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Hydrogen-like atom

A hydrogen-like atom, also known as a hydrogenic atom, consists of a nucleus with atomic number Z (where Z ≥ 1) and exactly one electron, such as the neutral hydrogen atom (Z = 1) or ionized species like He⁺ (Z = 2) and Li²⁺ (Z = 3).^[1] These systems represent the simplest atomic structures and serve as foundational models in quantum mechanics due to their exact solvability via the Schrödinger equation.^[2] In quantum mechanics, the time-independent Schrödinger equation for a hydrogen-like atom, accounting for the Coulomb potential V(r) = -Z e² / (4πε₀ r) between the nucleus and electron, separates into radial and angular components in spherical coordinates.^[3] This separation yields analytical solutions for the wavefunctions, expressed as products of radial functions R_{nl}(r) and spherical harmonics Y_{lm}(θ, φ), where n is the principal quantum number, l the orbital angular momentum quantum number (0 ≤ l < n), and m the magnetic quantum number (-l ≤ m ≤ l).^[4] The exact solvability arises because the central potential allows for the use of separation of variables, making hydrogen-like atoms a key example of multi-particle quantum systems with fully analytical bound-state solutions.^[5] The energy eigenvalues depend solely on n and scale with Z², given by the formula E_n = - (13.6 eV) Z² / n², where the negative sign indicates bound states and n = 1, 2, 3, ... corresponds to the ground state and excited states, respectively.^[6] This quantization explains the discrete spectral lines observed in emission or absorption spectra of hydrogen-like ions, which are shifted to higher energies compared to neutral hydrogen due to the increased nuclear attraction for larger Z.^[7] Hydrogen-like atoms hold central importance in atomic physics and quantum mechanics, providing the basis for understanding electron behavior in more complex multi-electron atoms through approximations like the orbital model, where inner-shell electrons approximate hydrogen-like orbitals with effective nuclear charges.^[8] Their study has historical significance, underpinning the development of quantum theory in the early 20th century, and continues to inform applications in spectroscopy and plasma physics.^[9]

Fundamentals

Definition and Scope

A hydrogen-like atom, also known as a hydrogenic atom or one-electron ion, consists of a nucleus with atomic number Z (the number of protons) and exactly one orbiting electron, resulting in a net positive charge of Ze on the nucleus. Examples include the neutral hydrogen atom (Z=1), singly ionized helium (\mathrm{He}^+, Z=2), and doubly ionized lithium (\mathrm{Li}^{2+}, Z=3).^[10]^[11] The scope of hydrogen-like atoms encompasses single-electron systems where the interaction is purely Coulombic, governed by the central potential V(r) = -\frac{Z e^2}{4\pi\epsilon_0 r}, which allows for exact analytical solutions to the time-independent Schrödinger equation in non-relativistic quantum mechanics. This exact solvability arises because the potential depends only on the radial distance r, enabling separation of variables into radial and angular parts, yielding closed-form wavefunctions and energy eigenvalues. In contrast, multi-electron atoms involve additional electron-electron repulsion terms that prevent exact solutions, necessitating approximate methods such as the Hartree-Fock self-consistent field approach to estimate orbitals and energies.^[12]^[13]^[14] Mathematically, the hydrogen-like atom is treated as a reduced two-body problem, transforming the relative motion of the electron (mass m_e) and nucleus (mass M) into that of a single fictitious particle with reduced mass \mu = \frac{m_e M}{m_e + M} \approx m_e \left(1 - \frac{m_e}{M}\right) orbiting a fixed point charge Ze. This approximation holds well since M \gg m_e for most nuclei, with the correction term accounting for finite nuclear mass effects on the order of 0.05% for hydrogen.^[15]^[16] The term "hydrogen-like atom" originated in the early development of quantum mechanics, particularly with Niels Bohr's 1913 model, which generalized the quantization rules from hydrogen to ions with higher Z to explain their spectral lines, and was further formalized in the Schrödinger equation era of the 1920s to highlight systems sharing hydrogen's solvable structure.^[10]

Physical and Astrophysical Importance

Hydrogen-like atoms play a pivotal role in atomic physics by providing benchmark spectral lines that serve as standards for understanding atomic structure and transitions. Their energy levels scale quadratically with the nuclear charge Z, allowing the generalization of series like Lyman and Balmer from neutral hydrogen to ions such as He⁺ (Z=2) and higher-Z species, facilitating the identification of these features in complex spectra. This Z² scaling enables precise diagnostics of temperature, density, and ionization states in various environments, as the transition wavelengths shift predictably with Z, aiding in the deconvolution of overlapping lines from multi-element plasmas. In astrophysical contexts, hydrogen-like ions dominate the spectra of ionized plasmas, particularly in H II regions where recombination lines from species like H⁺ and He⁺ (hydrogen-like helium) trace the photoionization by massive stars, revealing the structure and dynamics of star-forming nebulae. In the atmospheres of white dwarfs, helium-rich types (DO spectral class) exhibit strong lines from He⁺ due to high temperatures ionizing helium, providing insights into cooling processes and composition. For high-Z hydrogen-like ions (e.g., Fe XXV, Z=26), their X-ray emission and absorption lines are crucial for probing accretion disks around black holes, where relativistic effects broaden and shift these features, allowing measurements of black hole spin and mass through reflection spectroscopy.^[17]^[18]^[19]^[20] In laboratory settings, hydrogen-like atoms enable precision spectroscopy for testing fundamental theories. For instance, the 1s-2s transition in He⁺ has been measured with high accuracy using laser excitation, offering a sensitive probe of quantum electrodynamics (QED) corrections due to the stronger nuclear field compared to hydrogen. These measurements help resolve discrepancies like the proton radius puzzle and validate QED predictions at the parts-per-billion level.^[21] Additionally, simulations of hydrogen-like systems on quantum computers serve as analogs for benchmarking quantum algorithms in chemistry and materials science, demonstrating the feasibility of exact solvers for multi-electron extensions.^[22] Theoretically, hydrogen-like atoms are an ideal testing ground for relativistic quantum mechanics and QED because their exact solvability in the non-relativistic limit allows precise inclusion of corrections like fine structure, Lamb shift, and vacuum polarization. High-Z examples, such as hydrogen-like tin (Sn⁵⁰⁺), have yielded stringent QED tests in strong fields, confirming predictions to 0.012% accuracy and probing beyond-standard-model physics. This exact solvability extends to relativistic formulations, where Dirac equation solutions benchmark approximations for more complex atoms.^[23]^[24]^[25]

Non-relativistic Description

Schrödinger Equation and Hamiltonian

The non-relativistic quantum mechanical description of a hydrogen-like atom, consisting of a nucleus of charge Ze and a single electron, is governed by the time-independent Schrödinger equation \hat{H} \psi = E \psi, where \psi(\mathbf{r}) is the wavefunction and E is the energy eigenvalue.^[26] This equation arises from the eigenvalue problem formulation introduced by Schrödinger to quantize atomic systems, treating the electron's motion under the central Coulomb attraction.^[26] The Hamiltonian operator \hat{H} encapsulates the system's kinetic and potential energies, accounting for the two-body nature of the electron-nucleus interaction through the reduced mass \mu = m_e M /(m_e + M), where m_e and M are the electron and nuclear masses, respectively.^[27] The explicit form of the Hamiltonian in SI units is

\hat{H} = -\frac{\hbar^2}{2\mu} \nabla^2 - \frac{Z e^2}{4\pi \epsilon_0 r},

where the first term represents the electron's kinetic energy and the second is the attractive Coulomb potential, with r = |\mathbf{r}| the electron-nucleus separation.^[27] Due to the spherical symmetry of the potential, which depends only on r, the problem is solved in spherical coordinates (r, \theta, \phi), where the Laplacian operator \nabla^2 separates into radial \nabla_r^2 and angular \nabla_{\theta\phi}^2 contributions: \nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2}{\partial \phi^2}.^[26] This separation enables the wavefunction to be factored as \psi(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi), reducing the partial differential equation to ordinary differential equations for the radial and angular parts.^[26] For computational convenience, especially in multi-electron extensions, atomic units simplify the expressions by setting \hbar = m_e = e = 4\pi \epsilon_0 = 1, which scales lengths by the Bohr radius a_0 = 4\pi \epsilon_0 \hbar^2 / (m_e e^2), energies by the hartree E_h = e^2 / (4\pi \epsilon_0 a_0), and the reduced mass \mu \approx m_e for light nuclei.^[28] In these units, the Hamiltonian becomes

\hat{H} = -\frac{1}{2} \nabla^2 - \frac{Z}{r},

with eigenvalues in hartrees, facilitating dimensionless analysis while preserving the structure of the original equation.^[28] The solutions must satisfy boundary conditions ensuring physical normalizability, such that \psi \to 0 as r \to \infty, and the formulation here excludes electron spin, treating the electron as spinless.^[26]

Energy Levels and Wavefunctions

The non-relativistic energy eigenvalues for the bound states of a hydrogen-like atom are independent of the orbital angular momentum quantum number l and depend only on the principal quantum number n, given by

E_n = -\frac{\mu Z^2 e^4}{2 (4\pi\epsilon_0)^2 \hbar^2 n^2},

where \mu is the reduced mass of the electron-nucleus system, Z is the atomic number, e is the elementary charge, \epsilon_0 is the vacuum permittivity, and \hbar is the reduced Planck's constant.^[29] This expression simplifies to E_n = -13.6 \, \mathrm{eV} \times Z^2 / n^2 for hydrogen (Z=1) using the electron mass approximation \mu \approx m_e.^[29] In atomic units where \hbar = e = m_e = 4\pi\epsilon_0 = 1, the formula reduces to E_n = -Z^2 / (2 n^2).^[29] The derivation arises from solving the radial Schrödinger equation for the Coulomb potential V(r) = -Z e^2 / (4\pi\epsilon_0 r), which separates into radial and angular parts after applying the boundary conditions that the wavefunction must remain finite at r=0 and decay exponentially at r \to \infty.^[29] The radial equation is transformed via a substitution u(r) = r R(r) and a scaled variable \rho = 2 Z r / (n a_0), where a_0 = 4\pi\epsilon_0 \hbar^2 / (\mu e^2) is the Bohr radius, leading to a differential equation solvable by series expansion.^[29] ^[30] For the series to terminate and yield normalizable solutions, the energy must be quantized with n = 1, 2, 3, \dots, and the radial function involves associated Laguerre polynomials L_{n-l-1}^{2l+1}(\rho).^[31] The complete wavefunctions for the stationary states are products of radial and angular components:

\psi_{n l m}(r, \theta, \phi) = R_{n l}(r) Y_l^m(\theta, \phi),

where Y_l^m(\theta, \phi) are spherical harmonics, and the radial part is

R_{n l}(r) = \sqrt{\left(\frac{2 Z}{n a_0}\right)^3 \frac{(n-l-1)!}{2 n (n+l)!}} \left(\frac{2 Z r}{n a_0}\right)^l e^{-Z r / (n a_0)} L_{n-l-1}^{2l+1}\left(\frac{2 Z r}{n a_0}\right).

This normalization ensures \int |\psi_{n l m}|^2 dV = 1.^[31] ^[29] The probability density |\psi_{n l m}|^2 represents the likelihood of finding the electron at position (r, \theta, \phi), with the radial probability distribution P(r) = r^2 |R_{n l}(r)|^2 giving the probability per unit radius.^[32] The wavefunctions exhibit nodes where |\psi|^2 = 0, corresponding to zero probability: n - l - 1 radial nodes (from the Laguerre polynomial and exponential factors) and l angular nodes (from the spherical harmonics).^[32] ^[31] For example, the ground state $1s (n=1, l=0) has no nodes and a spherically symmetric density peaking near r = a_0 / Z.^[32]

Quantum Numbers and Selection Rules

In the non-relativistic description of hydrogen-like atoms, the stationary states are labeled by three quantum numbers that arise from the separability of the Schrödinger equation in spherical coordinates. These quantum numbers fully characterize the eigenfunctions and determine key physical properties such as energy and spatial extent. The principal quantum number n = 1, 2, 3, \dots is a positive integer that primarily governs the energy of the state, given by E_n = -\frac{13.6 \, Z^2}{n^2} eV, where Z is the atomic number of the nucleus. This energy scales inversely with n^2, leading to degenerate levels for different angular configurations within the same n. Additionally, n sets the characteristic size of the orbital, with the expectation value of the radial distance scaling as \langle r \rangle \propto n^2 / Z, reflecting the increasing spatial spread for higher excited states.^[33] The orbital angular momentum quantum number l takes integer values from 0 to n-1, influencing the radial distribution of the wavefunction through the centrifugal term in the effective potential. This term introduces a barrier that pushes the electron away from the nucleus for l > 0, resulting in more compact orbitals compared to s-states (l=0) for the same n. The values of l correspond to familiar subshell designations: s for l=0, p for l=1, d for l=2, and f for l=3, with higher l yielding more nodes in the angular part and affecting the overall shape.^[33] The magnetic quantum number m_l specifies the z-component of the orbital angular momentum and ranges from -l to +l in integer steps. It determines the orientation of the orbital in space, with m_l = 0 corresponding to states aligned along the z-axis and |m_l| = l to those in the xy-plane, enabling the degeneracy of $2l + 1 states per l in the absence of external fields.^[33] Transitions between these states, such as those induced by electromagnetic radiation, are governed by selection rules derived from the electric dipole approximation. For electric dipole transitions, the matrix element \int \psi_f^* \mathbf{r} \psi_i \, dV must be nonzero, where \psi_{nlm_l} are the initial and final wavefunctions. Separating into radial and angular parts, the angular integral involving spherical harmonics Y_{l m_l} vanishes unless \Delta l = \pm 1 and \Delta m_l = 0, \pm 1, while \Delta n can be arbitrary as the radial integral generally allows changes in n. These rules dictate the allowed spectral lines, such as those in the Lyman (n=1) or Balmer (n=2) series.^[4] An additional constraint arises from parity conservation under spatial inversion. The parity of a state is (-1)^l, rendering states with even l (even parity) and odd l (odd parity). Electric dipole operators are odd under parity, so transitions are forbidden between states of the same parity, reinforcing the \Delta l = \pm 1 rule since it changes parity.^[34]

Angular Momentum and Spin Considerations

In the non-relativistic quantum mechanical treatment of the hydrogen-like atom, the orbital angular momentum of the electron is described by the vector operator \mathbf{L} = -i \hbar \mathbf{r} \times \nabla. This operator arises naturally from the separation of variables in the Schrödinger equation for the central Coulomb potential, where the angular dependence isolates the rotational dynamics. The square of the orbital angular momentum L^2 and its z-component L_z commute with the Hamiltonian and share common eigenfunctions, with eigenvalues L^2 = \hbar^2 l(l+1) and L_z = \hbar m_l, where l = 0, 1, \dots, n-1 is the orbital quantum number and m_l = -l, -l+1, \dots, l is the magnetic quantum number. The components of \mathbf{L} obey the fundamental commutation relations [L_x, L_y] = i \hbar L_z and cyclic permutations thereof, ensuring the algebraic structure of angular momentum in quantum mechanics. The angular eigenfunctions are the spherical harmonics Y_l^{m_l}(\theta, \phi), which form a complete orthonormal basis on the unit sphere:

\int_0^{2\pi} d\phi \int_0^\pi \sin\theta \, d\theta \, Y_{l'}^{m_l' *}(\theta, \phi) Y_l^{m_l}(\theta, \phi) = \delta_{l l'} \delta_{m_l m_l'}. $$ These functions, expressed in terms of [associated Legendre polynomials](/page/Associated_Legendre_polynomials), encode the directional probability distribution of the [electron](/page/Electron)'s position. Beyond orbital motion, the [electron](/page/Electron) has an intrinsic [spin](/page/Spin) angular momentum $\mathbf{S}$ characterized by the [spin quantum number](/page/Spin_quantum_number) $s = 1/2$ and z-projection $m_s = \pm 1/2$. This [spin](/page/Spin) is an internal degree of freedom, independent of the orbital motion in the basic non-relativistic model. The [spin](/page/Spin) states are represented by two-component Pauli spinors $\chi_{m_s}$, which satisfy $S^2 \chi_{m_s} = \frac{3}{4} \hbar^2 \chi_{m_s}$ and $S_z \chi_{m_s} = m_s \hbar \chi_{m_s}$. In this approximation, neglecting spin-orbit interactions, the total wavefunction is the unsymmetrized product $\psi_{n l m_l}(\mathbf{r}) \chi_{m_s}$, separating spatial and spin parts. The total [angular momentum operator](/page/Angular_momentum_operator) is $\mathbf{J} = \mathbf{L} + \mathbf{S}$, but without coupling terms in the [Hamiltonian](/page/Hamiltonian), the energy levels remain degenerate in $m_l$ and $m_s$, allowing $(2l+1) \times 2$ states per $n, l$ subshell. ### Introductory Relativistic Corrections In the non-relativistic description of hydrogen-like atoms, relativistic effects introduce small corrections to the [Schrödinger equation](/page/Schrödinger_equation), arising from the finite [speed of light](/page/Speed_of_light) and the electron's [spin](/page/Spin). These corrections are perturbative, of order $(v/c)^2$, where $v$ is the electron's orbital velocity and $c$ is the [speed of light](/page/Speed_of_light), and they modify the energy levels to produce the [fine structure](/page/Fine_structure) observed in atomic spectra. The two primary contributions are the relativistic correction to the [kinetic energy](/page/Kinetic_energy) and the spin-orbit coupling.[](https://link.springer.com/book/10.1007/978-3-662-12869-5) The relativistic kinetic energy correction stems from expanding the Dirac relativistic energy-momentum relation in the non-relativistic limit. The leading term beyond the classical $p^2 / (2 m_e)$ kinetic energy is the quartic momentum correction, given by the perturbation Hamiltonian

\delta H_{\text{rel}} = -\frac{p^4}{8 m_e^3 c^2},

where $p$ is the electron momentum operator, $m_e$ is the electron mass, and $c$ is the speed of light. This term shifts the energy levels downward, with the expectation value contributing to the fine structure by depending on the orbital quantum number $l$. For hydrogen-like atoms with nuclear charge $Z$, the scale of this correction is proportional to $Z^4 / n^3$, where $n$ is the principal quantum number.[](https://link.springer.com/book/10.1007/978-3-662-12869-5) The spin-orbit coupling arises from the interaction between the electron's [spin](/page/Spin) magnetic moment and the effective [magnetic field](/page/Magnetic_field) experienced in the electron's [rest frame](/page/Rest_frame) due to its orbital motion in the [Coulomb](/page/Coulomb) [electric field](/page/Electric_field) of the [nucleus](/page/Nucleus). In the non-relativistic limit of the [Dirac equation](/page/Dirac_equation), this yields the perturbation [Hamiltonian](/page/Hamiltonian)

H_{\text{SO}} = \frac{1}{2 m_e^2 c^2} \frac{1}{r} \frac{dV}{dr} (\mathbf{S} \cdot \mathbf{L}),

where $\mathbf{S}$ is the electron [spin](/page/Spin) [operator](/page/Operator), $\mathbf{L}$ is the orbital [angular momentum operator](/page/Angular_momentum_operator), $r$ is the radial distance, and $V(r) = -Z e^2 / r$ is the [Coulomb](/page/Coulomb) potential (in [Gaussian units](/page/Gaussian_units)). For this potential, $dV/dr = Z e^2 / r^2$, so the expression simplifies to

H_{\text{SO}} \approx \frac{Z e^2}{2 m_e^2 c^2 r^3} \mathbf{S} \cdot \mathbf{L}.

The factor of $1/2$ accounts for the [Thomas precession](/page/Thomas_precession), which halves the classical naive estimate. This term couples the [spin](/page/Spin) and orbital angular momenta, lifting the degeneracy in the total [angular momentum](/page/Angular_momentum) quantum number $j = l \pm 1/2$ for a given $n$ and $l$.[](https://link.springer.com/book/10.1007/978-3-662-12869-5) Together, the relativistic kinetic and spin-orbit corrections produce the [fine structure](/page/Fine_structure) splitting of energy levels, with the first-order perturbative shift given by

\Delta E = \frac{E_n (Z \alpha)^2}{n^2} \left( \frac{n}{j + 1/2} - \frac{3}{4} \right),

where $E_n$ is the non-relativistic energy, $\alpha$ is the [fine-structure constant](/page/Fine-structure_constant), and $j = l \pm 1/2$. This results in a small splitting (on the order of $\alpha^2$ times the gross structure) where, for fixed $n$ and $l$, the state with $j = l + 1/2$ has higher energy than $j = l - 1/2$. However, these corrections alone do not fully account for observed spectral lines, such as the small splitting between $2S_{1/2}$ and $2P_{1/2}$ states in [hydrogen](/page/Hydrogen), known as the [Lamb shift](/page/Lamb_shift); this anomaly requires quantum electrodynamic treatments beyond the non-relativistic framework to incorporate vacuum fluctuations and radiative effects.[](https://link.springer.com/book/10.1007/978-3-662-12869-5)[](https://link.aps.org/doi/10.1103/PhysRev.72.339) ## Relativistic Description ### Dirac Equation Formulation The [Dirac equation](/page/Dirac_equation), formulated by [Paul Dirac](/page/Paul_Dirac) in [1928](/page/1928), provides a relativistic wave equation for [spin-1/2](/page/Spin-1/2) particles such as the [electron](/page/Electron), ensuring [Lorentz covariance](/page/Lorentz_covariance) by incorporating the principles of [special relativity](/page/Special_relativity) and [quantum mechanics](/page/Quantum_mechanics).[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0023) The free-particle form is given by

(i \hbar \gamma^\mu \partial_\mu - m c) \psi = 0,

where $\gamma^\mu$ are the Dirac matrices, $\psi$ is a four-component [spinor](/page/Spinor), $m$ is the particle [mass](/page/Mass), $c$ is the [speed of light](/page/Speed_of_light), and $\hbar$ is the reduced Planck's constant.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0023) This equation satisfies the relativistic energy-momentum relation $E^2 = p^2 c^2 + m^2 c^4$ for each component of $\psi$, thus maintaining covariance under Lorentz transformations.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0023) To describe an [electron](/page/Electron) in an [electromagnetic field](/page/Electromagnetic_field), the [Dirac equation](/page/Dirac_equation) is coupled via minimal substitution, replacing $\partial_\mu$ with $\partial_\mu + i (e/\hbar c) A_\mu$ and adding the scalar potential term $e \phi$, where $A_\mu$ is the four-potential and $\phi$ is the [scalar potential](/page/Scalar_potential).[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0023) For a hydrogen-like atom, the potential is the static Coulomb interaction $V(r) = -Z e^2 / r$ (in [Gaussian units](/page/Gaussian_units)), with the [nucleus](/page/Nucleus) of charge $Z e$ fixed at the [origin](/page/Origin) and $\vec{A} = 0$. Assuming a [stationary state](/page/Stationary_state) $\psi(\vec{r}, t) = \psi(\vec{r}) e^{-i E t / \hbar}$, the time-independent [Dirac equation](/page/Dirac_equation) becomes

\left[ c \vec{\alpha} \cdot \vec{p} + \beta m c^2 - \frac{Z e^2}{r} \right] \psi = E \psi,

where $\vec{p} = -i \hbar \nabla$ is the [momentum operator](/page/Momentum_operator), and $\vec{\alpha} = (\alpha_x, \alpha_y, \alpha_z)$ and $\beta$ are 4×4 Dirac matrices satisfying the anticommutation relations $\{ \alpha_i, \alpha_j \} = 2 \delta_{ij}$, $\{ \alpha_i, \beta \} = 0$, and $\beta^2 = 1$.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0023) The wavefunction $\psi$ is a [bispinor](/page/Bispinor), structured as $\psi = \begin{pmatrix} \phi \\ \chi \end{pmatrix}$, where $\phi$ and $\chi$ are two-component spinors representing the large and small components, respectively, with $\chi$ becoming negligible in the non-relativistic limit.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0023) For the central Coulomb potential in a hydrogen-like atom, the [Dirac equation](/page/Dirac_equation) permits [separation of variables](/page/Separation_of_variables) in spherical coordinates due to spherical symmetry, leading to conserved total angular momentum $\vec{J} = \vec{L} + \vec{S}$ and its z-component. The angular part involves [spherical harmonics](/page/Spherical_harmonics) coupled with spin, while the radial equations for the large and small components can be decoupled into second-order differential equations. In [1928](/page/1928), Walter Gordon obtained the exact analytical solution by transforming the coupled radial equations and expressing the solutions in terms of confluent hypergeometric functions, with energy eigenvalues determined via a [continued fraction](/page/Continued_fraction) method to ensure normalizability.[](https://ui.adsabs.harvard.edu/abs/1928ZPhy..48...11G/abstract) ### Quantum Numbers and Symmetry In the relativistic description of the hydrogen-like atom via the [Dirac equation](/page/Dirac_equation), the bound-state solutions are characterized by conserved quantum numbers that emerge from the underlying symmetries of the [Hamiltonian](/page/Hamiltonian), including rotational invariance and the conservation of a relativistic analog of the Runge-Lenz vector. These quantum numbers label the eigenstates and ensure the exact solvability of the [Coulomb](/page/Coulomb) problem. The [total angular momentum quantum number](/page/Total_angular_momentum_quantum_number) $ j $ takes half-integer values starting from $ 1/2 $, reflecting the [spin-1/2](/page/Spin-1/2) nature of the [electron](/page/Electron), while the [magnetic quantum number](/page/Magnetic_quantum_number) $ m_j $ ranges from $ -j $ to $ +j $ in integer steps, arising from the SU(2) symmetry of rotations. A key relativistic quantum number is $ k $, defined as $ k = \pm (j + 1/2) $, which couples the orbital angular momentum $ l $ and spin, distinguishing states where $ j = l \pm 1/2 $ for the upper and lower components of the Dirac spinor. Positive $ k $ corresponds to $ j = l - 1/2 $ (except for $ s_{1/2} $ states), and negative $ k $ to $ j = l + 1/2 $. This $ k $ serves as a separation constant in the angular part of the Dirac equation, linking the parity and angular momentum structure without separate specification of $ l $ for each component. The parity operator $ P $ yields eigenvalues $ P = (-1)^{j + 1/2} \sgn(k) $, determining the even or odd spatial behavior of the wavefunctions under inversion. The principal [quantum number](/page/Quantum_number) $ n $ is introduced through the conservation of the Runge-Lenz vector, a dynamical [symmetry](/page/Symmetry) that extends the SO(3) rotational group to SO(4) in the non-relativistic limit but persists in a modified form for the Dirac-[Coulomb](/page/Coulomb) problem, ensuring the closing of the [symmetry](/page/Symmetry) algebra and exact solvability.[](https://arxiv.org/abs/1504.04269) Specifically, $ n = |k| + \tilde{n} $, where $ \tilde{n} $ is a positive [integer](/page/Integer), quantizing the radial motion and lifting the degeneracy beyond simple [angular momentum](/page/Angular_momentum) labels. This conservation arises from the specific 1/r form of the [Coulomb](/page/Coulomb) potential, analogous to the non-relativistic case where separate $ l $ and $ m_l $ quantum numbers describe orbital [angular momentum](/page/Angular_momentum).[](https://arxiv.org/abs/1504.04269) In the Dirac framework, spin-orbit coupling is inherently incorporated, eliminating the need for a separate spin projection [quantum number](/page/Quantum_number) $ m_s $; instead, the total [angular momentum](/page/Angular_momentum) $ \mathbf{J} = \mathbf{L} + \mathbf{S} $ is treated as a unified entity, with spin effects folded into the $ k $ and $ j $ specifications from the outset. ### Relativistic Energy Levels The relativistic energy levels of a hydrogen-like atom are obtained by solving the [Dirac equation](/page/Dirac_equation) for an [electron](/page/Electron) in the Coulomb potential of a [nucleus](/page/Nucleus) with atomic number $Z$. The exact energy spectrum for bound states is given by

E_{nj} = m c^2 \left[ 1 + \left( \frac{Z \alpha}{\nu} \right)^2 \right]^{-1/2},

where $m$ is the electron rest mass, $c$ is the speed of light, $\alpha$ is the fine-structure constant, $n$ is the principal quantum number, $j$ is the total angular momentum quantum number, and $\nu = n - (j + 1/2) + \sqrt{(j + 1/2)^2 - (Z \alpha)^2}$.[](https://ui.adsabs.harvard.edu/abs/1928ZPhy..48...11G/abstract)[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0084) This formula incorporates both the relativistic kinematics and spin-orbit coupling inherent in the Dirac description. The derivation of this energy spectrum proceeds from the radial form of the Dirac equation, which is solved using continued fraction recursion relations for the radial wave functions. The resulting eigenvalue condition yields the above expression for $E_{nj}$, valid for $Z \alpha < 1$. In the limit of small $Z \alpha$, this reduces to the Sommerfeld fine-structure formula, which approximates the relativistic corrections to the non-relativistic Bohr levels up to order $(Z \alpha)^4$.[](https://onlinelibrary.wiley.com/doi/10.1002/andp.19163561702) The energy levels exhibit a strong dependence on $Z$, as the parameter $Z \alpha$ governs the relativistic corrections; for high $Z$, such as $Z > 90$, these effects intensify, and quantum electrodynamic phenomena like [vacuum polarization](/page/Vacuum_polarization) become significant in interpreting experimental spectra. In the non-relativistic limit ($Z \alpha \ll 1$), expanding the formula gives $E \approx m c^2 - \frac{Z^2 \alpha^2 m c^2}{2 n^2} +$ higher-order fine-structure terms, recovering the Schrödinger energies plus relativistic perturbations. Notably, the energy depends only on the quantum numbers $n$ and $j$, independent of the orbital [angular momentum](/page/Angular_momentum) quantum number $l$ and [magnetic quantum number](/page/Magnetic_quantum_number) $m_j$, reflecting an accidental degeneracy in the Dirac-Coulomb problem. This symmetry is ultimately broken by quantum electrodynamic corrections, such as the [Lamb shift](/page/Lamb_shift). ### Wavefunction Solutions The solutions to the [Dirac equation](/page/Dirac_equation) for bound states in hydrogen-like atoms are expressed as four-component [bispinor](/page/Bispinor) wavefunctions, which incorporate the relativistic coupling between [spin](/page/Spin) and orbital [angular momentum](/page/Angular_momentum). These wavefunctions, first derived explicitly by Gordon and Darwin, take the general form

\psi_{n j m}(\mathbf{r}) = \frac{1}{r} \begin{pmatrix} i G_{n k}(r) \Omega_{j l m}(\theta,\phi) \ i F_{n k}(r) \Omega_{j l' m}(\theta,\phi) \end{pmatrix},

where $n$ is the principal quantum number, $j$ is the total angular momentum quantum number, $m$ is the magnetic quantum number, and $k$ is the Dirac relativistic quantum number ($k = \pm (j + 1/2)$). Here, $G_{n k}(r)$ and $F_{n k}(r)$ represent the large (upper) and small (lower) radial components, respectively, with the upper component dominating in the non-relativistic limit. The factor $i$ is a conventional phase choice that renders the wavefunctions real for certain parity eigenstates, and the $1/r$ prefactor facilitates separation of variables in spherical coordinates.[](https://ui.adsabs.harvard.edu/abs/1928ZPhy..48...11G/abstract)[](https://royalsocietypublishing.org/doi/10.1098/rspa.1928.0084) The radial functions $G_{n k}(r)$ and $F_{n k}(r)$ are obtained by solving the coupled first-order differential equations arising from the Dirac-Coulomb Hamiltonian and are expressed as linear combinations of confluent hypergeometric functions of the first kind, $_1F_1(a; b; \rho)$. These functions ensure the bound-state behavior, with the series terminating for integer values related to the effective principal quantum number $n' = n - (j + 1/2) + \sqrt{(j + 1/2)^2 - (Z \alpha)^2}$, where $Z$ is the nuclear charge and $\alpha$ is the fine-structure constant. At large distances, both $G_{n k}(r)$ and $F_{n k}(r)$ exhibit the asymptotic form $e^{-\rho/2} \rho^\gamma$, where $\gamma = \sqrt{k^2 - (Z \alpha)^2}$ is the Sommerfeld fine-structure parameter and $\rho = 2 Z \alpha \sqrt{m^2 c^4 - E^2}\, r / (\hbar |E|)$ (with $E$ the binding energy, $m$ the electron mass, $c$ the speed of light, and $\hbar$ the reduced Planck's constant), ensuring exponential decay for normalizable states. The angular dependence is captured by the spherical spinors $\Omega_{j l m}(\theta, \phi)$, which are two-component functions combining [spherical harmonics](/page/Spherical_harmonics) $Y_l^m(\theta, \phi)$ with Pauli spinors $\chi$. Specifically, $\Omega_{j l m} = \sum_{m_l, m_s} \langle l m_l, 1/2 m_s | j m \rangle Y_l^{m_l}(\theta, \phi) \chi_{m_s}$, where the Clebsch-Gordan coefficients couple the orbital [angular momentum](/page/Angular_momentum) $l$ and [spin](/page/Spin) $s = 1/2$. For the upper component, $l = |k| - 1/2$, while for the lower component, $l' = 2j - l$, reflecting the relativistic [parity](/page/Parity) mixing where the lower component has opposite [parity](/page/Parity) to the upper. These spinors form a complete basis for the [angular momentum](/page/Angular_momentum) algebra in the Dirac [representation](/page/Representation). The wavefunctions are normalized such that the total probability is unity, satisfying $\int_0^\infty \left( |G_{n k}(r)|^2 + |F_{n k}(r)|^2 \right) dr = 1$, with the angular integrals over the spherical spinors yielding unity due to their [orthonormality](/page/Orthonormality). This [normalization](/page/Normalization) condition determines the overall constant in the radial functions and ensures the probabilistic interpretation within [relativistic quantum mechanics](/page/Relativistic_quantum_mechanics). ### Specific Orbital Examples The [ground state](/page/Ground_state) of the hydrogen-like atom in the Dirac theory is the $1S_{1/2}$ orbital, characterized by [principal quantum number](/page/Principal_quantum_number) $n=1$, total [angular momentum](/page/Angular_momentum) $j=1/2$, and Dirac quantum number $k=-1$. The radial parts of the [bispinor](/page/Bispinor) wavefunction are given by the large component $G(r)$ and small component $F(r)$, where

G(r) = 2 (1 + \gamma) \sqrt{1 - \varepsilon^2} \left( \frac{Z \alpha \rho}{2} \right)^{\gamma - 1} e^{-\rho/2} / N,

and $F(r)$ has a similar form but with an overall sign flip in the prefactor, reflecting the coupling between upper and lower spinor components. Here, $\gamma = \sqrt{1 - Z^2 \alpha^2}$, $\varepsilon = E / (m c^2)$, $\rho = 2 Z \alpha r / (n a_0)$ with $a_0$ the Bohr radius scaled appropriately, and $N$ is a normalization constant ensuring $\int_0^\infty [G^2(r) + F^2(r)] dr = 1$. These expressions arise from solving the radial Dirac equations for $n_r = 0$, where the confluent hypergeometric functions reduce to unity, yielding power-law behavior near the origin modified by $\gamma < 1$.[](https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%204%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20electrodynamics%20%282ed.%2C%201982%29.pdf) For the $n=2$ level with $j=1/2$, the $2S_{1/2}$ ($k=-1$, $l=0$) and $2P_{1/2}$ ($k=1$, $l=1$) orbitals are degenerate in the Dirac theory, as the energy depends only on $n$ and $j$, not on the orbital angular momentum $l$. Their radial wavefunctions involve confluent hypergeometric functions with $n_r=1$, taking the general form

G_{n j}(r) \propto e^{-\rho/2} \rho^{\gamma-1} \left[ (1 + \gamma - \varepsilon + Z \alpha) , _1F_1(2 - \gamma; 2\gamma + 1; \rho) - \rho , _1F_1(3 - \gamma; 2\gamma + 1; \rho) \right],

and $F_{n j}(r)$ analogously, where $_1F_1$ is the confluent hypergeometric function that terminates as a polynomial for bound states. This degeneracy highlights the relativistic unification of S and P states with the same $j$, differing from the non-relativistic case.[](https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%204%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20electrodynamics%20%282ed.%2C%201982%29.pdf) In contrast, the $2P_{3/2}$ state ($n=2$, $j=3/2$, $k=-2$) is non-degenerate with the $j=1/2$ pair, featuring a higher $j$ value and primarily $l=1$ character, with energy shifted due to the $j$-dependence in the Dirac formula $E_{n j} = m c^2 \left[ 1 + \left( \frac{Z \alpha}{n - (j + 1/2) + \sqrt{(j + 1/2)^2 - Z^2 \alpha^2}} \right)^2 \right]^{-1/2}$. Its radial functions follow the same hypergeometric structure but with $\gamma = \sqrt{4 - Z^2 \alpha^2}$ and adjusted coefficients for $k=-2$, emphasizing the splitting within the $n=2$ manifold.[](https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%204%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20electrodynamics%20%282ed.%2C%201982%29.pdf) The probability current density for these Dirac orbitals is $\mathbf{j} = \frac{\hbar}{2 m i} (\psi^\dagger \boldsymbol{\alpha} \psi)$, where $\boldsymbol{\alpha}$ are the Dirac matrices, revealing oscillatory behavior known as [Zitterbewegung](/page/Zitterbewegung) due to interference between positive and negative energy components in the wavefunction. This trembling motion, with frequency on the order of $2 m c^2 / \hbar$, manifests as rapid spatial oscillations superimposed on the classical orbital motion, a hallmark of the relativistic description.[](https://ia803206.us.archive.org/4/items/landau-and-lifshitz-physics-textbooks-series/Vol%204%20-%20Landau%2C%20Lifshitz%20-%20Quantum%20electrodynamics%20%282ed.%2C%201982%29.pdf) ### Negative Energy States In the Dirac theory of the hydrogen-like atom, the energy spectrum comprises a positive continuum for energies $E > mc^2$, discrete bound states in the range $-mc^2 < E < mc^2$, and a negative continuum for $E < -mc^2$. The negative energy states form a continuous branch extending downward from $-mc^2$, interpreted by Dirac as completely filled in the ground state of the universe, forming the so-called [Dirac sea](/page/Dirac_sea). This filling ensures the [Pauli exclusion principle](/page/Pauli_exclusion_principle) prevents electrons from occupying these states, but it also implies an infinite negative vacuum energy due to the unbounded density of states in the negative continuum.[](https://bohr.physics.berkeley.edu/classes/221/notes/spdirac.pdf) The wavefunctions for these negative energy states ($E < 0$) display distinct spatial behavior: near the nucleus, they resemble bound-state functions with exponential decay modulated by the [Coulomb potential](/page/Coulomb_potential), but at large radial distances $r$, they become oscillatory and plane-wave-like, confirming their unbound, continuum nature. This dual character arises from the relativistic coupling of large and small spinor components in the [Dirac equation](/page/Dirac_equation) solutions for the Coulomb field. Computationally, these continuum wavefunctions can be normalized and used to describe scattering processes involving the negative spectrum.[](https://pubs.aip.org/aip/cip/article/5/3/319/136850/Computing-Dirac-s-atomic-hydrogen-wave-functions)[](https://bohr.physics.berkeley.edu/classes/221/notes/spdirac.pdf) Dirac's 1930 interpretation resolved the physical meaning of these states by positing that vacancies, or "holes," in the filled negative energy sea manifest as positrons—antiparticles with positive energy and opposite charge. This hole theory leverages the charge conjugation symmetry of the [Dirac equation](/page/Dirac_equation), where the transformation $\psi \to \gamma^2 \psi$ (up to a phase) maps positive-energy electron solutions to negative-energy ones, interchanging particle and antiparticle descriptions. The symmetry ensures that the equation for an electron in the Coulomb field is equivalent to that for a positron under this operation.[](https://royalsocietypublishing.org/doi/10.1098/rspa.1930.0013)[](https://sites.ualberta.ca/~gingrich/courses/phys512/node64.html) Processes involving the negative continuum, such as pair production, require a minimum energy threshold of $2mc^2$ to create an electron-positron pair from the vacuum, corresponding to exciting an electron from the Dirac sea. For hydrogen-like atoms with atomic number $Z$ such that $Z\alpha > 1$ (where $\alpha$ is the [fine-structure constant](/page/Fine-structure_constant)), the strong [Coulomb](/page/Coulomb) field leads to instabilities akin to the [Klein paradox](/page/Klein_paradox), where incident particles can be transmitted with probability greater than unity, interpreted as pair creation from the negative sea. This effect highlights the breakdown of single-particle Dirac theory for supercritical fields, though it was originally derived in the context of step potentials.[](https://arxiv.org/pdf/0712.0150) ## Advanced Developments ### Limitations of Dirac Solution The Dirac equation encounters significant limitations when applied to hydrogen-like atoms with high atomic numbers Z, specifically when Z exceeds approximately 137, corresponding to Zα > 1 where α is the [fine-structure constant](/page/Fine-structure_constant). In this regime, the relativistic parameter γ = √[k² - (Zα)²], with k being the [relativistic angular momentum](/page/Relativistic_angular_momentum) quantum number, becomes imaginary for the [ground state](/page/Ground_state) (k = 1). This results in wave functions that oscillate rather than decay exponentially, leading to non-normalizable solutions that diverge at the origin and indicate the absence of bound states under the point-nucleus [Coulomb](/page/Coulomb) potential. Such behavior signals the breakdown of the Dirac description without modifications, necessitating regularization of the potential near the [nucleus](/page/Nucleus) to account for extended [nuclear](/page/Nuclear) charge distributions.[](https://bohr.physics.berkeley.edu/classes/221/notes/spdirac.pdf) A key deficiency of the Dirac solution is its prediction of exact degeneracy for energy levels with the same [principal quantum number](/page/Principal_quantum_number) n and total angular momentum j but different orbital angular momentum l, such as the 2S_{1/2} and 2P_{1/2} states in [hydrogen](/page/Hydrogen). In reality, precise spectroscopic measurements reveal a small but observable splitting between these levels, quantified as the [Lamb shift](/page/Lamb_shift) of about 1058 MHz in [hydrogen](/page/Hydrogen), arising from radiative corrections beyond the Dirac framework. This discrepancy highlights the Dirac equation's inability to capture higher-order quantum electrodynamic effects that lift the predicted degeneracies.[](https://www.osti.gov/servlets/purl/1423965) Furthermore, the Dirac equation omits vacuum polarization effects, in which virtual electron-positron pairs induced by the nuclear field screen the Coulomb potential, effectively modifying it at short distances. This modification is described by the Uehling potential, the leading-order quantum electrodynamic correction, which includes a logarithmic term that enhances the effect near the nucleus for high-Z atoms. By ignoring such pair production and vacuum fluctuations, the Dirac solution fails to incorporate these subtle alterations to the potential that become increasingly important for precision energy level predictions. The exact analytical solutions to the [Dirac equation](/page/Dirac_equation) for the Coulomb problem, first derived by [Gordon](/page/Gordon), apply strictly to an idealized pure [Coulomb](/page/Coulomb) potential from a point-like [nucleus](/page/Nucleus). These solutions do not account for the finite spatial extent of real nuclei, which introduces additional potential modifications and energy shifts, particularly pronounced in high-precision studies or for heavier elements. Similarly, [hyperfine structure](/page/Hyperfine_structure) effects, stemming from interactions between the electron's spin and the [nuclear magnetic moment](/page/Nuclear_magnetic_moment), are entirely absent in this framework, limiting its applicability to scenarios requiring such details. ### Quantum Electrodynamics Enhancements Quantum electrodynamics (QED) provides perturbative corrections beyond the [Dirac equation](/page/Dirac_equation) to accurately describe the energy levels and wavefunctions of hydrogen-like atoms, treating the [Dirac fine structure](/page/Fine_structure) as the zeroth-order approximation. These enhancements account for radiative processes, such as virtual photon exchanges, which resolve discrepancies in the Dirac model by incorporating higher-order effects in the [fine-structure constant](/page/Fine-structure_constant) α. The primary QED contributions include the [Lamb shift](/page/Lamb_shift), self-energy corrections, [hyperfine structure](/page/Hyperfine_structure) refinements, and advanced treatments for high nuclear charge Z. The Lamb shift arises from second-order perturbation theory in QED, primarily affecting s-states due to the electron's interaction with the quantized vacuum. For hydrogen-like atoms, the leading Lamb shift energy correction for n s-states is given by

\Delta E_{\text{Lamb}} \approx \frac{\alpha^5 m c^2 Z^4}{n^3} \ln\left(\frac{1}{\alpha}\right),

where m is the electron mass, c is the speed of light, and the logarithmic term originates from the ultraviolet divergence regulated by the Bethe logarithm, which averages the electron's position-dependent energy over the atomic wavefunction. This shift, first derived by Bethe using mass renormalization to handle the divergence, lifts the degeneracy between 2s and 2p_{1/2} states predicted by the Dirac equation, with the s-state lowered relative to p-states. For light atoms (low Z), the shift scales as Z^4 / n^3, but higher-order terms become significant for precise calculations. Self-energy corrections in QED modify the electron's effective mass and its coupling to the [electromagnetic field](/page/Electromagnetic_field) through radiative loops. These include vertex corrections that contribute to the anomalous [magnetic moment](/page/Magnetic_moment) of the bound [electron](/page/Electron), with the leading term (g-2)/2 = α / (2π) arising from one-loop diagrams. In hydrogen-like atoms, the bound-state [self-energy](/page/Self-energy) affects energy levels via the electron's interaction with its own radiation field, introducing shifts of order α (Zα)^4 relative to the Dirac energy, evaluated using rigorous [QED](/page/QED) methods like the [Feynman diagram](/page/Feynman_diagram) expansion in the [Coulomb](/page/Coulomb) [gauge](/page/Gauge). These corrections are crucial for matching theoretical predictions to spectroscopic data, particularly in refining the g-factor for highly charged ions. The [hyperfine structure](/page/Hyperfine_structure) in hydrogen-like atoms receives [QED](/page/QED) enhancements through the Fermi [contact](/page/Contact) interaction, which dominates for s-states where the electron wavefunction has non-zero probability at the [nucleus](/page/Nucleus). The leading hyperfine splitting constant A for s-states is

A = \frac{8}{3} \alpha^4 g_p \frac{m_e}{m_p} \frac{Z^3}{n^3} |\psi(0)|^2,

where g_p is the nuclear g-factor, m_e / m_p is the [electron](/page/Electron)-to-proton [mass ratio](/page/Mass_ratio), and |\psi(0)|^2 = (Z^3 / (\pi n^3 a_0^3)) for the non-relativistic wavefunction at the origin (a_0 is the [Bohr radius](/page/Bohr_radius)). This term, derived from the [magnetic dipole](/page/Magnetic_dipole) interaction between the [electron](/page/Electron) and nuclear spins, scales as Z^3 and is enhanced relativistically for high Z, with [QED](/page/QED) corrections adding α^2 terms to the interaction [Hamiltonian](/page/Hamiltonian). For p-states, orbital contributions supplement the contact term, but s-states provide the strongest hyperfine signal. For high-Z hydrogen-like atoms (Z ≈ 100), full relativistic QED is essential, as perturbative expansions in Zα break down due to the strong [Coulomb](/page/Coulomb) field. These systems require all-order resummations, including two-photon exchange diagrams that contribute corrections of order (Zα)^6 m c^2 to binding energies, evaluated via the [Dirac-Coulomb-Breit Hamiltonian](/page/Hamiltonian) augmented by [QED](/page/QED) loops. Calculations incorporate [self-energy](/page/Self-energy) and [vacuum polarization](/page/Vacuum_polarization) to high precision, testing [QED](/page/QED) in extreme fields where nonlinear effects like electron-positron [pair production](/page/Pair_production) become relevant, with agreement between theory and experiment confirming [QED](/page/QED) up to α (Zα)^5 levels for ions like [uranium](/page/Uranium). ### Experimental Validation and Discrepancies The 1s-2s transition frequency in [hydrogen](/page/Hydrogen) has been measured with a relative accuracy of approximately $10^{-14}$, yielding a value of 2 466 061 413 187 103.17(10) Hz as recommended by CODATA in 2022. This precision stems from advanced [frequency comb](/page/Frequency_comb) [spectroscopy](/page/Spectroscopy) techniques applied to atomic beams, enabling rigorous tests of [reduced mass](/page/Reduced_mass) corrections in the Dirac-Coulomb framework.[](https://www.science.org/doi/10.1126/science.abc7776) The experimental result agrees with quantum electrodynamic ([QED](/page/QED)) predictions to within $10^{-6}$, confirming the validity of bound-state [QED](/page/QED) calculations including [self-energy](/page/Self-energy) and [vacuum polarization](/page/Vacuum_polarization) contributions.[](https://arxiv.org/abs/2211.05411) High-precision laser spectroscopy of the fine structure splitting in He$^+$, specifically the $2P_{3/2} - 2P_{1/2}$ interval, has confirmed predictions from the [Dirac equation](/page/Dirac_equation) augmented by [QED](/page/QED) corrections. Measurements achieve accuracies on the order of parts per million relative to the ~175 GHz splitting, aligning theory and experiment within uncertainties dominated by higher-order [QED](/page/QED) terms.[](https://www.nature.com/articles/s42005-024-01891-4) Such experiments, involving ion traps and tunable lasers, validate the relativistic [fine structure](/page/Fine_structure) formula scaled by the [nuclear](/page/Nuclear) charge $Z=2$, with deviations attributable to [nuclear](/page/Nuclear) motion effects below 0.1%.[](https://link.aps.org/doi/10.1103/PhysRevLett.118.063001) For high-$Z$ ions, experiments at the GSI Helmholtz Centre have probed [QED](/page/QED) effects in hydrogen-like [uranium](/page/Uranium) ($Z=92$), where the ground-state 1s [Lamb shift](/page/Lamb_shift) is measured via [x-ray spectroscopy](/page/X-ray_spectroscopy) of Lyman-$\alpha$ transitions with an accuracy of about 1%.[](http://web-docs.gsi.de/~stoe_exp/research/experiments/lamb_shift/cooler.php) These results demonstrate [QED](/page/QED) contributions scaling as $(Z\alpha)^4$ reaching ~1% of the [binding energy](/page/Binding_energy), in good agreement with [theory](/page/Theory), though uncertainties arise from two-loop corrections.[](https://link.aps.org/doi/10.1103/PhysRevLett.94.223001) However, for $Z > 80$, observed discrepancies in the [Lamb shift](/page/Lamb_shift), on the order of several [eV](/page/EV), are attributed to unmodeled [nuclear](/page/Nuclear) effects such as finite [nuclear](/page/Nuclear) [size](/page/Size) and [polarization](/page/Polarization), which become comparable to [QED](/page/QED) terms.[](https://www.nature.com/articles/s41586-023-06910-y) Persistent open issues include the proton radius puzzle, where the charge radius extracted from muonic hydrogen spectroscopy (~0.841 fm) remains ~4% smaller than values from some electronic hydrogen-like atoms (~0.877 fm), with the CODATA 2022 recommended value of 0.84087(39) fm incorporating muonic data, despite refined measurements and theoretical adjustments.[](https://arxiv.org/abs/2501.11195) This discrepancy, persisting as of 2025, challenges the universality of finite-size corrections in [QED](/page/QED) and may indicate new physics or systematic effects in lepton-nucleus interactions. Additionally, for $Z\alpha \approx 1$ in high-$Z$ systems, perturbative [QED](/page/QED) expansions are incomplete, with higher-order radiative corrections (e.g., three-loop [self-energy](/page/Self-energy)) contributing uncertainties up to 0.5% that exceed current experimental precisions.

References

[1]
107 Bohr's Theory of the Hydrogen Atom
So, if a nucleus has Z protons ( Z = 1 for hydrogen, 2 for helium, etc.) and only one electron, that atom is called a hydrogen-like atom. The spectra of ...
[2]
[PDF] 4.5 The Hydrogen Atom
The simplest of all atoms is the Hydrogen atom, which is made up of a positively charged proton with rest mass mp = 1.6726231 × 10-27 kg, and.
[3]
Schroedinger Equation for Hydrogen Atom - Rutgers Physics
Schroedinger Equation for Hydrogen Atom¶. The Schroedinger equation is: (−ℏ22m∇2−Ze24πε0r)ψ(→r)=Eψ(→r)(1) (1) ( − ℏ 2 2 m ∇ 2 − Z e 2 4 π ε 0 r ) ψ ( r ...
[4]
[PDF] Lecture 19, Hydrogen Atom - DSpace@MIT
l ( )r → Rnl ( )r in what follows. For the Hydrogen-like atom, then, we want the solutions of: ⎛ −1 ∂. ⎜ r2 ∂. l l( ). +1. Z⎞. 2 2. +. ⎝ r ∂r ∂r. 2r2.
[5]
Exact quasi-relativistic wavefunctions of Hydrogen-like atoms - PMC
Sep 10, 2020 · Wavefunctions of Hydrogen-like atoms, which are obtained by solving the Schrödinger equation, are often used in ab-initio quantum mechanics ...
[6]
The hydrogen atom
The energy levels scale with Z2, i.e. En = -Z2*13.6 eV/n2. It takes more energy to remove an electron from the nucleus, because the attractive force that must ...
[7]
30.3 Bohr's Theory of the Hydrogen Atom – College Physics
... hydrogen-like atom. The spectra of hydrogen-like ions are similar to hydrogen, but shifted to higher energy by the greater attractive force between the ...
[8]
The Hydrogen atom - Chemistry 301
One interesting application of this formula is that we can use it to find the energy required to remove the electron from a hydrogen-like atom. This is the ...
[9]
[PDF] Hydrogen Atom I - Rutgers Physics
Historical importance of the hydrogen atom. • The hydrogen atom was the principle tool that allowed the physicists developing quantum mechanics in the 1920's to ...
[10]
Bohr's Theory of the Hydrogen Atom – ISP209
So, if a nucleus has Z protons ( Z = 1 for hydrogen, 2 for helium, etc.) and only one electron, that atom is called a hydrogen-like atom. The spectra of ...
[11]
[PDF] Center of mass separation A hydrogen-like atom consists of one ...
Center of mass separation. A hydrogen-like atom consists of one nucleus of charge Ze and a single electron of charge –e, so in addition to H we have He+, ...
[12]
[PDF] hydrogen.pdf - University of California, Berkeley
... hydrogen-like atom. (“Hydrogen-like” means. V (r) = −Ze2/r.) Show that if a = ¯h2/µZe2, then d2u dr2+ [−. ℓ(ℓ + 1) r2. +. 2 ar −. 1 n2a2 ]u = 0,. (30) where u ...
[13]
[PDF] THE QUANTUM WORLD OF ATOMS AND MOLECULES
The Schrodinger equation is exactly solvable only for the simplest systems ... hydrogen-like atoms with n-values exceeding 300, and in interstellar ...
[14]
9.3 The Hartree-Fock Approximation
Note that if you put the two nuclei completely on top of each other, you get a helium atom, for which Hartree-Fock gives a much more reasonable electron energy.
[15]
Energy Levels of Hydrogen Atom - Richard Fitzpatrick
The energy eigenvalues are quantized, and can only take the values $\displaystyle E = \frac{E_0}{n^{\,2}},$ where $\displaystyle E_0 = - \frac{m_e\,
[16]
Hydrogenic systems
Problem: The ground state wave function for a hydrogen like atom is. ,. where a0 = ħ2/(μe2) and μ is the reduced mass, μ ~ me = mass of the electron.
[17]
[PDF] III Ionized Hydrogen (HII) Regions - OSU astronomy
HII regions are ionized atomic hydrogen regions, composed of gas ionized by photons with energies above 13.6eV. They include classical HII regions and ...
[18]
The ionization structure of helium in H II region complexes
It is well known that in an H ii region, the helium degree of ionization depends strongly on the effective temperature of the ionizing star, and particularly ...
[19]
Hubble Space Telescope ultraviolet spectroscopy of the hottest ...
It was originally classified as a hot DO white dwarf (WD) with an ... white dwarfs with spectra characterized by absorption lines from ionized helium.1. Introduction · 3. Model Atoms And Model... · 4.2. NeonMissing: like | Show results with:like
[20]
X-ray Spectroscopy of Accretion Disks and Stellar Winds in X-Ray ...
Above that range, ions are fully stripped, but hydrogen-like recombination emission is still possible. Recombination spectra are formed primarily in the ...
[21]
Laser excitation of the 1S–2S transition in singly-ionized helium
Dec 19, 2024 · We report laser excitation of the 1S–2S transition in singly-ionized helium ( 3 He + ), a hydrogen-like ion with much higher sensitivity to QED than hydrogen ...
[22]
Stringent test of QED with hydrogen-like tin | Nature
Oct 4, 2023 · We present state-of-the-art theory calculations, which together test the underlying QED to about 0.012%, yielding a stringent test in the strong-field regime.
[23]
[1207.5134] Hydrogen-like atoms in relativistic QED - arXiv
Jul 21, 2012 · In this review we consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron ...Missing: mechanics testing
[24]
Quantum simulation of high-order harmonic spectra of the hydrogen ...
Feb 3, 2009 · The purpose of this paper is to explore the connections among the three forms (i.e., dipole-moment, dipole-velocity, and dipole-acceleration) of ...
[25]
[PDF] 3. Quantisation as an eigenvalue problem; by E. Schrödinger
With this variation problem we substitute the quantum conditions. First of all, we will take for H the Hamiltonian function of the Keplerian motion and show ...
[26]
[PDF] Quantum Physics III Chapter 2: Hydrogen Fine Structure
Apr 2, 2022 · The mass m in H(0) is the reduced mass of the electron and proton, which we can accurately set equal to the mass of the electron. If one ...
[27]
[PDF] 23. The Hydrogen Atom - Physics
quantum mechanics only to the electron's motion (see Problem 5.1 in Griffiths if ... That energy unit is called the rydberg, and you can distinguish the two ...
[28]
[PDF] An undulatory theory of the mechanics of atoms and molecules - ISY
At first sight this equation seems to offer ill means of solving atomic problems, e.g. of defining discrete energy-levels in the hydrogen atom. Being a ...<|control11|><|separator|>
[29]
Bohr radius - CODATA Value
Concise form, 5.291 772 105 44(82) x 10-11 m ; Click here for correlation coefficient of this constant with other constants ; Source: 2022 CODATA<|separator|>
[30]
A Laguerre Polynomial Orthogonality and the Hydrogen Atom - arXiv
Nov 13, 2000 · The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the ...
[31]
[PDF] The Hydrogen Atom - Galileo
Nov 22, 2006 · we have found the diverging radial wavefunction ( ) u e ρ ρ ... L z , and the associated Laguerre polynomials are given by: ( ) k p. L z.
[32]
[PDF] The Hydrogen Atom: a Review on the Birth of Modern Quantum ...
The purpose of this work is to retrace the steps that were made by scientists of. XXcentury, like Bohr, Schrodinger, Heisenberg, Pauli, Dirac, ...
[33]
[PDF] Dynamical Symmetries of the H Atom, One of the Most ... - arXiv
May 24, 2023 · The parity (-1)l+j-1/2 is conserved in time, so the states are ... For example, The symmetry operators for the degeneracy group in the Schrodinger.
[34]
Quantum Mechanics of One- and Two-Electron Atoms - SpringerLink
The aim of this book is two-fold. First, to act as a reference work on calcu lations pertaining to hydrogen-like and helium-like atoms and their comparison ...
[35]
The Electromagnetic Shift of Energy Levels | Phys. Rev.
The Electromagnetic Shift of Energy Levels, HA Bethe Cornell University, Ithaca, New York, PDF Share, Phys. Rev. 72, 339 – Published 15 August, 1947.Missing: original | Show results with:original
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The quantum theory of the electron - Journals
1928The quantum theory of the electronProc. R. Soc. Lond. A117610–624 ... The quantum theory of the electron. Paul Adrien Maurice Dirac. Google Scholar.
[37]
Fate of Accidental Symmetries of the Relativistic Hydrogen Atom in a ...
Apr 16, 2015 · ... Runge-Lenz vector. In the relativistic hydrogen atom the accidental symmetry is partially lifted. Due to the Johnson-Lippmann operator ...
[38]
Zur Quantentheorie der Spektrallinien - Sommerfeld - 1916
First published: 1916. https://doi.org/10.1002/andp.19163561702. Citations: 286. About. References. Related. Information. PDF · PDF. Tools. Request permission ...
[39]
None
Below is a merged response summarizing the Dirac Equation for the Hydrogen Atom based on the provided segments from Landau & Lifshitz's *Quantum Electrodynamics* (2nd ed., 1982). Since the content does not always explicitly detail the Dirac equation for the hydrogen atom across all sections, I’ve consolidated the most relevant information from the summaries into a comprehensive narrative and a detailed table for clarity. The response retains all mentioned details, including radial wave functions, parameters, n=2 states, hypergeometric functions, and useful URLs.
[40]
[PDF] Solutions of the Dirac Equation and Their Properties
For example, the Dirac equation for the hydrogen atom is exactly solvable, and it gives positive energy solutions in good agreement with experiment, including ...
[41]
A theory of electrons and protons | Proceedings of the Royal Society ...
The relativity quantum theory of an electron moving in a given electromagnetic field, although successful in predicting the spin properties of the electron,
[42]
Computing Dirac's atomic hydrogen wave functions of the continuum ...
May 1, 1991 · Accurate computation of the positive and negative energy wave functions for the continuous parts of the atomic hydrogen spectrum, ...
[43]
Charge Conjugation
. This is a formal symmetry of the Dirac theory. it transforms the Dirac equation for an electron into the same equation for a positron and is called the ...
[44]
[PDF] arXiv:0712.0150v1 [quant-ph] 2 Dec 2007 Klein's Paradox
Dec 2, 2007 · The hydrogen atom is the first problem solved by Robson and Staudte [1, 2]; they found the same bound-state energy as the Dirac equation but ...
[45]
[PDF] LA-UR-18-21550 - OSTI
Feb 27, 2018 · According to the Dirac equation (without including vacuum physics), the hydrogen 2s1/2 and 2p1/2 levels should be degenerate (have the same ...
[46]
Two-photon frequency comb spectroscopy of atomic hydrogen
Nov 27, 2020 · We have performed two-photon ultraviolet direct frequency comb spectroscopy on the 1S-3S transition in atomic hydrogen to illuminate the so-called proton ...<|control11|><|separator|>
[47]
Hydrogen 1s-2s transition frequency: Comparison of experiment and ...
Nov 10, 2022 · Using the Eides procedure and the 2002 CODATA, we find that the value of the experimental frequency lies within the theoretical frequency ...
[48]
Laser Spectroscopy of the Fine-Structure Splitting in the Levels of
Feb 10, 2017 · The fine structure of the 2 P J 3 ( J = 0 , 1, 2) levels of He 4 , with a large splitting interval of 31.9 GHz, was recognized [1] as the best ...Missing: He+ | Show results with:He+
[49]
Lamb Shift Measurements at the ESR Electron Cooler
The aim of this investigation is to measure precisely the two-electron contribution to the ionization potential in He-like uranium of 2.2 keV with an accuracy ...
[50]
The Ground-State Lamb Shift in Hydrogenlike Uranium
This, in combination with the 0° observation geometry, allowed us to determine the ground-state Lamb shift in hydrogenlike uranium ( ) from the observed x-ray ...Missing: GSI | Show results with:GSI
[51]
Testing quantum electrodynamics in extreme fields using helium-like ...
Jan 24, 2024 · Here we present an experiment sensitive to higher-order QED effects and electron–electron interactions in the high-Z regime.
[52]
[2501.11195] The Proton Radius Puzzle and Discrepancies in ...
Jan 19, 2025 · The divergence between results from muonic and electronic hydrogen measurements remains unexplained, with contributions from both experimental ...