Quantum Numbers – Definition, Rules, and Examples

Quantum numbers are numerical values that describe the allowed quantum states of electrons in atoms, including their energy level, orbital type, orbital orientation, and spin. Together, quantum numbers provide an address-like description of each electron in an atom and connect the mathematical solutions of quantum mechanics with familiar chemical ideas such as atomic orbitals, electron configurations, the periodic table, and chemical bonding.

For an electron in an atom, the standard set consists of four quantum numbers: the principal quantum number (n), azimuthal or angular momentum quantum number (l), magnetic quantum number (mₗ), and spin magnetic quantum number (mₛ). The allowed values are restricted rather than continuous, which is the origin of the word quantum.

Quantum numbers arise naturally from the mathematical description of particles in quantum mechanics. For hydrogen and hydrogen-like ions, the first three quantum numbers emerge from solutions of the Schrödinger equation, while electron spin requires an additional quantum number. In multi-electron atoms, quantum numbers remain essential labels for atomic orbitals and electron states, although electron-electron interactions make orbital energies more complicated than in hydrogen.

Key Takeaways: Quantum Numbers

• Quantum numbers describe the quantum state of an electron in an atom.
• The four electron quantum numbers are n, l, mₗ, and mₛ.
• The principal quantum number n identifies the principal shell and is closely related to orbital size and energy.
• The angular momentum quantum number l identifies the subshell and orbital type: s, p, d, f, and higher types.
• The magnetic quantum number mₗ distinguishes orbitals within a subshell.
• The spin magnetic quantum number mₛ has two possible values: +½ and −½.
• Not every combination of quantum numbers is allowed.
• The Pauli exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers.
• Quantum numbers explain the organization of shells, subshells, orbitals, and electron configurations.
• The Aufbau principle and Hund’s rule help determine how electrons occupy the states identified by quantum numbers.
• In hydrogen-like atoms, orbitals with the same n are degenerate. In multi-electron atoms, orbital energy generally depends on both n and l.

Why Quantum Numbers Are Important

Quantum numbers form a bridge between quantum mechanics and chemistry. They translate the mathematical states described by wavefunctions into labels that describe atomic structure.

Quantum numbers are important because they:

• Identify allowed electron states.
• Organize electrons into shells, subshells, and orbitals.
• Determine how many orbitals occur in each shell and subshell.
• Establish the maximum number of electrons in shells and subshells.
• Provide the foundation for writing electron configurations.
• Help explain the structure of the periodic table.
• Connect atomic orbitals with orbital angular momentum.
• Help predict atomic spectra and spectroscopic transitions.
• Underlie explanations of magnetism and electron spin.
• Contribute to understanding chemical bonding and periodic trends.

The rules of quantum numbers also reveal something fundamental about nature: atomic electrons cannot possess arbitrary combinations of energy and angular momentum. Only particular quantum states are allowed.

History of Quantum Numbers

Quantum numbers developed gradually as physicists attempted to explain atomic spectra and the stability of atoms.

Early Quantum Theory

In 1900, Max Planck proposed that electromagnetic energy is exchanged in discrete quantities called quanta. Then, in 1905, Albert Einstein extended the quantum concept to light while explaining the photoelectric effect.

In 1913, Niels Bohr proposed a model of the hydrogen atom in which the electron could occupy only certain allowed orbits. Bohr’s model introduced a principal quantum number associated with these allowed energy states.

The model successfully reproduced the major features of the hydrogen spectrum, but it could not adequately describe multi-electron atoms or many details of atomic spectra.

Additional Quantum Numbers

Spectroscopic observations revealed fine structure and splitting of spectral lines that required additional quantum numbers. The old quantum theory developed by researchers including Arnold Sommerfeld extended Bohr’s model by introducing additional quantization conditions associated with angular momentum and orbital orientation.

The terminology and interpretation of these early quantum numbers differed from their modern meanings, but they helped establish the idea that several discrete quantities were needed to specify an atomic state.

Modern Quantum Mechanics

Between 1925 and 1926, Werner Heisenberg, Erwin Schrödinger, Max Born, and others developed modern quantum mechanics.

Schrödinger’s wave equation replaced classical electron trajectories with wavefunctions. When the Schrödinger equation is solved for hydrogen, three quantum numbers naturally appear:

• n
• l
• mₗ

Rather than describing an electron’s classical orbit, these numbers describe properties of its quantum state. More precisely, these quantum numbers arise when the hydrogen-atom Hamiltonian is solved by separating the Schrödinger equation into radial and angular parts.

Electron Spin and the Fourth Quantum Number

In 1925, George Uhlenbeck and Samuel Goudsmit proposed that electrons possess an intrinsic angular momentum called spin. Wolfgang Pauli had already recognized the need for an additional two-valued quantum property to explain atomic spectra and electron arrangements.

The spin quantum number completed the familiar set of four quantum numbers used to describe electrons in atoms.

The Pauli exclusion principle then provided a fundamental rule governing electron configurations: no two electrons in the same atom can have identical values for all four quantum numbers.

What Are Quantum Numbers?

A quantum number is a discrete numerical label associated with a quantum state or a measurable property of that state.

For atomic electrons, four quantum numbers are commonly used:

1. Principal quantum number (n)
2. Angular momentum or azimuthal quantum number (l)
3. Magnetic quantum number (mₗ)
4. Spin magnetic quantum number (mₛ)

The first three characterize an atomic orbital in the familiar nonrelativistic description. Adding mₛ specifies an electron state within that orbital.

A useful simplified analogy is an address:

• n identifies the shell.
• l identifies the subshell.
• mₗ identifies an orbital within that subshell.
• mₛ distinguishes the two possible spin states associated with that orbital.

However, electrons are not little objects sitting at fixed addresses. Quantum numbers label quantum states and properties of wavefunctions rather than classical positions or trajectories.

Quantum Numbers and the Hamiltonian

In quantum mechanics, the Hamiltonian operator, usually written Ĥ, represents the total energy of a quantum system. The time-independent Schrödinger equation is:

Ĥψ = Eψ

where ψ is the wavefunction and E is an allowed energy eigenvalue.

For the hydrogen atom, solving this equation produces wavefunctions characterized by the quantum numbers n, l, and mₗ. These quantum numbers arise because the wavefunction must satisfy specific mathematical and physical conditions, such as being finite, continuous, single-valued, and normalizable. Only certain solutions meet these requirements.

The principal quantum number n is associated with the allowed energy states of hydrogen, while l and mₗ arise from the angular part of the wavefunction and describe orbital angular momentum and its projection along a chosen axis. Electron spin does not emerge from the nonrelativistic Schrödinger equation and requires the additional spin quantum numbers s and mₛ.

For multi-electron atoms, the Hamiltonian also includes electron-electron interactions. Exact solutions generally are not available, so atomic orbitals and their energies are obtained using approximation methods. Even so, quantum numbers remain essential labels for describing electron states.

The Four Quantum Numbers

Principal Quantum Number (n)

The principal quantum number, n, identifies the principal electron shell.

Its possible values are:

n = 1, 2, 3, 4, …

The value of n must be a positive integer.

Increasing n generally corresponds to:

• Greater average electron distance from the nucleus
• Larger orbitals
• Higher energy

For hydrogen-like atoms, the electron’s energy depends only on n. In multi-electron atoms, however, orbital energy also depends strongly on l because shielding, penetration, and electron-electron interactions remove much of the degeneracy found in hydrogen.

A shell with principal quantum number n contains:

• n subshells
• n² orbitals
• A maximum of 2n² electrons

For example, the n = 3 shell contains three subshells, nine orbitals, and up to 18 electrons.

Angular Momentum Quantum Number (l)

The angular momentum quantum number, also called the azimuthal quantum number, l, identifies the subshell and determines the magnitude of an electron’s orbital angular momentum.

For a given n:

l = 0, 1, 2, …, n − 1

The values correspond to orbital letters:

Thus:

• 1s is allowed.
• 1p is not allowed because n = 1 permits only l = 0.
• 2s and 2p are allowed.
• 2d is not allowed.
• 3s, 3p, and 3d are allowed.

The magnitude of orbital angular momentum is:

L = √[l(l + 1)] ħ

where ħ is the reduced Planck constant.

Notice that the magnitude is not simply lħ.

Magnetic Quantum Number (mₗ)

The magnetic quantum number, mₗ, describes the allowed projection of orbital angular momentum along a chosen axis.

For a particular value of l:

mₗ = −l, …, −1, 0, +1, …, +l

Therefore, there are:

2l + 1

possible mₗ values for each subshell.

Examples:

• s (l = 0): mₗ = 0 → 1 orbital
• p (l = 1): mₗ = −1, 0, +1 → 3 orbitals
• d (l = 2): mₗ = −2, −1, 0, +1, +2 → 5 orbitals
• f (l = 3): 7 possible values → 7 orbitals

This is the origin of the familiar numbers of orbitals in s, p, d, and f subshells.

Spin Magnetic Quantum Number (mₛ)

The spin magnetic quantum number, mₛ, specifies the allowed projection of an electron’s intrinsic spin angular momentum.

For an electron:

mₛ = +½ or −½

These values are commonly called spin-up and spin-down.

An electron is a spin-½ particle, so the magnitude of its intrinsic spin angular momentum is:

S = √[s(s + 1)] ħ

with s = ½.

The arrows ↑ and ↓ commonly used in orbital diagrams represent the two possible mₛ values.

Rules for Quantum Numbers

The easiest way to determine whether a set of quantum numbers is allowed is to check the values in order:

1. Check n. It must be a positive integer: 1, 2, 3, …
2. Check l. It must range from 0 through n − 1.
3. Check mₗ. It must be an integer from −l through +l.
4. Check mₛ. It must equal +½ or −½.

Each restriction depends on the quantum number before it, except for electron spin.

For example:

(n, l, mₗ, mₛ) = (3, 2, −1, +½)

is allowed because:

• n = 3 is valid.
• l = 2 is within 0 to 2.
• mₗ = −1 is within −2 to +2.
• mₛ = +½ is valid.

But:

(2, 2, 0, +½)

is not allowed because when n = 2, l can only equal 0 or 1.

Allowed Combinations of Quantum Numbers

Here are the orbital quantum numbers for the first four principal shells:

Every listed mₗ value can be paired with either mₛ = +½ or −½.

This leads directly to the maximum electron capacities:

Shells, Subshells, and Orbitals

Quantum numbers establish the hierarchy of atomic electron states.

Shell

A shell contains states sharing the same principal quantum number n.

Examples include:

• n = 1 → first shell
• n = 2 → second shell
• n = 3 → third shell

Subshell

A subshell consists of orbitals sharing the same n and l.

Examples include:

• 2s
• 2p
• 3s
• 3p
• 3d

Orbital

In the usual orbital description, an atomic orbital is characterized by n, l, and mₗ.

For example, a 3d orbital has:

n = 3 and l = 2.

There are five possible mₗ values, so the 3d subshell contains five orbitals.

Each orbital can accommodate at most two electrons because there are only two possible values of mₛ.

How Many Orbitals and Electrons Are Allowed?

Several useful relationships follow directly from quantum numbers.

For a subshell with angular momentum quantum number l:

Number of orbitals = 2l + 1

and:

Maximum number of electrons = 2(2l + 1)

Therefore:

• s → 1 orbital → 2 electrons
• p → 3 orbitals → 6 electrons
• d → 5 orbitals → 10 electrons
• f → 7 orbitals → 14 electrons

For a shell with principal quantum number n:

Number of orbitals = n²

and:

Maximum number of electrons = 2n²

So the n = 4 shell contains 16 orbitals and can accommodate up to 32 electrons.

Quantum Numbers and Orbital Shape

Quantum numbers also relate to the mathematical forms and familiar shapes of atomic orbitals.

The principal quantum number n influences orbital size and the number of nodes. The angular momentum quantum number l determines the number of angular nodes and is associated with orbital shape.

Generally:

• s orbitals are spherically symmetric.
• p orbitals have two principal lobes.
• d orbitals usually have more complex four-lobed or two-lobed-and-ring forms.
• f orbitals have still more complicated angular distributions.

The magnetic quantum number mₗ distinguishes different angular states within a subshell.

The total number of nodes for a hydrogenic atomic orbital is:

Total nodes = n − 1

The number of angular nodes is:

Angular nodes = l

The number of radial nodes is:

Radial nodes = n − l − 1

For example, a 3p orbital has n = 3 and l = 1. It therefore has two total nodes: one angular node and one radial node.

Relative Energies of Atomic Orbitals

Orbital energy requires an important distinction between hydrogen-like atoms and multi-electron atoms.

Hydrogen and Hydrogen-Like Ions

For a one-electron atom or ion, such as H, He⁺, or Li²⁺, the energy of an orbital depends only on n in the nonrelativistic Coulomb model.

As a result, states with the same n but different l are degenerate.

For example:

E(2s) = E(2p)

and:

E(3s) = E(3p) = E(3d)

within this approximation.

Multi-Electron Atoms

In atoms containing multiple electrons, electron-electron repulsion, shielding, and orbital penetration alter the relative energies.

Orbitals with the same n but different l generally no longer have equal energies.

Within the same principal shell, the approximate trend is commonly:

s < p < d < f

because s orbitals generally penetrate closer to the nucleus and experience a greater effective nuclear charge.

Across different shells, the ordering becomes more complicated.

The n + l Rule

The Madelung rule, often called the n + l rule, predicts the approximate order in which orbitals fill in many ground-state atoms.

The orbital with the smaller value of n + l generally fills first. If two orbitals have the same n + l, the orbital with the smaller n generally fills first.

For example:

• 4s: n + l = 4 + 0 = 4
• 3d: n + l = 3 + 2 = 5

So the Aufbau filling scheme places 4s before 3d.

Another comparison is:

• 3d: n + l = 5
• 4p: n + l = 5

Because 3d has the lower n, it comes first in the usual filling order.

A commonly used approximate sequence is:

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s …

However, this sequence is a filling guideline rather than a universal ranking of fixed orbital energies. Orbital energies depend on electron occupancy, atomic number, ionization state, and electron-electron interactions. This distinction helps explain electron-configuration exceptions and why transition metals usually lose 4s electrons before 3d electrons when forming cations.

Quantum Numbers and Electron Configurations

An electron configuration describes how electrons occupy atomic orbitals. Quantum numbers provide the underlying labels for every electron represented by the configuration.

For carbon:

1s² 2s² 2p²

The two 1s electrons share:

n = 1, l = 0, mₗ = 0

but have opposite spin quantum numbers.

The same applies to the two 2s electrons.

The two 2p electrons occupy different p orbitals in the ground state according to Hund’s rule.

Three major principles determine how electron states are occupied.

Aufbau Principle

The Aufbau principle states that electrons generally occupy lower-energy available orbitals before higher-energy orbitals when constructing a ground-state electron configuration.

The principle produces the familiar approximate filling sequence, but exceptions occur, particularly among transition and inner-transition elements.

Pauli Exclusion Principle

The Pauli exclusion principle states:

No two electrons in an atom can have the same values for all four quantum numbers.

Because two electrons in the same orbital already share n, l, and mₗ, they must have different mₛ values.

Thus, one orbital accommodates no more than two electrons.

Hund’s Rule

Hund’s rule states that electrons occupy degenerate orbitals singly with parallel spins before pairing.

Consider the three degenerate p orbitals.

For nitrogen, the 2p³ configuration is represented schematically as:

↑ ↑ ↑

rather than:

↑↓ ↑ _

The singly occupied arrangement has lower energy for the isolated atom.

Degenerate Orbitals

Degenerate orbitals are orbitals or states having the same energy.

In an isolated multi-electron atom without an external field, the orbitals within a given subshell are ordinarily degenerate in the basic nonrelativistic orbital model. Thus, the three 2p orbitals have the same energy, as do the five 3d orbitals.

This degeneracy is important for Hund’s rule because electrons distribute among equal-energy orbitals before pairing.

Degeneracy can be removed by interactions or external conditions. Examples include:

• Magnetic fields
• Electric fields
• Spin-orbit coupling
• Chemical bonding
• Crystal or ligand fields

The splitting of formerly degenerate energy levels is responsible for important phenomena such as the Zeeman effect and much of transition-metal spectroscopy.

Quantum Numbers and the Periodic Table

Quantum numbers help explain the structure of the periodic table.

The familiar blocks correspond primarily to the type of subshell being filled:

• s-block: l = 0
• p-block: l = 1
• d-block: l = 2
• f-block: l = 3

The capacities of these subshells explain the characteristic widths of the blocks:

• s-block → 2 columns
• p-block → 6 columns
• d-block → 10 columns
• f-block → 14 columns

These numbers are not arbitrary. They follow directly from the number of possible mₗ values and the two allowed electron spin states.

Therefore, quantum numbers provide a quantum-mechanical basis for much of the periodic table’s organization.

Quantum Numbers and Spectroscopy

Quantum numbers are especially important in atomic spectroscopy because transitions between quantum states produce or absorb photons.

Not every mathematically imaginable transition occurs with equal probability. Selection rules specify which transitions are allowed or strongly favored for a particular interaction.

For electric-dipole transitions in the simplest atomic treatment, important selection rules include:

Δl = ±1

and:

Δmₗ = 0, ±1

Spin is normally unchanged:

Δs = 0

More complete treatments use total angular momentum quantum numbers and account for spin-orbit coupling. Selection rules explain why some spectral lines are strong, others are weak, and some transitions are forbidden in the electric-dipole approximation.

Beyond the Four Introductory Quantum Numbers

The familiar four quantum numbers are extremely useful, but they are not the only quantum numbers used in physics.

For orbital angular momentum:

• l gives its magnitude.
• mₗ gives its projection along a chosen axis.

For electron spin:

• s = ½ gives the electron’s intrinsic spin.
• mₛ = ±½ gives its projection.

When orbital and spin angular momentum couple, physicists use the total angular momentum quantum number j:

j = l ± ½

and its projection quantum number mⱼ.

For multi-electron atoms, additional quantum numbers such as L, S, and J describe the combined angular momenta of multiple electrons. These quantities appear in atomic term symbols and are especially important in spectroscopy and more advanced quantum mechanics.

So, the statement that an electron “has four quantum numbers” is a useful introductory framework, not a claim that quantum mechanics contains only four possible quantum numbers.

Example Problem 1: Is a Set of Quantum Numbers Allowed?

Question: Is the following set of quantum numbers allowed?

n = 3, l = 2, mₗ = −2, mₛ = +½

Solution:

For n = 3:

l may equal 0, 1, or 2.

So l = 2 is allowed.

For l = 2:

mₗ may equal −2, −1, 0, +1, or +2.

So mₗ = −2 is allowed.

Finally:

mₛ = +½

is an allowed spin value.

Answer: Yes. The set is allowed and describes an electron associated with a 3d orbital.

Example Problem 2: How Many Electrons Fit in a 4d Subshell?

Question: What is the maximum number of electrons in the 4d subshell?

Solution:

For a d subshell:

l = 2.

Therefore:

mₗ = −2, −1, 0, +1, +2.

There are five possible orbitals.

Each orbital accommodates two electrons with opposite values of mₛ.

So:

5 orbitals × 2 electrons/orbital = 10 electrons

Answer: The 4d subshell holds a maximum of 10 electrons.

Example Problem 3: Quantum Numbers for a 3p Electron

Question: What values of the quantum numbers are possible for an electron in a 3p orbital?

Solution:

The number 3 gives:

n = 3.

The letter p means:

l = 1.

For l = 1:

mₗ = −1, 0, or +1.

For every one of these orbital states:

mₛ = +½ or −½.

Therefore, there are:

3 × 2 = 6

possible combinations, consistent with the fact that a 3p subshell can accommodate six electrons.

Common Misconceptions About Quantum Numbers

Misconception: Quantum numbers tell you exactly where an electron is.

Quantum numbers describe properties of quantum states. They do not specify an electron’s precise position or classical trajectory.

Misconception: The four quantum numbers are simply four coordinates for an electron.

The analogy of an electron “address” is useful for learning, but quantum numbers represent eigenvalues or labels associated with quantum states, not spatial coordinates.

Misconception: mₗ literally tells you which familiar orbital shape you have.

The relationship is more subtle. The mₗ states are angular-momentum eigenstates. Familiar real-valued orbitals such as pₓ and pᵧ can be constructed as linear combinations of states with particular mₗ values. So identifying mₗ = −1, 0, and +1 directly with pₓ, pᵧ, and p_z is generally an oversimplification.

Misconception: Electron spin means the electron literally spins like a tiny ball.

Spin is an intrinsic quantum property. Although it behaves mathematically like angular momentum in important ways, it should not be interpreted as the classical rotation of a tiny spherical electron.

Misconception: The principal quantum number completely determines orbital energy.

This is true for ideal nonrelativistic hydrogen-like atoms, but not generally for multi-electron atoms.

Misconception: 4s is always lower in energy than 3d.

The Aufbau filling diagram is not a universal orbital-energy diagram. Orbital energies change with atomic number, electron occupancy, and ionization.

Misconception: An orbital can contain two electrons because there are two spin directions.

More precisely, two electrons can occupy the same spatial orbital because the two electrons can have different spin projections while obeying the Pauli exclusion principle.

Misconception: The Aufbau principle has no exceptions.

The familiar filling sequence is an extremely useful approximation, but actual ground-state electron configurations include exceptions, particularly among transition metals and heavier elements.

Frequently Asked Questions

What are the four quantum numbers?

The four quantum numbers commonly used to describe an electron in an atomic orbital are the principal quantum number n, angular momentum quantum number l, magnetic quantum number mₗ, and spin magnetic quantum number mₛ.

What does the principal quantum number describe?

The principal quantum number n identifies the principal shell and is related to orbital size and energy.

What does the angular momentum quantum number describe?

The angular momentum quantum number l identifies the subshell and determines the magnitude of orbital angular momentum. Its values correspond to s, p, d, f, and higher subshells.

What values can l have?

For a given value of n:

0 ≤ l ≤ n − 1

For example, when n = 4, l may equal 0, 1, 2, or 3.

What values can mₗ have?

The magnetic quantum number takes integer values from −l through +l, including zero.

What are the possible electron spin quantum numbers?

For an electron:

mₛ = +½ or −½

Which quantum number determines orbital shape?

The angular momentum quantum number l is most directly associated with the angular form of an orbital and the number of angular nodes.

Which quantum number determines orbital orientation?

The magnetic quantum number mₗ describes the allowed projection of orbital angular momentum along a chosen axis. Introductory chemistry often describes this as determining orbital orientation, although the exact quantum-mechanical interpretation is more nuanced.

How many orbitals are in a subshell?

A subshell with quantum number l contains:

2l + 1 orbitals

How many orbitals are in a shell?

A shell with principal quantum number n contains:

n² orbitals

How many electrons can occupy a shell?

The maximum number is:

2n² electrons

Can two electrons have the same four quantum numbers?

Not within the same atom. The Pauli exclusion principle forbids two electrons from having identical sets of all four quantum numbers.

Why does a p subshell hold six electrons?

For p orbitals, l = 1, giving three possible values of mₗ: −1, 0, and +1. Each orbital state permits two spin projections, giving 3 × 2 = 6 electron states.

Are quantum numbers unique to electrons?

No. Quantum numbers occur throughout quantum mechanics. They describe states of atoms, nuclei, elementary particles, molecules, and other quantum systems.

Quick Reference

The four quantum numbers can be remembered as a hierarchy:

Shell → Subshell → Orbital → Spin

which corresponds to:

n → l → mₗ → mₛ

The allowed values are:

Principal quantum number:
n = 1, 2, 3, …

Angular momentum quantum number:
l = 0, 1, 2, …, n − 1

Magnetic quantum number:
mₗ = −l, …, 0, …, +l

Spin magnetic quantum number:
mₛ = +½ or −½

These simple restrictions generate much of the familiar structure of atoms. They determine how many orbitals occur in each subshell, how many electrons each shell can accommodate, how electron configurations are constructed, and ultimately why the periodic table has the structure it does.

References and Further Reading

• Dirac, Paul A. M. (1982). Principles of Quantum Mechanics. Oxford University Press. ISBN 0-19-852011-5.
• Eisberg, Robert Martin; Resnick, Robert (1985). Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd ed.). John Wiley & Sons. ISBN 978-0-471-87373-0.
• Kragh, Helge (2012). Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925. Oxford University Press. doi:10.1093/acprof:oso/9780199654987.003.0007. ISBN 978-0-19-965498-7.
• Krane, Kenneth S. (1988). Introductory Nuclear Physics. John Wiley & Sons. ISBN 0-471-80553-X.
• Schrödinger, Erwin (1926). “Quantisation as an Eigenvalue Problem”. Annalen der Physik. 81 (18): 109–139. doi:10.1002/andp.19263861802