Quantum Number · aiwiki.page

A quantum number is a numerical label used to characterize a state or property of a physical system in quantum mechanics. It commonly specifies an eigenvalue of an observable, either directly or through a mathematical relation. A suitable set of quantum numbers distinguishes the states in a chosen basis; one quantum number alone may not uniquely identify a state. Familiar examples include the principal, orbital, magnetic, and spin-projection quantum numbers used to describe an electron in an atom. (ocw.mit.edu)

Mathematical meaning

A measurable quantity, or observable, is represented by an operator. If a state $|\psi\rangle$ satisfies

\hat A|\psi\rangle=a|\psi\rangle,

it is an eigenstate of $\hat A$ , and measuring that observable yields the eigenvalue $a$ with certainty. Quantum numbers label these eigenvalues and eigenstates. The label need not equal the measured quantity: an angular-momentum quantum number $l$ , for example, corresponds to an eigenvalue $l(l+1)\hbar^2$ of squared orbital angular momentum. Here $\hbar=h/(2\pi)$ , with $h$ the Planck constant. (ocw.mit.edu)

When several independent states share an eigenvalue, that eigenvalue is degenerate. Additional compatible observables can distinguish them. A complete set of commuting observables has simultaneous eigenvalues that identify each common basis state uniquely, apart from an overall phase. The choice of such a set is not necessarily unique. (ocw.mit.edu)

This does not mean that every quantum state has definite values of every quantum number. A superposition may contain several basis states with different labels; its description requires their complex amplitudes, rather than a single set of labels. Quantum numbers therefore classify basis states without replacing the general wave function or state vector. (ocw.mit.edu)

Atomic quantum numbers

For a bound electron in the nonrelativistic Coulomb model of hydrogen, neglecting spin-dependent interactions, a conventional basis is labeled by

n,\quad l,\quad m_l,\quad m_s.

The first three specify the spatial orbital, while the fourth specifies the electron’s spin projection. These labels remain central to atomic notation, although interactions in more complicated atoms can require different coupling schemes. (nist.gov)

Quantum number	Symbol	Allowed values	Meaning
Principal	$n$	$1,2,3,\ldots$	Identifies the shell and, in the basic hydrogen model, the energy
Orbital angular momentum	$l$	$0,1,\ldots,n-1$	Determines squared orbital angular momentum
Orbital magnetic	$m_l$	$-l,-l+1,\ldots,l$	Determines orbital angular momentum along a chosen axis
Spin projection	$m_s$	$-\tfrac12,+\tfrac12$	Determines electron spin along that axis

These ranges apply to the conventional bound-state atomic basis. (nist.gov)

Principal quantum number. In the nonrelativistic one-electron Coulomb problem, the bound-state energy depends on $n$ as

E_n\propto-\frac{Z^2}{n^2},

where $Z$ is the nuclear charge number. States with the same $n$ but different $l$ and $m_l$ consequently have the same energy in this approximation. This degeneracy is not a universal property of atoms: electron interactions and relativistic corrections introduce additional structure. (ocw.mit.edu)

Orbital and magnetic quantum numbers. Their defining relations are

\hat L^2|l,m_l\rangle =\hbar^2l(l+1)|l,m_l\rangle, \qquad \hat L_z|l,m_l\rangle =\hbar m_l|l,m_l\rangle.

Thus $l$ determines the magnitude of orbital angular momentum, while $m_l$ determines one component. Subshells with $l=0,1,2,3$ are called $s,p,d,f$ , respectively. The term “magnetic” reflects the relevance of the projection to magnetic interactions; the label is meaningful even without an applied magnetic field. (nist.gov)

Spin quantum numbers. The electron has intrinsic spin quantum number $s=\tfrac12$ . This fixed value must be distinguished from the two possible projection values $m_s=\pm\tfrac12$ :

\hat S^2=\hbar^2s(s+1), \qquad S_z=\hbar m_s.

Elementary accounts sometimes call $m_s$ the “spin quantum number,” but “spin-projection quantum number” avoids confusing it with $s$ . (nist.gov)

State counting and electron configurations

For a specified $l$ , there are $2l+1$ possible values of $m_l$ . Including the two spin projections gives $2(2l+1)$ one-electron states in a subshell. The Pauli exclusion principle permits each such state to be occupied by at most one electron. The maximum subshell occupancies are therefore $2,6,10,$ and $14$ for $s,p,d,$ and $f$ . (nist.gov)

Summing over the subshells of shell $n$ gives

\sum_{l=0}^{n-1}2(2l+1)=2n^2.

For example, $n=2$ contains two $2s$ states and six $2p$ states, for a total capacity of eight electrons. These counts underlie electron configurations, but do not alone determine the energetic order in which subshells are occupied. In many-electron atoms, energy depends on interactions beyond the principal quantum number. (nist.gov)

Coupled angular momenta and spectroscopic notation

When spin–orbit coupling is significant, combining orbital and spin angular momentum is often more useful than labeling their projections separately:

\mathbf J=\mathbf L+\mathbf S.

For one electron, $j=l\pm\tfrac12$ , except that $l=0$ allows only $j=\tfrac12$ . The corresponding projection quantum number takes values $m_j=-j,-j+1,\ldots,j$ . An alternative one-electron basis is therefore labeled by $n,l,j,m_j$ . (nist.gov)

In many-electron atoms, LS coupling combines the electrons’ orbital angular momenta into total $L$ , and their spins into total $S$ , before combining these into $J$ . A term symbol is written

{}^{2S+1}L_J.

The superscript is the spin multiplicity, and the letters $S,P,D,F,\ldots$ represent $L=0,1,2,3,\ldots$ . Thus ${}^{3}P_2$ denotes $S=1$ , $L=1$ , and $J=2$ . LS coupling is particularly useful when electrostatic interactions dominate spin-dependent interactions. (physics.nist.gov)

In real spectra, configurations and terms may be mixed. A label can describe the leading component of an eigenstate rather than an exact assignment. Strong mixing can make a configuration label primarily a bookkeeping designation. (pml.nist.gov)

Good quantum numbers and selection rules

A good quantum number is associated with a quantity preserved by the system’s dynamics. For a time-independent operator $\hat Q$ , the condition

[\hat H,\hat Q]=0

expresses compatibility with the Hamiltonian. A state initially possessing a definite eigenvalue of $\hat Q$ retains it during evolution. Whether a quantum number is good therefore depends on the Hamiltonian, including the interactions retained in the model. (mitocw.ups.edu.ec)

Quantum numbers also express selection rules in spectroscopy. For atomic electric-dipole transitions, parity must change and

\Delta J=0,\pm1,

with $J=0\leftrightarrow J=0$ excluded. In pure LS coupling, additional rules include $\Delta S=0$ and $\Delta L=0,\pm1$ , excluding $L=0\leftrightarrow L=0$ . A transition forbidden by electric-dipole rules may still occur through another multipole process or through state mixing. (physics.nist.gov)

Beyond atomic electrons

Quantum numbers are not limited to the four familiar atomic labels. In the one-dimensional quantum harmonic oscillator, a nonnegative integer $n$ labels energy eigenstates:

E_n=\hbar\omega\left(n+\frac12\right), \qquad n=0,1,2,\ldots.

Here the numbering begins at zero, unlike the principal quantum number of a bound hydrogen state. The same symbol can therefore have different meanings and ranges in different systems. (mit.edu)

In particle physics, quantum numbers classify particles and their internal properties. Examples include spin, electric charge, parity, baryon number, and quark-flavor quantum numbers. A quark has spin $\tfrac12$ and baryon number $\tfrac13$ ; an antiquark has baryon number $-\tfrac13$ . Flavor labels help classify hadrons, while weak interactions can change quark flavor. A classification label is therefore not automatically conserved in every interaction. (pdg.lbl.gov)

Historical development

Atomic quantum numbers developed from efforts to explain discrete spectra and atomic structure. In Niels Bohr’s atomic model, numbers distinguished permitted electron states within an orbital picture. In 1925, Wolfgang Pauli formulated the exclusion principle, introducing a two-valued degree of freedom needed to characterize electron states. The subsequent quantum-mechanical treatment connected state labels to operator eigenvalues rather than classical electron trajectories. (nobelprize.org)

References

Quantum Physics II, Lecture Notes 5ocw.mit.edu
Lecture notes, Chapter 2. Introduction to Quantum Mechanicsocw.mit.edu
Complete System of Commuting Operatorsocw.mit.edu
Atomic Spectroscopy - Atomic States, Shells, and Configurationsnist.gov
The Hydrogen Atomocw.mit.edu
Atomic Spectroscopy: An Introductionphysics.nist.gov
NIST: Atomic Spectra Database - Energy Levels Help Filepml.nist.gov
Lecture 30: Time-Dependent Perturbation Theory I. H is Time-Independent, Zewail Wavepacketmitocw.ups.edu.ec
NIST: Atomic Spectros. - Spectral Linesphysics.nist.gov
Harmonic oscillator (QM)mit.edu
Quark Modelpdg.lbl.gov
Weak Interactionpdg.lbl.gov