azimuthal quantum number

The term "azimuthal quantum number" was introduced by Arnold Sommerfeld in 1915{{Cite book |last=Whittaker |first=Edmund Taylor |title=A history of the theories of aether and electricity |date=1989 |publisher=Dover |isbn=978-0-486-26126-3 |series=Dover classics of science and mathematics |location=New York}}{{rp|II:132}} as part of an ad hoc description of the energy structure of atomic spectra. Only later with the quantum model of the atom was it understood that this number, {{mvar|ℓ}}, arises from quantization of orbital angular momentum. Some textbooks{{Cite book |last=Schiff |first= Leonard |title=Quantum mechanics |publisher= McGraw-Hill |year= 1949}}{{rp|199}} and the ISO standard 80000-10:2019{{Cite web |title=ISO Online Browsing Platform |url=https://www.iso.org/obp/ui/en/#iso:std:iso:80000:-10:ed-2:v1:en |access-date=2024-02-20 |at=10-13.3}} call {{mvar|ℓ}} the orbital angular momentum quantum number.

The energy levels of an atom in an external magnetic field depend upon the {{mvar|m{{sub|ℓ}}}} value so it is sometimes called the magnetic quantum number.{{Cite book |last=Eisberg |first=Robert M. |title=Quantum physics of atoms, molecules, solids, nuclei, and particles |last2=Resnick |first2=Robert |date=2009 |publisher=Wiley |isbn=978-0-471-87373-0 |edition=2. ed., 37. [Nachdr.] |location=New York}}{{rp|240}}

The lowercase letter {{mvar|ℓ}}, is used to denote the orbital angular momentum of a single particle. For a system with multiple particles, the capital letter {{mvar|L}} is used.

There are four quantum numbers{{mdash}}n, ℓ, m_ℓ, m_s{{mdash}} connected with the energy states of an isolated atom's electrons. These four numbers specify the unique and complete quantum state of any single electron in the atom, and they combine to compose the electron's wavefunction, or orbital.

When solving to obtain the wave function, the Schrödinger equation resolves into three equations that lead to the first three quantum numbers, meaning that the three equations are interrelated. The azimuthal quantum number arises in solving the polar part of the wave equation{{mdash}}relying on the spherical coordinate system, which generally works best with models having sufficient aspects of spherical symmetry.

File:Vector model of orbital angular momentum.svg

An electron's angular momentum, {{math|L}}, is related to its quantum number {{math|ℓ}} by the following equation:

$$\backslash mathbf\{L\}^2\backslash Psi\; =\; \backslash hbar^2\; \backslash ell(\backslash ell\; +\; 1)\; \backslash Psi,$$

where {{math|ħ}} is the reduced Planck constant, {{math|L}} is the orbital angular momentum operator and $\backslash Psi$ is the wavefunction of the electron. The quantum number {{math|ℓ}} is always a non-negative integer: 0, 1, 2, 3, etc. (Notably, {{math|L}} has no real meaning except in its use as the angular momentum operator; thus, it is standard practice to use the quantum number {{math|ℓ}} when referring to angular momentum).

Atomic orbitals have distinctive shapes, (see top graphic) in which letters, s, p, d, f, etc., (employing a convention originating in spectroscopy) denote the shape of the atomic orbital. The wavefunctions of these orbitals take the form of spherical harmonics, and so are described by Legendre polynomials. The several orbitals relating to the different (integer) values of ℓ are sometimes called sub-shells{{mdash}}referred to by lowercase Latin letters chosen for historical reasons{{mdash}}as shown in the table "Quantum subshells for the azimuthal quantum number".

{{Clear}}

Each of the different angular momentum states can take 2(2ℓ + 1) electrons. This is because the third quantum number m_ℓ (which can be thought of loosely as the quantized projection of the angular momentum vector on the z-axis) runs from −ℓ to ℓ in integer units, and so there are 2ℓ + 1 possible states. Each distinct n, ℓ, m_ℓ orbital can be occupied by two electrons with opposing spins (given by the quantum number m_s = ±{{1/2}}), giving 2(2ℓ + 1) electrons overall. Orbitals with higher ℓ than given in the table are perfectly permissible, but these values cover all atoms so far discovered.

For a given value of the principal quantum number n, the possible values of ℓ range from 0 to {{math|1=n − 1}}; therefore, the {{math|1=n = 1}} shell only possesses an s subshell and can only take 2 electrons, the {{math|1=n = 2}} shell possesses an s and a p subshell and can take 8 electrons overall, the {{math|1=n = 3}} shell possesses s, p, and d subshells and has a maximum of 18 electrons, and so on.

A simplistic one-electron model results in energy levels depending on the principal number alone. In more complex atoms these energy levels split for all {{math|n > 1}}, placing states of higher ℓ above states of lower ℓ. For example, the energy of 2p is higher than of 2s, 3d occurs higher than 3p, which in turn is above 3s, etc. This effect eventually forms the block structure of the periodic table. No known atom possesses an electron having ℓ higher than three (f) in its ground state.

The angular momentum quantum number, ℓ and the corresponding spherical harmonic govern the number of planar nodes going through the nucleus. A planar node can be described in an electromagnetic wave as the midpoint between crest and trough, which has zero magnitudes. In an s orbital, no nodes go through the nucleus, therefore the corresponding azimuthal quantum number ℓ takes the value of 0. In a p orbital, one node traverses the nucleus and therefore ℓ has the value of 1. $L$ has the value $\backslash sqrt\{2\}\backslash hbar$.

Depending on the value of n, there is an angular momentum quantum number ℓ and the following series. The wavelengths listed are for a hydrogen atom:

{{unbulleted list | style = padding-left: 1.3em;

| $n\; =\; 1,\; L\; =\; 0$, Lyman series (ultraviolet)

| $n\; =\; 2,\; L\; =\; \backslash sqrt\{2\}\backslash hbar$, Balmer series (visible)

| $n\; =\; 3,\; L\; =\; \backslash sqrt\{6\}\backslash hbar$, Ritz–Paschen series (near infrared)

| $n\; =\; 4,\; L\; =\; 2\backslash sqrt\{3\}\backslash hbar$, Brackett series (short-wavelength infrared)

| $n\; =\; 5,\; L\; =\; 2\backslash sqrt\{5\}\backslash hbar$, Pfund series (mid-wavelength infrared).

}}

{{further|Angular momentum coupling}}

Given a quantized total angular momentum $\backslash mathbf\{j\}$ that is the sum of two individual quantized angular momenta $\backslash boldsymbol\{\backslash ell\}\_1$ and $\backslash boldsymbol\{\backslash ell\}\_2$,

$$\backslash mathbf\{j\}\; =\; \backslash boldsymbol\{\backslash ell\}\_1\; +\; \backslash boldsymbol\{\backslash ell\}\_2$$

the quantum number $j$ associated with its magnitude can range from $|\backslash ell\_1\; -\; \backslash ell\_2|$ to $\backslash ell\_1\; +\; \backslash ell\_2$ in integer steps

where $\backslash ell\_1$ and $\backslash ell\_2$ are quantum numbers corresponding to the magnitudes of the individual angular momenta.

File:LS coupling.svg between measuring angular momentum component.]]

Due to the spin–orbit interaction in an atom, the orbital angular momentum no longer commutes with the Hamiltonian, nor does the spin. These therefore change over time. However the total angular momentum {{math|J}} does commute with the one-electron Hamiltonian and so is constant. {{math|J}} is defined as

$$\backslash mathbf\{J\}\; =\; \backslash mathbf\{L\}\; +\; \backslash mathbf\{S\}$$

{{math|L}} being the orbital angular momentum and {{math|S}} the spin. The total angular momentum satisfies the same commutation relations as orbital angular momentum, namely

$$[J\_i,\; J\_j\; ]\; =\; i\; \backslash hbar\; \backslash varepsilon\_\{ijk\}\; J\_k$$

from which it follows that

$$\backslash left[J\_i,\; J^2\; \backslash right]\; =\; 0$$

where {{math|J_i}} stand for {{math|J_x}}, {{math|J_y}}, and {{math|J_z}}.

The quantum numbers describing the system, which are constant over time, are now {{math|j}} and {{math|m_j}}, defined through the action of {{math|J}} on the wavefunction $\backslash Psi$

$$\backslash begin\{align\}$$

\mathbf{J}^2\Psi &= \hbar^2 j(j+1) \Psi \\[1ex]

\mathbf{J}_z\Psi &= \hbar m_j\Psi

\end{align}

So that {{math|j}} is related to the norm of the total angular momentum and {{math|m_j}} to its projection along a specified axis. The j number has a particular importance for relativistic quantum chemistry, often featuring in subscript in for deeper states near to the core for which spin-orbit coupling is important.

As with any angular momentum in quantum mechanics, the projection of {{math|J}} along other axes cannot be co-defined with {{math|J_z}}, because they do not commute.

The eigenvectors of {{math|j}}, {{math|s}}, {{math|m_j}} and parity, which are also eigenvectors of the Hamiltonian, are linear combinations of the eigenvectors of {{math|ℓ}}, {{math|s}}, {{math|m_ℓ}} and {{math|m_s}}.

{{See also|Density functional theory|History of spectroscopy}}

File:Electron_energy_loss_spectroscopy_coreloss_lsmo.svg.]]

The angular momentum quantum numbers strictly refer to isolated atoms. However, they have wider uses for atoms in solids, liquids or gases. The {{math|ℓ m}} quantum number corresponds to specific spherical harmonics and are commonly used to describe features observed in spectroscopic methods such as X-ray photoelectron spectroscopy{{Cite book |last=Hüfner |first=Stefan |url=http://link.springer.com/10.1007/978-3-662-09280-4 |title=Photoelectron Spectroscopy |date=2003 |publisher=Springer Berlin Heidelberg |isbn=978-3-642-07520-9 |series=Advanced Texts in Physics |location=Berlin, Heidelberg |doi=10.1007/978-3-662-09280-4}} and electron energy loss spectroscopy.{{Cite book |last=Egerton |first=R.F. |url=https://link.springer.com/10.1007/978-1-4419-9583-4 |title=Electron Energy-Loss Spectroscopy in the Electron Microscope |date=2011 |publisher=Springer US |isbn=978-1-4419-9582-7 |location=Boston, MA |language=en |doi=10.1007/978-1-4419-9583-4}} (The notation is slightly different, with X-ray notation where K, L, M are used for excitations out of electron states with $n=0,\; 1,\; 2$.)

The angular momentum quantum numbers are also used when the electron states are described in methods such as Kohn–Sham density functional theory{{Cite journal |last=Kohn |first=W. |last2=Sham |first2=L. J. |date=1965 |title=Self-Consistent Equations Including Exchange and Correlation Effects |url=https://link.aps.org/doi/10.1103/PhysRev.140.A1133 |journal=Physical Review |language=en |volume=140 |issue=4A |pages=A1133–A1138 |doi=10.1103/PhysRev.140.A1133 |issn=0031-899X}} or with gaussian orbitals.{{Citation |last=Gill |first=Peter M.W. |title=Molecular integrals Over Gaussian Basis Functions |date=1994 |work=Advances in Quantum Chemistry |volume=25 |pages=141–205 |url=https://linkinghub.elsevier.com/retrieve/pii/S0065327608600192 |access-date=2024-02-20 |publisher=Elsevier |language=en |doi=10.1016/s0065-3276(08)60019-2 |isbn=978-0-12-034825-1}} For instance, in silicon the electronic properties used in semiconductor device are due to the p-like states with $l=1$ centered at each atom, while many properties of transition metals depend upon the d-like states with $l=2$.{{Cite book |last=Pettifor |first=D. G. |title=Bonding and structure of molecules and solids |date=1996 |publisher=Clarendon Press |isbn=978-0-19-851786-3 |edition=Reprint with corr |series=Oxford science publications |location=Oxford}}

The azimuthal quantum number was carried over from the Bohr model of the atom, and was posited by Arnold Sommerfeld.{{cite book|last=Eisberg|first=Robert|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles | year=1974|publisher=John Wiley & Sons Inc|location=New York|isbn=978-0-471-23464-7|pages=114–117}} The Bohr model was derived from spectroscopic analysis of atoms in combination with the Rutherford atomic model. The lowest quantum level was found to have an angular momentum of zero. Orbits with zero angular momentum were considered as oscillating charges in one dimension and so described as "pendulum" orbits, but were not found in nature.{{cite journal|title=Note on "pendulum" orbits in atomic models |journal=Proc. Natl. Acad. Sci. |year=1927 |volume=13 |pages=413–419 |pmid=16587189 |author=R.B. Lindsay |issue=6 |author-link=Robert Bruce Lindsay |doi=10.1073/pnas.13.6.413 |bibcode=1927PNAS...13..413L |pmc=1085028|doi-access=free }} In three-dimensions the orbits become spherical without any nodes crossing the nucleus, similar (in the lowest-energy state) to a skipping rope that oscillates in one large circle.

• Introduction to quantum mechanics
• Particle in a spherically symmetric potential
• Angular momentum coupling
• Angular momentum operator
• Clebsch–Gordan coefficients

{{reflist}}

• [http://galileo.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html Development of the Bohr atom]
• [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html#c1 The azimuthal equation explained]

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