Quantum number

Short description: Notation for conserved quantities in physics and chemistry

"Q-number" redirects here. For the Q-theory concept, see Q-analog. For the number format, see Q (number format).

Single electron orbitals for hydrogen-like atoms with quantum numbers n = 1, 2, 3 (blocks), ℓ (rows) and m (columns). The spin s is not visible, because it has no spatial dependence. Part of a series on

Quantum mechanics
iℏ∂∂t|ψ(t)⟩=H^|ψ(t)⟩

Schrödinger equation

* Introduction
* Glossary
* History

Background

* Classical mechanics
* Old quantum theory
* Bra–ket notation

* Hamiltonian
* Interference

Fundamentals

* Coherence
* Decoherence
* Complementarity
* Energy level
* Entanglement
* Hamiltonian
* Uncertainty principle
* Ground state
* Interference
* Measurement
* Nonlocality
* Observable
* Operator
* Quantum
* Quantum fluctuation
* Quantum foam
* Quantum levitation
* Quantum number
* Quantum noise
* Quantum realm
* Quantum state
* Quantum system
* Quantum teleportation
* Qubit
* Spin
* Superposition
* Symmetry
* (Spontaneous) symmetry breaking
* Vacuum state
* Wave propagation
* Wave function
* Wave function collapse
* Wave–particle duality
* Matter wave

Effects

* Zeeman effect
* Stark effect
* Aharonov–Bohm effect
* Landau quantization
* Quantum Hall effect
* Quantum Zeno effect
* Quantum tunnelling
* Photoelectric effect
* Casimir effect

Experiments

* Afshar
* Bell's inequality
* Davisson–Germer
* Double-slit
* Elitzur–Vaidman
* Franck–Hertz
* Leggett–Garg inequality
* Mach–Zehnder
* Popper

* Quantum eraser (delayed-choice)

* Schrödinger's cat
* Quantum suicide and immortality
* Stern-Gerlach
* Wheeler's delayed-choice

Formulations

* Overview

* Heisenberg
* Interaction
* Matrix
* Phase-space
* Schrödinger
* Sum-over-histories (path-integral)
* Hellmann–Feynman theorem

Equations

* Dirac
* Klein–Gordon
* Pauli
* Rydberg
* Schrödinger

Interpretations

* Overview

* Consistent histories
* Copenhagen
* de Broglie–Bohm
* Ensemble
* Hidden-variable
* Many-worlds
* Objective collapse
* Bayesian
* Quantum logic
* Relational
* Stochastic
* Scale relativity
* Transactional

Advanced topics

* Quantum annealing
* Quantum chaos
* Quantum computing
* Density matrix
* Quantum field theory
* Fractional quantum mechanics
* Quantum gravity
* Quantum information science
* Quantum machine learning
* Perturbation theory (quantum mechanics)
* Relativistic quantum mechanics
* Scattering theory
* Spontaneous parametric down-conversion
* Quantum statistical mechanics

Scientists

* Aharonov
* Bell
* Blackett
* Bloch
* Bohm
* Bohr
* Born
* Bose
* de Broglie
* Candlin
* Compton
* Dirac
* Davisson
* Debye
* Ehrenfest
* Einstein
* Everett
* Fock
* Fermi
* Feynman
* Glauber
* Gutzwiller
* Heisenberg
* Hilbert
* Jordan
* Kramers
* Pauli
* Lamb
* Landau
* Laue
* Moseley
* Millikan
* Onnes
* Planck
* Rabi
* Raman
* Rydberg
* Schrödinger
* Sommerfeld
* von Neumann
* Weyl
* Wien
* Wigner
* Zeeman
* Zeilinger
* Goudsmit
* Uhlenbeck
* Yang

* v
* t
* e

In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantum numbers includes the principal, azimuthal, magnetic, and spin quantum numbers. To describe other systems, different quantum numbers are required. For subatomic particles, one needs to introduce new quantum numbers, such as the flavour of quarks, which have no classical correspondence.

Quantum numbers are closely related to eigenvalues of observables. When the corresponding observable commutes with the Hamiltonian of the system, the quantum number is said to be "good", and acts as a constant of motion in the quantum dynamics.

## Contents

* 1 History
* 1.1 Electronic quantum numbers
* 1.2 Nuclear quantum numbers
* 1.3 Connection to symmetry
* 2 Properties
* 2.1 Good quantum numbers
* 3 Electron in a hydrogen-like atom
* 3.1 Principal quantum number
* 3.2 Azimuthal quantum number
* 3.3 Magnetic quantum number
* 3.4 Spin magnetic quantum number
* 3.5 Aufbau principle and Hund's rules
* 4 Spin–orbit coupled systems
* 5 Elementary particles
* 6 Multiplicative quantum numbers
* 7 See also
* 8 References
* 9 Further reading

## History

See also: Physics:History of quantum mechanics

### Electronic quantum numbers

In the era of the old quantum theory, starting from Max Planck's proposal of quanta in his model of blackbody radiation (1900) and Albert Einstein's adaptation of the concept to explain the photoelectric effect (1905), and until Erwin Schrödinger published his eigenfunction equation in 1926,[1] the concept behind quantum numbers developed based on atomic spectroscopy and theories from classical mechanics with extra ad hoc constraints.[2]: 106 Many results from atomic spectroscopy had been summarized in the Rydberg formula involving differences between two series of energies related by integer steps. The model of the atom, first proposed by Niels Bohr in 1913, relied on a single quantum number. Together with Bohr's constraint that radiation absorption is not classical, it was able to explain the Balmer series portion of Rydberg's atomic spectrum formula.[3]

As Bohr notes in his subsequent Nobel lecture, the next step was taken by Arnold Sommerfeld in 1915.[4] Sommerfeld's atomic model added a second quantum number and the concept of quantized phase integrals to justify them.[5]: 207 Sommerfeld's model was still essentially two dimensional, modeling the electron as orbiting in a plane; in 1919 he extended his work to three dimensions using 'space quantization' in place of the quantized phase integrals.[6]: 152 Karl Schwarzschild and Sommerfeld's student, Paul Epstein, independently showed that adding third quantum number gave a complete account for the Stark effect results.

A consequence of space quantization was that the electron's orbital interaction with an external magnetic field would be quantized. This seemed to be confirmed when the results of the Stern-Gerlach experiment reported quantized results for silver atoms in an inhomogeneous magnetic field. The confirmation would turn out to be premature: more quantum numbers would be needed.[7]

The fourth and fifth quantum numbers of the atomic era arose from attempts to understand the Zeeman effect. Like the Stern-Gerlach experiment, the Zeeman effect reflects the interaction of atoms with a magnetic field; in a weak field the experimental results were called "anomalous", they diverged from any theory at the time. Wolfgang Pauli's solution to this issue was to introduce another quantum number taking only two possible values, ±ℏ/2.[8] This would ultimately become the quantized values of the projection of spin, an intrinsic angular momentum quantum of the electron. In 1927 Ronald G. J. Fraser (de) demonstrated that the quantization in the Stern-Gerlach experiment was due to the magnetic moment associated with the electron spin rather than its orbital angular momentum.[7] Pauli's success in developing the arguments for a spin quantum number without relying on classical models set the stage for the development of quantum numbers for elementary particles in the remainder of the 20th century.[8]

Bohr, with his Aufbau or "building up" principle, and Pauli with his exclusion principle connected the atom's electronic quantum numbers in to a framework for predicting the properties of atoms.[9] When Schrödinger published his wave equation and calculated the energy levels of hydrogen, these two principles carried over to become the basis of atomic physics.

### Nuclear quantum numbers

With successful models of the atom, the attention of physics turned to models of the nucleus. Beginning with Heisenberg's initial model of proton-neutron binding in 1932, Eugene Wigner introduced isospin in 1937, the first 'internal' quantum number unrelated to a symmetry in real spacetime.[10]: 45

### Connection to symmetry

As quantum mechanics developed, abstraction increased and models based on symmetry and invariance played increasing roles. Two years before his work on the quantum wave equation, Schrödinger applied the symmetry ideas originated by Emmy Noether and Hermann Weyl to the electromagnetic field.[11]: 198 As quantum electrodynamics developed in the 1930s and 1940s, group theory became an important tool. By 1953 Chen Ning Yang had become obsessed with the idea that group theory could be applied to connect the conserved quantum numbers of nuclear collisions to symmetries in a field theory of nucleons.[11]: 202 With Robert Mills, Yang developed a non-abelian gauge theory based on the conservation of the nuclear isospin quantum numbers.

## Properties

The tally of quantum numbers varies from system to system and has no universal answer. Hence these parameters must be found for each system to be analyzed. A quantized system requires at least one quantum number. The dynamics (i.e. time evolution) of any quantum system are described by a quantum operator in the form of a Hamiltonian, H. There is one quantum number of the system corresponding to the system's energy; i.e., one of the eigenvalues of the Hamiltonian. There is also one quantum number for each linearly independent operator O that commutes with the Hamiltonian. A complete set of commuting observables (CSCO) that commute with the Hamiltonian characterizes the system with all its quantum numbers. There is a one-to-one relationship between the quantum numbers and the operators of the CSCO, with each quantum number taking one of the eigenvalues of its corresponding operator. As a result of the different basis that may be arbitrarily chosen to form a complete set of commuting operators, different sets of quantum numbers may be used for the description of the same system in different situations.

### Good quantum numbers

Main article: Physics:Good quantum number

A specific category of quantum numbers are called "good" quantum numbers; they correspond to eigenvalues of operators that commute with the Hamiltonian, quantities that can be known with precision at the same time as the system's energy. Specifically, observables that commute with the Hamiltonian are simultaneously diagonalizable with it and so the eigenvalues a and the energy (eigenvalues of the Hamiltonian) are not limited by an uncertainty relation arising from non-commutativity. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together. Many observables have discrete spectra (sets of eigenvalues) in quantum mechanics, so the quantities can only be measured in discrete values. In particular, this leads to quantum numbers that take values in discrete sets of integers or half-integers; although they could approach infinity in some cases.

## Electron in a hydrogen-like atom

* Principal quantum number (n)
* Azimuthal quantum number (ℓ)
* Magnetic quantum number (mℓ)
* Spin magnetic quantum number (ms)

These quantum numbers are also used in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of molecular orbitals requires other quantum numbers, because the symmetries of the molecular system are different.

### Principal quantum number

The principal quantum number describes the electron shell of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom, that is[12] n=1,2,…

For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron in caesium can have an n value from 1 to 6. The average distance between the electron and the nucleus increases with n.

### Azimuthal quantum number

The azimuthal quantum number, also known as the orbital angular momentum quantum number, describes the subshell, and gives the magnitude of the orbital angular momentum through the relation L2=ℏ2ℓ(ℓ+1).

In chemistry and spectroscopy, ℓ = 0 is called s orbital, ℓ = 1, p orbital, ℓ = 2, d orbital, and ℓ = 3, f orbital.

The value of ℓ ranges from 0 to n − 1, so the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on:[13] ℓ=0,1,2,…,n−1

A quantum number beginning in n = 3, ℓ = 0, describes an electron in the s orbital of the third electron shell of an atom. In chemistry, this quantum number is very important, since it specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles. The azimuthal quantum number can also denote the number of angular nodes present in an orbital. For example, for p orbitals, ℓ = 1 and thus the amount of angular nodes in a p orbital is 1.

### Magnetic quantum number

The magnetic quantum number describes the specific orbital within the subshell, and yields the projection of the orbital angular momentum along a specified axis: Lz=mℓℏ

The values of mℓ range from −ℓ to ℓ, with integer intervals.[14] The s subshell (ℓ = 0) contains only one orbital, and therefore the mℓ of an electron in an s orbital will always be 0. The p subshell (ℓ = 1) contains three orbitals, so the mℓ of an electron in a p orbital will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with mℓ values of −2, −1, 0, 1, and 2.

### Spin magnetic quantum number

The spin magnetic quantum number describes the intrinsic spin angular momentum of the electron within each orbital and gives the projection of the spin angular momentum S along the specified axis: Sz=msℏ

In general, the values of ms range from −s to s, where s is the spin quantum number, associated with the magnitude of particle's intrinsic spin angular momentum:[15] ms=−s,−s+1,−s+2,⋯,s−2,s−1,s

An electron state has spin number s = ⁠1/2⁠, consequently ms will be +⁠1/2⁠ ("spin up") or −⁠1/2⁠ "spin down" states. Since electron are fermions, they obey the Pauli exclusion principle: each electron state must have different quantum numbers. Therefore, every orbital will be occupied with at most two electrons, one for each spin state.

### Aufbau principle and Hund's rules

Main articles: Physics:Aufbau principle and Hund's rules

A multi-electron atom can be modeled qualitatively as a hydrogen like atom with higher nuclear charge and correspondingly more electrons. The occupation of the electron states in such an atom can be predicted by the Aufbau principle and Hund's empirical rules for the quantum numbers. The Aufbau principle fills orbitals based on their principal and azimuthal quantum numbers (lowest n \+ ℓ first, with lowest n breaking ties; Hund's rule favors unpaired electrons in the outermost orbital). These rules are empirical but they can be related to electron physics.[16]: 10 [17]: 260

## Spin–orbit coupled systems

When one takes the spin–orbit interaction into consideration, the L and S operators no longer commute with the Hamiltonian, and the eigenstates of the system no longer have well-defined orbital angular momentum and spin. Thus another set of quantum numbers should be used. This set includes[18]Cite error: Closing `</ref>` missing for `<ref>` tag

11H | I = (⁠1/2⁠)+ |  | 96C | I = (⁠3/2⁠)− |  | 2011Na | I = 2+

|
|
|
|
|
|
|

21H | I = 1+ |  | 106C | I = 0+ |  | 2111Na | I = (⁠3/2⁠)+
31H | I = (⁠1/2⁠)+ |  | 116C | I = (⁠3/2⁠)− |  | 2211Na | I = 3+
|  |  | 126C | I = 0+ |  | 2311Na | I = (⁠3/2⁠)+
|  |  | 136C | I = (⁠1/2⁠)− |  | 2411Na | I = 4+
|  |  | 146C | I = 0+ |  | 2511Na | I = (⁠5/2⁠)+
|  |  | 156C | I = (⁠1/2⁠)+ |  | 2611Na | I = 3+

The reason for the unusual fluctuations in I, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of NMR spectroscopy in organic chemistry,[19] and MRI in nuclear medicine,[20] due to the nuclear magnetic moment interacting with an external magnetic field.

## Elementary particles

For a more complete description of the quantum states of elementary particles, see Physics:Standard Model.

Elementary particles contain many quantum numbers which are usually said to be intrinsic to them. However, the elementary particles are quantum states of the Standard Model of particle physics, and hence the quantum numbers of these particles bear the same relation to the Hamiltonian of this model as the quantum numbers of the Bohr atom does to its Hamiltonian. In other words, each quantum number denotes a symmetry of the problem. It is more useful in quantum field theory to distinguish between spacetime and internal symmetries.

Typical quantum numbers related to spacetime symmetries are spin (related to rotational symmetry), the parity, C-parity and T-parity (related to the Poincaré symmetry of spacetime). Typical internal symmetries[clarification needed] are lepton number and baryon number or the electric charge. (For a full list of quantum numbers of this kind see the article on flavour.)

## Multiplicative quantum numbers

Most conserved quantum numbers are additive, so in an elementary particle reaction, the sum of the quantum numbers should be the same before and after the reaction. However, some, usually called parities, are multiplicative; i.e., their product is conserved. All multiplicative quantum numbers belong to a symmetry (like parity) in which applying the symmetry transformation twice is equivalent to doing nothing (involution).

## See also

* Electron configuration

## References

1. ↑ Schrödinger, Erwin (1926). "Quantisation as an Eigenvalue Problem". Annalen der Physik 81 (18): 109–139. doi:10.1002/andp.19263861802. Bibcode: 1926AnP...386..109S.
2. ↑ Whittaker, Edmund T. (1989). A history of the theories of aether & electricity. 2: The modern theories, 1900 - 1926 (Repr ed.). New York: Dover Publ. ISBN 978-0-486-26126-3.
3. ↑ Heilbron, John L. (June 2013). "The path to the quantum atom" (in en). Nature 498 (7452): 27–30. doi:10.1038/498027a. ISSN 0028-0836. PMID 23739408. https://www.nature.com/articles/498027a.
4. ↑ Niels Bohr – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Sun. 25 Feb 2024.
5. ↑ Eckert, Michael; Eckert, Michael; Artin, Tom (2013). Arnold Sommerfeld: science, life and turbulent times 1868-1951. New York: Springer. ISBN 978-1-4614-7461-6.
6. ↑ Kragh, Helge (2012-05-17). Niels Bohr and the Quantum Atom: The Bohr Model of Atomic Structure 1913–1925. Oxford University Press. doi:10.1093/acprof:oso/9780199654987.003.0004. ISBN 978-0-19-965498-7. http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987.
7. ↑ 7.0 7.1 Friedrich, Bretislav; Herschbach, Dudley (2003-12-01). "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics" (in en). Physics Today 56 (12): 53–59. doi:10.1063/1.1650229. ISSN 0031-9228. Bibcode: 2003PhT....56l..53F. https://pubs.aip.org/physicstoday/article/56/12/53/632269/Stern-and-Gerlach-How-a-Bad-Cigar-Helped-Reorient.
8. ↑ 8.0 8.1 Giulini, Domenico (2008-09-01). "Electron spin or "classically non-describable two-valuedness"". Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 39 (3): 557–578. doi:10.1016/j.shpsb.2008.03.005. ISSN 1355-2198. Bibcode: 2008SHPMP..39..557G. https://www.sciencedirect.com/science/article/pii/S1355219808000269.
9. ↑ Kragh, Helge (2012-05-17) (in en). 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. http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199654987.001.0001/acprof-9780199654987.
10. ↑ Brown, L.M. (1988). "Remarks on the history of isospin". in Winter, Klaus (in en). Festi-Val: Festschrift for Val Telegdi; essays in physics in honour of his 65th birthday; [a symposium ... was held at CERN, Geneva on 6 July 1987]. Amsterdam: North-Holland Physics Publ. ISBN 978-0-444-87099-5. https://archive.org/details/festivalfestschr0000unse/page/40/mode/2up?q=isospin.
11. ↑ 11.0 11.1 Baggott, J. E. (2013). The quantum story: a history in 40 moments (Impression: 3 ed.). Oxford: Oxford Univ. Press. ISBN 978-0-19-956684-6.
12. ↑ Beiser, A. (1987). Concepts of Modern Physics (4th ed.). McGraw-Hill (International). ISBN 0-07-100144-1.
13. ↑ Atkins, P. W. (1977). Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry. 1. Oxford University Press. ISBN 0-19-855129-0.
14. ↑ Eisberg & Resnick 1985.
15. ↑ Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics. Schuam's Outlines (2nd ed.). McGraw Hill (USA). ISBN 978-0-07-162358-2.
16. ↑ Jolly, William L. (1984). Modern Inorganic Chemistry (1st ed.). McGraw-Hill. pp. 10–12. ISBN 0-07-032760-2. https://archive.org/details/trent_0116300649799/page/10.
17. ↑ Levine, Ira N. (1983). Physical chemistry (2 ed.). New York: McGraw-Hill. ISBN 978-0-07-037421-8.
18. ↑ Atkins, P. W. (1977). Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry. 1. Oxford University Press. ISBN 0-19-855129-0.
19. ↑ Cite error: Invalid `<ref>` tag; no text was provided for refs named `Atkins 1977`
20. ↑ Krane, Kenneth S. (1988). Introductory Nuclear Physics. John Wiley & Sons. ISBN 0-471-80553-X.

## Further reading

* Dirac, Paul A. M. (1982). Principles of Quantum Mechanics. Oxford University Press. ISBN 0-19-852011-5.
* Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. https://archive.org/details/introductiontoel00grif_0.
* Halzen, Francis; Martin, Alan D. (1984). Quarks and Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2. https://archive.org/details/quarksleptonsint0000halz.
* 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. https://archive.org/details/quantumphysicsof00eisb.

* v
* t
* e

Electron configuration

* Electron shell
* Introduction to quantum mechanics
* Quantum mechanics

Quantum numbers|

* Principal quantum number (n)
* Azimuthal quantum number (ℓ)
* Magnetic quantum number (m)
* Spin quantum number (s)

Periodic table|

* Periodic table (electron configurations)

Principles|

* Pauli exclusion principle
* Hund's rule
* Aufbau principle

* v
* t
* e

Quantum mechanics

Background|

* Introduction
* History
* timeline
* Glossary
* Classical mechanics
* Old quantum theory

Fundamentals|

* Bra–ket notation
* Casimir effect
* Complementarity
* Density matrix
* Energy level
* ground state
* excited state
* degenerate levels
* Vacuum state
* Zero-point energy
* QED vacuum
* QCD vacuum
* Hamiltonian
* Operator
* Quantum coherence
* Quantum decoherence
* Measurement
* Quantum
* Quantum realm
* Quantum system
* Quantum state
* Quantum number
* Quantum entanglement
* Quantum superposition
* Quantum nonlocality
* Quantum tunnelling
* Quantum levitation
* Quantum fluctuation
* Quantum annealing
* Quantum foam
* Quantum noise
* Heisenberg uncertainty principle
* Spontaneous parametric down-conversion
* Von Neumann entropy
* Spin
* Scattering theory
* Symmetry in quantum mechanics
* Symmetry breaking
* Spontaneous symmetry breaking
* Wave propagation
* Quantum interference
* Wave function
* Wave function collapse
* Wave–particle duality
* Matter wave
* Qubit
* Qutrit
* Observable
* Probability distribution

Formulations|

* Formulations
* Heisenberg
* Interaction
* Matrix mechanics
* Schrödinger
* Path integral formulation
* Phase space

Equations|

* Dirac
* Klein–Gordon
* Pauli
* Rydberg
* Schrödinger

Interpretations|

* Interpretations
* Bayesian
* Consistent histories
* Copenhagen
* de Broglie–Bohm
* Ensemble
* Hidden variables
* Many worlds
* Objective collapse
* Quantum logic
* Relational
* Stochastic
* Transactional
* Cosmological

Experiments|

* Afshar
* Bell's inequality
* Cold Atom Laboratory
* Davisson–Germer
* Delayed-choice quantum eraser
* Double-slit
* Franck–Hertz experiment
* Leggett–Garg inequality
* Mach-Zehnder inter.
* Elitzur–Vaidman
* Popper
* Quantum eraser
* Schrödinger's cat
* Quantum suicide and immortality
* Stern–Gerlach
* Wheeler's delayed choice

Science|

* Q-analog
* List
* Quantum algebra
* Quantum calculus
* Quantum differential calculus
* Quantum geometry
* Quantum group
* Quantum Bayesianism
* Quantum biology
* Quantum chemistry
* Quantum chaos
* Quantum cognition
* Quantum cosmology
* Quantum dynamics
* Quantum economics
* Quantum evolution
* Quantum finance
* Quantum game theory
* Quantum measurement problem
* Quantum mind
* Quantum probability
* Quantum social science
* Quantum stochastic calculus
* Quantum spacetime

Technology|

* Quantum algorithms
* Quantum amplifier
* Quantum cellular automata
* Quantum finite automata
* Quantum electronics
* Quantum logic gates
* Quantum clock
* Quantum channel
* Quantum bus
* Quantum circuit
* Phase qubit
* Matrix isolation
* Quantum dot
* Quantum dot display
* Quantum dot solar cell
* Quantum dot cellular automaton
* Quantum dot single-photon source
* Quantum dot laser
* Quantum well
* Quantum computing
* Timeline
* Quantum cryptography
* Post-quantum cryptography
* Quantum error correction
* Quantum imaging
* Quantum image processing
* Quantum information
* Quantum key distribution
* Quantum machine
* Quantum machine learning
* Quantum metamaterial
* Quantum metrology
* Quantum network
* Quantum neural network
* Quantum optics
* Quantum programming
* Quantum sensors
* Quantum simulator
* Quantum teleportation
* Quantum levitation
* Time travel
* Quantum complexity theory

Extensions|

* Quantum statistical mechanics
* Relativistic quantum mechanics
* Quantum field theory
* Axiomatic quantum field theory
* Quantum field theory in curved spacetime
* Thermal quantum field theory
* Topological quantum field theory
* Local quantum field theory
* Conformal field theory
* Two-dimensional conformal field theory
* Liouville field theory
* History
* Quantum gravity

* Category
* Physics Portal
* Physics WikiProject
* Commons

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Quantum number. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Physics:Quantum_number&oldid=4372043"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template