reason why the energy is quantized is easy enough to ...<|control11|><|separator|>
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[PDF] Finite Square Well - UNCW
On the right is the odd bound state wavefunction for the electron in a square well of width a = 6a0. This corresponds to a bound state energy of E = 8.829 eV, ...
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[PDF] Lecture 13: March 10, 2021 Heisenberg's Uncertainty Principle
(I) Zero Point Energy: According to Heisenberg uncertainty principle, even at absolute zero temperature, particles cannot be at rest when confined in a finite ...Missing: definition | Show results with:definition
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nglos324 - zeropointenergy
Zero point energy is the minimum energy of a quantum oscillator at v=0, resulting from Heisenberg's Uncertainty Principle.
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[PDF] 9. Harmonic Oscillator - MIT OpenCourseWare
E = ω(a†a + 1 ) = ω(N + 1 ) and the minimum energy hω/2 is called the zero point energy. 2. 2. 9.1.3 Position representation. We have now started from a ...
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Quantum Harmonic Oscillator - HyperPhysics
The energy of the ground vibrational state is often referred to as "zero point vibration". The zero point energy is sufficient to prevent liquid helium-4 from ...Missing: mechanics | Show results with:mechanics
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Vacuum Energy - The Cosmological Constant - Sean M. Carroll
The cosmological constant turns out to be a measure of the energy density of the vacuum - the state of lowest energy.
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Vibrational Zero-Point Energies (ZPE) - CCCBDB
The vibrational zero-point energy is the energy difference between the lowest point on the potential energy surface (equilibrium energy) and the energy of the ...
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[PDF] Theory of Superconductivity - MRI Questions
CHANGE IN ZERO-POINT ENERGY. OF LATTICE VIBRATIONS. The contribution tothe condensation energy from the change in zero-point energy of the lattice can be.
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[PDF] RELAXATION DYNAMICS OF SOLID HELIUM-4
However, X-ray scattering has shown that vibrations in solid 4He, due to the the quantum-mechanical zero-point motion (ZPM) of the atoms, can be up to 25% of ...<|control11|><|separator|>
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Sinha Group/Research/Helium
Nov 8, 2007 · In fact, the atomic zero point motion is so large that classically, the solid should melt. Nonetheless, the solid exists!
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An Undulatory Theory of the Mechanics of Atoms and Molecules
The URL provided (https://www.googletagmanager.com) does not contain specific content related to a "Schrödinger" discussion, "harmonic oscillator," or "energy levels," as it is a domain primarily associated with Google Tag Manager, a tool for managing JavaScript and HTML tags on websites. It does not appear to host academic or scientific content about physics or the Schrödinger equation. Since the request seems to be a misunderstanding or a misdirection, and no relevant content is available at the specified URL, I cannot extract or summarize information about the harmonic oscillator or energy levels from it.
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The quantum theory of the emission and absorption of radiation
The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed ...
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Quantum control of molecular rotation | Rev. Mod. Phys.
Sep 18, 2019 · The molecule is assumed to be in its vibronic, i.e., electronic and vibrational, ground state ... quantum harmonic oscillator. It can be ...
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Phonon Pair Creation by Inflating Quantum Fluctuations in an Ion Trap
Oct 29, 2019 · ... quantum harmonic oscillator with time-dependent frequency Ω k ( t ) ... ground state and turns into an excited, squeezed state, cf. Figs ...
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[PDF] 1926-Schrodinger.pdf
It was stated in the beginning of this paper that in the present theory both the laws of motion and the quantum conditions can be deduced from one Hamiltonian ...
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Niels Bohr between physics and chemistry
May 1, 2013 · In building up electron structures, Bohr was governed by what he called the Aufbau, or construction, principle: The addition of electron number ...
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[PDF] On the Quantum Theory of Molecules - UFPR
It will be shown that the familiar components of the terms of a molecule; the energy of electronic motion, of the nuclear vibration and of the rotation, ...
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1.11: Molecular Orbital Theory - Chemistry LibreTexts
Aug 28, 2020 · The actual ground state of $H_{2}^{+}$ has the 1$\sigma_{g}$ orbital occupied. The equilibrium internuclear distance Re is 2.00 bohr and the ...
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[PDF] An Introduction to Configuration Interaction Theory - - Sherrill Group
One particularly nice feature of the CI method is that the calculated lowest energy eigenvalue is always an upper bound to the exact ground state energy. Our ...
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Variational Principles for Describing Chemical Reactions: The Fukui ...
Given ν0(r⃗), a molecule is most stable for the ground-state density. In the energetic formulation, one finds the ground-state density by minimizing the ...
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Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik
Download PDF ... Cite this article. Sommerfeld, A. Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik. Z. Physik 47, 1–32 (1928).
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Theory of Superconductivity | Phys. Rev.
A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive.
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Relations between absorption, emission, and excited state chemical ...
May 28, 2021 · We show that 2D spectra and these relations determine the thermodynamic standard chemical potential difference between the lowest excited and ground electronic ...
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Ground state potential energy surfaces around selected atoms from ...
Jan 29, 2016 · Here we demonstrate how sub-natural linewidth resonant inelastic soft x-ray scattering at vibrational resolution is utilized to determine ground state ...
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Electronic Spectroscopy and Excited States Analysis - Nature
By deciphering absorption and emission spectra, researchers gain insights into molecular structures, electron distribution, and dynamic processes that occur ...
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Measuring the quantum state of photoelectrons | Nature Photonics
Jan 29, 2025 · Absorption of an XUV photon promotes an electron from the ground state of a neutral argon atom (3p6) to the continuum, ionizing the atom. The ...
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A localized view on molecular dissociation via electron-ion partial ...
Mar 28, 2022 · Inner-shell photoelectron spectroscopy provides an element-specific probe of molecular structure, as core-electron binding energies are ...
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Following excited-state chemical shifts in molecular ultrafast x-ray ...
Jan 11, 2022 · X-ray photoelectron spectroscopy (XPS) is a proven tool to obtain information about local charge with atomic specificity in electronic ground ...Missing: ionization | Show results with:ionization
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Surface-Enhanced Impulsive Coherent Vibrational Spectroscopy
Nov 4, 2016 · The information about the initial vibrational phase allows the vibrations of the ground and excited electronic states to be separated from each ...
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High-throughput computation of Raman spectra from first principles
Feb 8, 2023 · Raman spectroscopy is a widely-used non-destructive material characterization method, which provides information about the vibrational modes ...
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Direct laser cooling of a symmetric top molecule - Science
Sep 11, 2020 · Laser cooling of atomic systems has enabled substantial advances in quantum simulation, precision clocks, and quantum many-body physics (1–4).
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Cooling of atoms using an optical frequency comb | Scientific Reports
Feb 21, 2019 · We report on laser cooling of neutral rubidium atoms by using a single mode of a frequency comb. Cooling is achieved on a dipole-allowed transition at 780 nm.
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Cooling positronium to ultralow velocities with a chirped laser pulse ...
Sep 11, 2024 · When laser radiation is skilfully applied, atoms and molecules can be cooled, allowing the precise measurements and control of quantum ...Methods · Cooling Laser · Doppler Spectroscopy
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Broadened quantum critical ground state in a disordered ... - Nature
Mar 16, 2024 · A plausible explanation is that the AM state originates from broadening of the SIT. An existence of such a broadened critical ground state, a so ...
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Unconventional valley-dependent optical selection rules and landau ...
Jun 10, 2020 · Selection rules are of vital importance in determining the basic optical properties of atoms, molecules and semiconductors.
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Broken selection rule in the quantum Rabi model | Scientific Reports
Jun 7, 2016 · We report the spectroscopic observation of a resonant transition that breaks a selection rule in the quantum Rabi model, implemented using an LC resonator and ...Results · Sign-Preserving Transition · Methods<|control11|><|separator|>
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Self-Consistent Equations Including Exchange and Correlation Effects
From a theory of Hohenberg and Kohn, approximation methods for treating an inhomogeneous system of interacting electrons are developed.
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Quantum Monte Carlo simulations of solids | Rev. Mod. Phys.
Jan 5, 2001 · This article describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calculate the properties of many- ...
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Neural Network Potentials: A Concise Overview of Methods
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