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Principal quantum number

The principal quantum number, denoted as n, is a fundamental quantum number in atomic physics that specifies the primary energy level, or shell, of an electron in an atom or ion.^[1] It takes positive integer values starting from 1 (n = 1, 2, 3, \dots), with no upper limit, and determines the average distance of the electron from the nucleus as well as the overall size of the electron's orbital.^[2] Higher values of n correspond to electrons occupying larger orbitals farther from the nucleus, resulting in increased electron energy and reduced binding to the atomic core.^[1] Introduced in Niels Bohr's 1913 model of the hydrogen atom, the principal quantum number originally described the quantized angular momentum and radius of electron orbits, laying the groundwork for understanding atomic spectra.^[3] In the full quantum mechanical framework developed by Erwin Schrödinger in 1926, n emerges as an eigenvalue of the Hamiltonian operator in the Schrödinger equation for the hydrogen-like atom, where the energy levels are given by E_n = -\frac{13.6 \, \text{eV}}{n^2} for hydrogen.^[2] This quantization ensures discrete energy states, explaining phenomena such as atomic emission and absorption lines.^[3] For multi-electron atoms, n still primarily governs the shell structure, though electron-electron interactions cause deviations from the simple $1/n^2 energy dependence seen in hydrogen.^[1] Each shell defined by n can hold up to $2n^2 electrons and contains n subshells, each characterized by a secondary azimuthal quantum number l ranging from 0 to n-1.^[2] The principal quantum number thus plays a central role in electron configuration, periodic table organization, and predicting chemical properties, as electrons in higher n shells are more easily removed, influencing ionization energies and reactivity.^[1]

Fundamentals

Definition and Notation

The principal quantum number, denoted by n, is a positive integer (n = 1, 2, 3, \dots) that specifies the energy level and relative size of an atomic orbital within the quantum mechanical description of atoms.^[1]^[3] It arises as one of the quantum numbers in the solutions to the time-independent Schrödinger equation for hydrogen-like atoms, where it primarily governs the radial extent and principal energy shell of the electron's wave function.^[4] In standard notation, [n](/page/N+) takes on discrete integer values beginning with 1, which corresponds to the ground state of the atom, and extends theoretically without an upper bound, allowing for increasingly higher energy levels.^[5]^[6] Larger values of [n](/page/N+) indicate orbitals that are farther from the nucleus on average and encompass greater spatial volume.^[1] The designation "principal quantum number" traces its origin to early quantum theory, where it served to quantize the primary or principal orbits of electrons in semi-classical atomic models.^[7] This numbering facilitates the organization of atomic states and relates directly to the quantization of energy in multi-electron atoms as well.^[8]

Role in Quantum Mechanics

In quantum mechanics, the principal quantum number n serves as the foundational descriptor for an electron's energy level and radial distribution in an atomic orbital, directly influencing the possible values of the other quantum numbers that specify the electron's state. Specifically, n determines the range of the azimuthal quantum number l, which can take integer values from 0 to n-1, thereby defining the subshell types (s for l=0, p for l=1, etc.).^[9] This constraint on l in turn sets the possible values for the magnetic quantum number m_l, ranging from -l to +l in integer steps, which specifies the orbital's orientation in space.^[1] The spin quantum number m_s, with values of +\frac{1}{2} or -\frac{1}{2}, remains independent of n but completes the set for distinguishing electron spins within the same spatial orbital.^[10] As the primary quantizer, n establishes the atomic shell structure, labeling shells as K (n=1), L (n=2), M (n=3), and so on, with each successive shell accommodating more subshells due to the increasing number of allowed l values.^[11] For instance, the K shell (n=1) has only one subshell (l=0), while the N shell (n=4) supports four subshells (l=0,1,2,3), enabling greater complexity in electron arrangements as n increases.^[1] This hierarchical organization underpins the filling of atomic orbitals according to the aufbau principle, where electrons occupy shells in order of rising n. In multi-electron atoms, the full specification of an electron's quantum state requires the quartet n, l, m_l, and m_s, ensuring compliance with the Pauli exclusion principle that prohibits two electrons from sharing identical quantum numbers.^[10] This complete set uniquely designates each electron's position and properties within the atom, facilitating the prediction of chemical behavior and spectral lines.^[9]

Historical Development

Bohr Model Introduction

In 1913, Niels Bohr developed a seminal atomic model for the hydrogen atom, introducing the concept of quantized electron orbits to explain the discrete spectral lines observed in its emission spectrum. This model built on Ernest Rutherford's nuclear atom by incorporating elements of Planck's quantum theory, proposing that electrons could occupy stable, non-radiating "stationary states" around the nucleus.^[12] Bohr's approach resolved a key paradox in classical physics, where orbiting electrons would continuously radiate energy and spiral into the nucleus, by restricting motion to specific discrete paths.^[13] A central postulate of Bohr's model was the quantization of the electron's angular momentum in these stationary states, expressed as mvr = n \hbar, where m is the electron mass, v its velocity, r the orbital radius, n a positive integer, and \hbar = h / 2\pi with h as Planck's constant.^[13] Here, n, later termed the principal quantum number, labels the allowed orbits with n = 1, 2, [3, \dots](/page/3_Dots), corresponding to circular paths of increasing radius and energy.^[12] This quantization condition ensured that only certain angular momenta—multiples of \hbar—were permissible, preventing continuous energy loss and stabilizing the atom.^[13] Bohr further postulated that radiation is emitted or absorbed only when the electron transitions between these quantized states, with the emitted photon's frequency given by the energy difference between levels.^[12] Applying the angular momentum quantization to the hydrogen atom, Bohr derived the energy of the n-th state as E_n = -\frac{13.6 \, \mathrm{eV}}{n^2}, where the negative value signifies bound states relative to the ionization limit at E = 0.^[14] This expression directly connects n to both the orbital radius, which scales as n^2, and the electron velocity, which scales as $1/n, providing a quantitative basis for the model's predictions of hydrogen's spectral series.^[15] The ground state at n=1 yields E_1 = -13.6 \, \mathrm{eV}, matching experimental ionization energies.^[14]

Quantum Mechanical Refinement

In 1916, Arnold Sommerfeld extended Niels Bohr's atomic model by incorporating elliptical electron orbits to account for the fine structure observed in hydrogen spectral lines. This refinement introduced a subsidiary quantum number, now known as the azimuthal quantum number l, which parameterized the orbit's eccentricity and angular momentum, while preserving the principal quantum number n as the key determinant of the semi-major axis and overall energy quantization. Sommerfeld's approach retained Bohr's integer values for n but allowed for relativistic corrections, enhancing the model's explanatory power for atomic spectra without altering n's fundamental status.^[16] The transition to modern quantum mechanics in the mid-1920s further refined the principal quantum number. In 1925, Werner Heisenberg formulated matrix mechanics, a non-commutative algebraic framework for quantum phenomena, in which n emerges naturally as the index labeling the discrete energy eigenvalues for stationary states in the hydrogen atom, generalizing Bohr's ad hoc integer to a rigorous spectral label derived from matrix diagonalization. Complementing this, Erwin Schrödinger's 1926 wave mechanics solved the time-independent Schrödinger equation for the hydrogen atom, yielding wavefunctions where n denotes the quantum number associated with energy levels, appearing as an integer eigenvalue that quantizes the radial extent and energy, thus embedding n within a continuous wave description of electron probability distributions.^[17]^[18] Discussions at the 1927 Solvay Conference on Electrons and Photons, attended by luminaries including Bohr, Heisenberg, Schrödinger, and Einstein, played a pivotal role in consolidating the principal quantum number's place in non-relativistic quantum theory. Participants debated the interpretive foundations of the new mechanics, affirming n's centrality in describing discrete energy states and transitions, thereby bridging old quantum ideas with the matrix and wave formulations and establishing n as an invariant feature of the emerging quantum paradigm.

Theoretical Derivation

Semiclassical Approach

The Bohr-Sommerfeld quantization rule provides a semiclassical framework for deriving the principal quantum number n by imposing quantization conditions on classical periodic orbits, serving as an intermediate step between classical mechanics and full quantum mechanics. This rule generalizes Bohr's original angular momentum quantization to more general motions using action-angle variables. For a system with periodic motion, the action integral over one complete cycle must satisfy

\oint \mathbf{p} \cdot d\mathbf{q} = n h,

where \mathbf{p} is the canonical momentum, d\mathbf{q} is the infinitesimal displacement in configuration space, n is a positive integer known as the principal quantum number, and h is Planck's constant. This condition ensures that the phase space area enclosed by the orbit is an integer multiple of h, quantizing the allowed states and introducing discreteness into the otherwise continuous classical trajectories.^[19] In its application to the hydrogen atom, the Bohr-Sommerfeld rule builds directly on Bohr's simpler circular orbit model. For circular orbits, the quantization of angular momentum yields m v r = n \hbar, where m is the electron mass, v is its speed, r is the orbital radius, and \hbar = h / 2\pi. Solving the classical equations of motion for the Coulomb potential V(r) = -e^2 / r (in atomic units) together with this condition gives discrete orbital radii r_n \propto n^2, with the constant of proportionality being the Bohr radius a_0 = \hbar^2 / (m e^2). Thus, r_n = n^2 a_0. Sommerfeld extended this to elliptical orbits by introducing separate quantization for radial and angular actions, J_r = n_r h and J_\phi = l h, where n_r and l are radial and azimuthal quantum numbers, respectively, leading to the principal quantum number n = n_r + l + 1. This derivation reproduces the discrete shell structure of atomic orbitals, with larger n corresponding to more distant, higher-energy orbits.^[19] Despite its successes, the semiclassical approach has inherent limitations, particularly for small values of n. It provides accurate energy levels for the hydrogen atom across all n due to the special properties of the $1/r potential, but the underlying orbital descriptions break down for n=1, where the ground state (l=0) has no classical analog—the electron cannot classically orbit without angular momentum without spiraling into the nucleus. The method neglects zero-point energy effects from quantum fluctuations, which in full quantum mechanics enforce a non-zero minimum kinetic energy and prevent collapse, rendering the classical turning points invalid near r=0 for low n and low l. This approximation improves for large n, where de Broglie wavelengths are small compared to orbital scales, allowing classical-like behavior.

Wave Mechanical Derivation

The wave mechanical derivation of the principal quantum number originates from solving the time-independent Schrödinger equation for the hydrogen atom, which describes the stationary states of the electron in the Coulomb potential. The equation is given by

-\frac{\hbar^2}{2\mu} \nabla^2 \psi + V \psi = E \psi,

where \psi(\mathbf{r}) is the wave function, \mu is the reduced mass of the electron-proton system, E is the energy eigenvalue, and V(r) = -\frac{e^2}{r} is the attractive Coulomb potential (in cgs units).^[6]^[20] This partial differential equation governs the bound states with E < 0, and its solutions yield the quantized energy levels indexed by the principal quantum number n. To solve this, the wave function is separated in spherical coordinates (r, \theta, \phi), assuming \psi(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi), due to the spherical symmetry of the potential. Substituting into the Schrödinger equation and dividing by R \Theta \Phi leads to three ordinary differential equations: one for the azimuthal part \Phi(\phi), which yields the magnetic quantum number m, one for the polar part \Theta(\theta), which combines with \Phi to form spherical harmonics Y_{l m}(\theta, \phi) introducing the orbital quantum number l, and the radial equation for R(r). The separation constants enforce l(l+1) in the centrifugal term.^[21]^[6]^[22] The radial equation takes the form

\frac{d}{dr} \left( r^2 \frac{dR}{dr} \right) + \left[ \frac{2\mu r^2}{\hbar^2} (E - V(r)) - l(l+1) \right] R = 0,

or, with the substitution u(r) = r R(r),

-\frac{\hbar^2}{2\mu} \frac{d^2 u}{dr^2} + \left[ V(r) + \frac{\hbar^2 l(l+1)}{2\mu r^2} \right] u = E u.

For the Coulomb potential, a change of variables \rho = \frac{2}{\ a_0 n} r (where a_0 relates to the Bohr radius) transforms this into the associated Laguerre differential equation. The solutions involve associated Laguerre polynomials L_{n_r}^{2l+1}(\rho), where n_r is a secondary index.^[6]^[22]^[20] The principal quantum number n emerges from the boundary conditions requiring the wave function to remain finite at r = 0 and normalizable as r \to \infty. Near r = 0, the solution behaves as R(r) \sim r^l to cancel the centrifugal singularity, while at large r, an exponential decay e^{-\rho/2} ensures square-integrability for bound states. The power series solution for the radial function terminates only if the coefficient of the recursive relation vanishes, imposing n_r = n - l - 1 where n = 1, 2, 3, \dots and n > l. This termination condition quantizes the allowed states, with n serving as the principal quantum number that indexes the energy eigenvalues.^[21]^[6]^[22]

Physical Properties

Energy Levels

In the quantum mechanical treatment of the hydrogen atom, the bound-state energy levels depend exclusively on the principal quantum number n, which takes positive integer values starting from 1. The energy E_n for the electron in a hydrogen atom (nuclear charge Z = 1) is given by

E_n = -\frac{13.6 \, \text{eV}}{n^2},

where the negative sign indicates bound states relative to the ionization threshold at E = 0. This expression arises from the exact solution of the time-independent Schrödinger equation for the Coulomb potential and holds independently of the azimuthal quantum number l and the magnetic quantum number m_l, leading to degeneracy in those quantum numbers for each n. The ground state corresponds to n = 1, with E_1 = -13.6 \, \text{eV}, while higher n values yield less negative energies, approaching zero as n \to \infty.^[23]^[24] For hydrogen-like ions, consisting of a nucleus of charge Z e and a single electron, the energy levels scale with the square of the nuclear charge due to the strengthened Coulomb attraction in the potential V(r) = -\frac{Z e^2}{4 \pi \epsilon_0 r}. The generalized formula becomes

E_n = -\frac{13.6 \, \text{eV} \, Z^2}{n^2},

again independent of l and m_l, with n = 1 as the ground state. For example, in singly ionized helium (Z = 2), the ground-state energy is -54.4 \, \text{eV}, four times deeper than in hydrogen. This scaling applies to all one-electron systems, such as He^+, Li^{2+}, and higher-Z ions.^[25] While the non-relativistic formula provides the dominant energy dependence on n, relativistic effects introduce fine-structure corrections that modify the levels weakly, with shifts scaling inversely with powers of n beyond $1/n^2 (such as $1/n^3 for the leading term). These adjustments, arising from the Dirac equation's treatment of electron spin and relativistic kinematics, partially lift the l-degeneracy but retain an overall primary dependence on n.^[26]

Orbital Characteristics

The principal quantum number n fundamentally governs the spatial extent of atomic orbitals in hydrogen-like atoms. The average radial distance of the electron from the nucleus, given by the expectation value \langle r \rangle, scales as \langle r \rangle \propto n^2 a_0, where a_0 is the Bohr radius (approximately 0.529 Å). This quadratic dependence means that orbitals with higher n form larger electron shells, with the size increasing dramatically—for instance, the average radius for n=2 is roughly four times that for n=1.^[27] A key structural feature tied to n is the number of radial nodes, which are spherical surfaces where the radial probability density vanishes. The number of these nodes is exactly n - l - 1, with l being the azimuthal quantum number (ranging from 0 to n-1). Radial nodes divide the orbital into regions of alternating probability, affecting how the electron density is distributed along the radius; for example, the 3p orbital (n=3, l=1) has one radial node, creating a more complex probability profile than the node-free 1s orbital./Quantum_Mechanics/09._The_Hydrogen_Atom/Radial_Nodes) Higher values of n result in more diffuse orbitals, where the electron probability is spread over a greater volume, leading to less concentrated density near the nucleus overall. For s-states (l=0), the radial wavefunction remains non-zero at the origin, enabling penetration to the nucleus regardless of n, though the increased diffuseness for larger n shifts much of the probability to greater distances./Book%3A_University_Physics_III_-Optics_and_Modern_Physics(OpenStax)/08%3A_Atomic_Structure/8.02%3A_The_Hydrogen_Atom)

Applications and Implications

In Atomic Spectra

The principal quantum number n governs the discrete energy levels in the hydrogen atom, enabling the prediction and interpretation of its atomic emission and absorption spectra through electron transitions between these levels. When an electron jumps from a higher level n_2 to a lower level n_1 (with n_2 > n_1), it emits a photon with energy equal to the difference between the levels, producing sharp spectral lines; absorption occurs in the reverse process. These transitions, where \Delta n \neq 0, follow the empirical Rydberg formula for the wavenumber:

\frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right),

where \lambda is the wavelength and R is the Rydberg constant for hydrogen (R \approx 1.097 \times 10^7 \, \mathrm{m}^{-1})./08%3A_Central_Potentials/8.04%3A_Rydberg_Formula)^[28] This formula, originally empirical, was theoretically justified by Niels Bohr in 1913 as arising from quantized stationary states labeled by n.^[28] The formula organizes the hydrogen spectrum into distinct series, each corresponding to a fixed lower level n_1 and varying higher levels n_2 > n_1. The Balmer series (n_1 = 2) produces visible lines, such as the red H\alpha line at 656.3 nm (n_2 = 3 \to 2) and blue H\beta at 486.1 nm (n_2 = 4 \to 2), first empirically described by Johann Balmer in 1885 using measurements of hydrogen's visible emission lines.^[29] The Lyman series (n_1 = 1), observed by Theodore Lyman from 1906 to 1914, lies in the ultraviolet region, with the strongest line at 121.6 nm (n_2 = 2 \to 1).^[30] The Paschen series (n_1 = 3) appears in the infrared, starting at 1875 nm (n_2 = 4 \to 3). Additional series like Brackett (n_1 = 4) and Pfund (n_1 = 5) extend further into the infrared. These series converge as n_2 increases, reflecting the increasing density of energy levels with higher n./08%3A_Central_Potentials/8.04%3A_Rydberg_Formula) As n_2 \to \infty, the energy difference approaches the ionization energy from the lower level, yielding the series limit where $1/\lambda = R / n_1^2; beyond this, the spectrum becomes a continuum corresponding to photoionization, with no discrete lines./Quantum_Mechanics/09._The_Hydrogen_Atom/Bohr's_Hydrogen_Atom) For the Lyman series, this limit is at 91.2 nm, marking the onset of the ultraviolet continuum absorption. Bohr's model predicted these limits theoretically, linking them to the zero-point energy as n \to \infty.^[28] Experimental verification of n-based predictions came from early spectroscopic observations, where hydrogen's discharge tube emissions matched the Rydberg formula's line positions precisely. Balmer's 1885 analysis of visible lines, later extended by Rydberg's 1888 generalization, aligned with Bohr's 1913 quantum interpretation, confirming the role of integer n values in producing the observed discrete spectrum without ad hoc assumptions.^[29]^[28] Modern precision measurements, such as those using diffraction gratings, continue to validate these predictions to high accuracy./08%3A_Central_Potentials/8.04%3A_Rydberg_Formula)

In Multi-Electron Systems

In multi-electron atoms, the principal quantum number n continues to define the primary energy shells, but electron-electron interactions significantly modify the simple hydrogen-like ordering. The Aufbau principle governs the sequential filling of orbitals, starting with the lowest energy levels, where orbitals are grouped by increasing n and, within each shell, by azimuthal quantum number l. For example, the ground-state configuration of carbon (Z=6) is $1s^2 2s^2 2p^2, filling the n=1 shell completely before occupying n=2. This principle, formalized in the early quantum era, ensures that electrons occupy subshells in order of rising n + l values, with ties resolved by lower n first, as per Madelung's empirical ordering rule. The maximum electron capacity of each shell is given by $2n^2, arising from the degeneracy of orbitals: for a given n, there are n possible l values (0 to n-1), each subshell holds $2(2l+1) electrons due to magnetic and spin quantum numbers, summing to $2n^2 total states per Pauli exclusion.^[1]/02:_Atomic_Structure/2.02:_The_Schrodinger_equation_particle_in_a_box_and_atomic_wavefunctions/2.2.03:_Aufbau_Principle) A key modification in multi-electron systems is the screening effect, where inner-shell electrons shield outer electrons from the full nuclear charge Z, reducing the effective nuclear charge Z_{\text{eff}} experienced by valence electrons in higher n shells. This shielding lowers the energy of outer orbitals relative to hydrogen-like expectations, compressing energy levels and influencing filling order. For an electron in shell n, Z_{\text{eff}} = Z - \sigma, where \sigma is the screening constant accounting for contributions from other electrons; electrons in the same shell screen less effectively (about 0.35) than those in inner shells (up to 0.85 or 1.00). Slater's rules provide a practical approximation for \sigma, grouping electrons into l units and assigning shielding factors based on their principal quantum numbers relative to the electron of interest—for instance, all electrons in shells with n-1 contribute 0.85 to \sigma for s or p electrons. This effective charge decrease stabilizes higher n shells less than expected, altering inter-shell spacings.^[31] Despite these guidelines, exceptions to the Aufbau filling order occur in transition metals, where the proximity in energy between ns and (n-1)d orbitals—due to partial screening and relativistic effects—favors configurations with half-filled or fully filled subshells for enhanced stability. Chromium (Z=24) adopts [\ce{Ar}] 4s^1 3d^5 instead of the expected [\ce{Ar}] 4s^2 3d^4, as the half-filled $3d^5 (lower n=3) provides greater exchange energy and symmetry despite the higher principal shell for $4s (n=4). Similarly, copper (Z=29) has [\ce{Ar}] 4s^1 3d^{10} over [\ce{Ar}] 4s^2 3d^9, prioritizing the closed $3d^{10} subshell for minimized unpaired electrons and increased stability. These anomalies highlight how n-dependent radial distributions influence orbital penetration and overall electronic stability in multi-electron atoms.^[32]

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