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Magnetic quantum number

The magnetic quantum number, denoted as m_\ell, is a fundamental quantum number in atomic physics that specifies the orientation in space of an electron's orbital relative to an applied magnetic field. It serves as the third quantum number in the set used to describe the unique state of an electron within an atom, following the principal quantum number n (which determines the energy level and size of the orbital) and the azimuthal quantum number \ell (which defines the shape and subshell of the orbital). The possible values of m_\ell range from -\ell to +\ell in integer steps, including zero, yielding $2\ell + 1 distinct orientations for each subshell. This quantum number arises from the solutions to the Schrödinger equation for the hydrogen atom and is crucial for understanding phenomena such as the Zeeman effect, where atomic spectral lines split in a magnetic field due to differing orbital orientations. In multi-electron atoms, the combination of n, \ell, and m_\ell uniquely identifies each atomic orbital, with the fourth quantum number, the spin magnetic quantum number m_s, further distinguishing electron spins within those orbitals.

Definition and Fundamentals

Definition and Role

The magnetic quantum number, denoted as m_\ell, specifies the of the vector along a chosen z-axis in a , describing the orientation of an electron's orbital in space. This arises in the context of atomic orbitals and is termed "magnetic" due to its influence on energy levels observed in external . In , m_\ell plays a crucial role by distinguishing orbitals that share the same n (which sets the ) and \ell (which defines the orbital's shape), but differ in their spatial orientation relative to the axis. For instance, the three p orbitals (\ell = 1) are differentiated by their m_\ell values, corresponding to lobes aligned along the x, y, or z directions. Along with n and \ell, m_\ell specifies the in hydrogen-like atoms, where the spatial depends on these numbers to describe the ; the full of an also requires the m_s. The \ell ranges from 0 to n-1, providing the framework for m_\ell's allowable range. The possible values of m_\ell are the integers m_\ell = -\ell, -\ell+1, \dots, 0, \dots, \ell-1, \ell, yielding $2\ell + 1 distinct orientations for each subshell.

Possible Values and Notation

The magnetic quantum number, denoted as m_\ell, takes on integer values ranging from -\ell to +\ell in integer steps, where \ell is the . This discrete set yields $2\ell + 1 possible values for each \ell. In standard notation, m_\ell specifically refers to the orbital magnetic quantum number, while the generic symbol m is sometimes used in broader contexts or to encompass both orbital and spin projections. For illustration, when \ell = 0 (s orbital), the only possible value is m_\ell = 0; for \ell = 1 (p orbital), the values are m_\ell = -1, 0, +1; and for \ell = 2 (d orbital), they are m_\ell = -2, -1, 0, +1, +2. These $2\ell + 1 states are degenerate, sharing the same energy in the absence of external perturbations that break .

Theoretical Derivation

From the

The time-independent for the , describing a single in the Coulomb potential of the proton, is solved in spherical coordinates (r, \theta, \phi) to account for the central symmetry of the system. The wave function \psi(r, \theta, \phi) is assumed to be separable as \psi(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi), where R(r) handles the radial dependence, \Theta(\theta) the polar angle, and \Phi(\phi) the azimuthal angle. Substituting this form into the and dividing by R(r) \Theta(\theta) \Phi(\phi) yields separate equations for each coordinate after introducing separation constants. The angular part of the Laplacian operator in spherical coordinates leads to coupled equations for \Theta(\theta) and \Phi(\phi). Specifically, the \phi-dependent term separates out, resulting in the azimuthal equation: \frac{d^2 \Phi}{d\phi^2} + m^2 \Phi = 0, where m^2 is the separation constant. The general solution to this is \Phi(\phi) = A e^{i m \phi} + B e^{-i m \phi}, which can be combined into \Phi(\phi) = \frac{1}{\sqrt{2\pi}} e^{i m \phi} for over the [0, 2\pi). For the wave function to be single-valued and physically meaningful, \Phi(\phi) must satisfy periodic boundary conditions: \Phi(\phi + 2\pi) = \Phi(\phi). This requires e^{i m 2\pi} = 1, implying that m must be an integer, denoted as the magnetic quantum number m_l. Thus, m_l = 0, \pm 1, \pm 2, \dots, quantizing the azimuthal dependence of the electron's wave function. The \theta-dependent equation, after incorporating m_l^2, takes the form of an associated Legendre equation, introducing another separation constant l(l+1), where l is a non-negative integer (the orbital quantum number). For the solutions to be regular and finite at \theta = 0 and \theta = \pi, the condition |m_l| \leq l must hold. This bounds the possible values of the magnetic quantum number for a given l. The radial equation, meanwhile, yields the principal quantum number n through boundary conditions on R(r), but does not directly affect the quantization of m_l.

Connection to Spherical Harmonics

The Y_l^m(\theta, \phi) constitute the complete set of solutions to the angular portion of the for a particle in a central potential, where the indices l and m are the orbital and magnetic quantum numbers, respectively. The magnetic quantum number m (often denoted m_l) specifically governs the azimuthal dependence of these functions, reflecting the quantization of the z-component of through the periodicity in the azimuthal \phi; the e^{i m \phi} ensures single-valuedness on , requiring m to be an ranging from -l to +l. The explicit structure of the spherical harmonics separates into a θ-dependent part involving associated Legendre polynomials and a φ-dependent exponential term: Y_l^m(\theta, \phi) = (-1)^m \sqrt{ \frac{(2l+1)(l - m)!}{4\pi (l + m)!} } \, P_l^{|m|}(\cos \theta) \, e^{i m \phi} for m \geq 0, with the associated Legendre functions P_l^{|m|}(x) defined as P_l^{|m|}(x) = (-1)^{|m|} (1 - x^2)^{|m|/2} \frac{d^{|m|}}{dx^{|m|}} P_l(x), where P_l(x) are the Legendre polynomials; for negative m, Y_l^m(\theta, \phi) = (-1)^m [Y_l^{-m}(\theta, \phi)]^*. This form arises naturally from the separation of variables in spherical coordinates, with the e^{i m \phi} factor directly encoding the eigenvalue m \hbar for the operator L_z = -i \hbar \frac{\partial}{\partial \phi}./07%3A_Orbital_Angular_Momentum/7.06%3A_Spherical_Harmonics) The normalization constant in the expression ensures that the functions are normalized over the unit sphere, while the phase factor (-1)^m for m > 0 adopts the Condon-Shortley convention, facilitating real combinations for atomic orbitals. The overall parity of Y_l^m(\theta, \phi) under spatial inversion (\theta, \phi) \to (\pi - \theta, \phi + \pi) is (-1)^l, independent of m, which determines whether the function is even or odd and influences selection rules in quantum transitions. The spherical harmonics form an orthonormal basis on the sphere, satisfying \int_0^{2\pi} d\phi \int_0^\pi \sin \theta \, d\theta \, Y_{l'}^{m'*}(\theta, \phi) Y_l^m(\theta, \phi) = \delta_{l l'} \delta_{m m'}, which guarantees that states with different m_l values are orthogonal and thus distinguishable in measurements of the z-component of angular momentum. This orthogonality, stemming from the completeness of the Legendre polynomials in θ and the Fourier basis in φ, underscores the distinct physical roles of each m_l state within a given l subshell./07%3A_Orbital_Angular_Momentum/7.06%3A_Spherical_Harmonics)

Relation to Angular Momentum

Orbital Angular Momentum Components

The orbital operator in is defined as the vector operator \mathbf{L} = -i \hbar \mathbf{r} \times \nabla, with Cartesian components L_x = -i \hbar (y \partial_z - z \partial_y), L_y = -i \hbar (z \partial_x - x \partial_z), and L_z = -i \hbar (x \partial_y - y \partial_x). These operators obey the commutation relations [L_x, L_y] = i \hbar L_z and cyclic permutations thereof, mirroring the algebra of classical . The z-component operator L_z admits the spherical harmonics Y_l^m(\theta, \phi) as eigenfunctions, satisfying the eigenvalue equation L_z Y_l^m = m_l \hbar Y_l^m, where m_l is the magnetic quantum number ranging from -l to +l in integer steps, and m_l \hbar quantifies the magnitude of the orbital angular momentum projection along the z-axis. The squared angular momentum operator L^2 = L_x^2 + L_y^2 + L_z^2 commutes with L_z, sharing the same eigenfunctions Y_l^m, and yields the eigenvalue equation L^2 Y_l^m = l(l+1) \hbar^2 Y_l^m, where l = 0, 1, 2, \dots is the orbital quantum number determining the total angular momentum scale. In such states, the angular momentum vector has fixed magnitude \sqrt{l(l+1)} \hbar and precesses around the z-axis, with its tip tracing a cone due to the nonzero projection m_l \hbar. The noncommutativity of the components implies that Y_l^m states specify L_z precisely but leave L_x and L_y uncertain, with the z-component alone fully determined while the transverse components exhibit spreads \Delta L_x \approx \hbar \sqrt{l(l+1) - m_l^2} (and similarly for \Delta L_y). This quantization arises because simultaneous eigenstates of L^2 and all three components do not exist except for l=0.

Magnetic Moment Association

The orbital magnetic moment \boldsymbol{\mu}_l of an electron arises from its orbital angular momentum \mathbf{L} and is given by \boldsymbol{\mu}_l = -\frac{e}{2m_e} \mathbf{L}, where e is the elementary charge and m_e is the electron mass. This relation reflects the classical analogy of a current loop generated by the electron's orbital motion, quantized in quantum mechanics. The z-component of this magnetic moment, \mu_{l,z}, is quantized and equals -\frac{e \hbar}{2m_e} m_l = -\mu_B m_l, where \mu_B = \frac{e \hbar}{2m_e} is the and m_l is the magnetic quantum number. This projection determines the component of \boldsymbol{\mu}_l along the quantization axis. The Landé g-factor for the orbital contribution is g_l = 1, in contrast to the spin g-factor g_s \approx 2, which accounts for the different origins of orbital and spin magnetic moments. In the vector model of the atom, the orbital magnetic moment \boldsymbol{\mu}_l is antiparallel to \mathbf{L} and precesses around the direction of \mathbf{L} due to the quantum uncertainty in the transverse components. However, in weak external magnetic fields, the z-projection of \boldsymbol{\mu}_l aligns with the value determined by m_l, stabilizing the observable component along the field direction. For multi-electron atoms, the total orbital magnetic moment is the vector sum over the individual orbital contributions from all electrons, \boldsymbol{\mu}_{l,\text{total}} = \sum_i \boldsymbol{\mu}_{l,i}, which couples with spin moments to yield the overall atomic magnetic moment. This summation follows the rules of angular momentum addition in the Russell-Saunders coupling scheme.

Effects in External Fields

Zeeman Splitting

The normal describes the splitting of atomic energy levels and corresponding spectral lines when an atom is placed in a weak external , arising from the interaction between the electron's orbital and the field. This phenomenon, first observed experimentally by in 1896, is explained in through applied to the unperturbed atomic . In the presence of a magnetic field \vec{B} aligned along the z-axis, the perturbation Hamiltonian for the orbital contribution is given by H' = -\vec{\mu}_l \cdot \vec{B} = \frac{\mu_B B}{\hbar} L_z, where \mu_B = \frac{e \hbar}{2 m_e} is the Bohr magneton, e is the elementary charge, m_e is the electron mass, and L_z is the z-component of the orbital angular momentum operator. Using first-order degenerate perturbation theory, the energy shift for a state characterized by the orbital quantum number l and magnetic quantum number m_l is \Delta E = \mu_B B m_l, where m_l = -l, -l+1, \dots, l. This results in the splitting of each degenerate (2l+1)-fold level into $2l+1 equally spaced sublevels separated by \mu_B B, with the orbital magnetic moment \mu_{l,z} = -\frac{\mu_B}{\hbar} L_z = -\mu_B m_l determining the direction and magnitude of the shift. For optical transitions between these split levels, the electric dipole approximation yields selection rules \Delta m_l = 0 (for \pi polarization, parallel to \vec{B}) and \Delta m_l = \pm 1 (for \sigma polarization, perpendicular to \vec{B}), leading to three equally spaced emission or absorption lines per original spectral line in the normal case. These rules arise from the matrix elements of the dipole operator in the basis of spherical harmonics, conserving angular momentum projection along the field direction. This description holds for weak magnetic fields where \mu_B B is much smaller than the fine-structure splitting, ensuring the perturbation is small and spin-orbit coupling dominates over the Zeeman interaction; in stronger fields or when electron is included, the anomalous Zeeman effect emerges with additional complexity. The normal effect is particularly observable in transitions between states (total S=0), where spin contributions vanish.

Stark Effect Interactions

The Stark effect describes the perturbation of atomic energy levels by an external , where the magnetic quantum number m_l plays a crucial role in determining the selection rules for state mixing and the resulting energy shifts. In the presence of a uniform \mathbf{E} along the z-axis, the perturbation is H' = - \mathbf{d} \cdot \mathbf{E} = e E z, with z = r \cos \theta, which preserves the azimuthal symmetry around the field direction. This conservation arises because the operator z commutes with L_z, the z-component of , ensuring that elements \langle n', l', m_l' | H' | n, l, m_l \rangle vanish unless \Delta m_l = m_l' - m_l = 0. In hydrogen atoms, the linear dominates for excited states due to the degeneracy of levels with the same n but different orbital quantum numbers l. The couples states within the same n manifold that have the same m_l but differing l (specifically \Delta l = \pm 1), leading to energy corrections proportional to the field strength E. For example, in the n=2 manifold, the states with m_l = 0 (a of $2s and $2p_{z}) mix, resulting in energy shifts of \pm 3 e E a_0, where a_0 is the , while the m_l = \pm 1 states (pure $2p_{\pm}) experience no shift and remain degenerate. This splitting is determined by the non-zero elements, such as \langle 2s | z | 2p_{m_l=0} \rangle = -3 a_0, which explicitly depend on the conserved m_l value. For non-hydrogenic atoms, where energy levels with the same n but different l are non-degenerate due to electron-electron interactions, the Stark effect is quadratic in E. The second-order energy shift for a state |n, l, m_l \rangle is given by \Delta E^{(2)} = e^2 E^2 \sum_{n' \neq n, l', m_l' = m_l} \frac{ |\langle n', l', m_l | z | n, l, m_l \rangle|^2 }{ E_{n l} - E_{n' l'} }, where the sum runs over unperturbed states with the same m_l, reflecting the \Delta m_l = 0 selection rule from the off-diagonal elements of the perturbation. This dependence on m_l arises because the connected states must share the same projection of angular momentum, leading to sublevel-specific shifts; for instance, states with higher |m_l| often exhibit smaller polarizabilities and thus reduced quadratic shifts due to fewer accessible mixing channels. In ground states like hydrogen's $1s (n=1, l=0, m_l=0), the shift is \Delta E = -\frac{9}{4} a_0^3 E^2 (in atomic units), uniform across m_l since l=0. The conservation of m_l in the Stark effect stems from the cylindrical symmetry imposed by the electric field, which reduces the full spherical symmetry of the atom to axial symmetry around the field axis, leaving L_z as a good quantum number. This symmetry dictates that perturbations like H' cannot mix sublevels with different m_l, ensuring that the azimuthal quantum number governs the field's interaction without altering the orbital angular momentum projection.

Historical and Experimental Context

Discovery and Development

The discovery of the Zeeman effect in 1896 by Pieter Zeeman provided early experimental evidence for quantized angular momentum projections, as the splitting of spectral lines in a magnetic field indicated discrete orientations of atomic orbits relative to the field direction. This observation, initially puzzling within classical electromagnetism, motivated subsequent theoretical efforts to quantify the component of angular momentum along the magnetic field axis, laying the groundwork for the magnetic quantum number concept. Zeeman's work, conducted at Leiden University, revealed a triplet structure in sodium D-lines under magnetic influence, suggesting a quantized "magnetic" aspect to electron orbits. In the old quantum theory, Arnold Sommerfeld advanced this idea through his 1916 relativistic extension of Niels Bohr's atomic model, replacing circular orbits with elliptical ones to account for fine structure in spectra. Sommerfeld introduced two additional quantum conditions: one for the azimuthal motion, yielding the quantum number k (precursor to the orbital angular momentum quantum number l), and another for the projection along a preferred axis, introducing the magnetic quantum number m (ranging from -k to +k) to describe the tilt of the orbital plane. This azimuthal quantization resolved inconsistencies in explaining the Zeeman effect within the Bohr-Sommerfeld framework, treating angular momentum as a precessing vector with discrete components. Sommerfeld's model, while semi-classical, marked the first systematic inclusion of a projection quantum number, influencing later quantum developments. The transition to full accelerated with Werner Heisenberg's 1925 formulation of , which reformulated atomic dynamics using non-commuting arrays for and , inherently implying quantized projections through commutation relations. Although Heisenberg's initial paper focused on transition amplitudes rather than explicit , the framework, developed with and , naturally accommodated discrete eigenvalues for components like L_z, aligning with the magnetic quantum number's role in spectral selection rules. This approach shifted from classical vector models to , providing a basis for understanding Zeeman splitting without assumptions. Erwin Schrödinger's 1926 wave mechanics completed the non-relativistic formalization, deriving the magnetic quantum number m_l (from -l to +l) through separation of the time-independent in spherical coordinates, where the \phi-dependent part yields e^{im_l\phi} solutions with m_l to ensure single-valued wavefunctions. This eigenvalue problem for the explicitly quantized the z-component of orbital as m_l \hbar, bridging wave and matrix formulations via the . Paul Dirac's 1928 relativistic further integrated m_l into the formula, combining it with projections to explain anomalous Zeeman effects and spectral doublets, solidifying the magnetic quantum number's place in modern . This progression from Sommerfeld's semi-classical hints to Dirac's synthesis resolved longstanding puzzles like the Zeeman triplet, establishing a rigorous quantum description of .

Experimental Observations

The , first observed in 1896 by through the broadening and subsequent splitting of sodium D-line spectral emissions in a , provided initial for the quantization of projections. Zeeman's revealed line components shifted by amounts proportional to the field strength, later resolved into triplets for certain transitions, consistent with the classical Lorentz model predicting energy shifts of \pm \mu_B B and zero for the \pi component, where \mu_B is the and B the field. In 1897, Thomas Preston's photographic observations of more intricate multiplet patterns in lines from elements like and highlighted the anomalous , where splittings deviated from simple triplets but still aligned with 2l+1 sublevels upon quantum , directly matching the possible values of the magnetic quantum number m_l = -l, \dots, +l. Hendrik Lorentz's contemporaneous theoretical attributed these shifts to the of orbital currents, laying the groundwork for associating the observed components with discrete m_l orientations, a prediction verified in subsequent high-resolution of alkali and alkaline-earth atoms. Atomic beam deflection experiments, exemplified by the 1922 Stern-Gerlach setup with silver atoms, demonstrated space quantization by splitting the beam into discrete paths corresponding to quantized projections of , originally proposed to test orbital contributions but revealing electron spin quantization with m_s = \pm 1/2. Although the ground-state silver atoms ($5s^1) lacked orbital angular momentum (l=0), the technique's principle extended to analogous tests for orbital m_l via resonance methods in the 1940s, where (EPR) spectra of transition metal complexes showed transitions between sublevels influenced by orbital contributions to the g-tensor, confirming m_l-dependent splittings in fields up to several tesla. Early EPR observations by Yevgeny Zavoisky in 1945 on salts resolved anisotropic resonances attributable to partial orbital quenching and m_l projections in d-orbital electrons, with linewidths and g-shifts matching predictions for l=2 or $3 shells, thus validating the magnetic quantum number's role in paramagnetic systems beyond pure spin. Microwave spectroscopy experiments on in 1947 by and Robert Retherford precisely measured the , resolving the small difference (about 1058 MHz) between the nominally degenerate $2S_{1/2} and $2P_{1/2} states, which lifted the degeneracy expected from relativistic Dirac theory and confirmed the distinct roles of orbital and contributions encoded in quantum numbers including m_l. In the $2P_{1/2} state, the fine-structure coupling mixes m_l = 0, \pm1 with m_s, but the experiment's radio-frequency excitation between hyperfine-resolved sublevels demonstrated that the shift persists across m_j projections, supporting the underlying m_l degeneracy in zero field that is lifted by spin-orbit interactions. This measurement, performed with beam deflection and cavity perturbation techniques achieving 0.1% precision, provided quantitative verification of m_l's influence on level structure, as subsequent calculations reproduced the splitting with m_l-dependent virtual photon corrections. Modern experiments leveraging and trapping since the mid-1980s have enabled direct state selection and manipulation of m_l sublevels in ultracold atomic ensembles. In magneto-optical traps (MOTs) developed around 1986, circularly polarized laser beams at the D2 transition of alkali atoms like selectively populate ground-state hyperfine levels with specific m_f, which incorporate m_l = 0 for s-states but extend to excited p-states where absorption probabilities depend on \Delta m_l = 0, \pm1 selection rules, allowing observation of m_l-resolved patterns. By the late 1980s, loading laser-cooled atoms into optical lattices—standing waves formed by counterpropagating lasers—facilitated the study of m_l-dependent site potentials and tunneling; for instance, early 1990 experiments with sodium atoms in 1D lattices revealed sublevel-specific Bragg , confirming m_l quantization through transfers matching $2\hbar k \sin\theta for different projections in weak fields. These techniques, achieving temperatures below 100 \muK and densities up to $10^{12} ^{-3}, have since verified m_l effects in coherent control schemes, such as Raman dressing of p-state orbitals.