Phase-space formulation (quantum mechanics)

From Wikipedia, the free encyclopedia

 

The phase-space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. 

The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl^[3] and Eugene Wigner.

The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two. Quantum mechanics in phase space is often favored in certain quantum optics applications, or in the study of decoherence and a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.

The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as algebraic deformation theory (see Kontsevich quantization formula) and noncommutative geometry.

Phase-space distribution

The phase-space distribution f(x, p) of a quantum state is a quasiprobability distribution. In the phase-space formulation, the phase-space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.

There are several different ways to represent the distribution, all interrelated. The most noteworthy is the Wigner representation, W(x, p), discovered first. Other representations (in approximately descending order of prevalence in the literature) include the Glauber–Sudarshan P, Husimi Q, Kirkwood–Rihaczek, Mehta, Rivier, and Born–Jordan representations. These alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified. 

The phase-space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta and over the position interval:

${\displaystyle \operatorname {P} [a\leq X\leq b]=\int _{a}^{b}\int _{-\infty }^{\infty }W(x,p)\,dp\,dx.}$

If Â(x, p) is an operator representing an observable, it may be mapped to phase space as A(x, p) through the Wigner transform. Conversely, this operator may be recovered by the Weyl transform.

The expectation value of the observable with respect to the phase-space distribution is

${\displaystyle \langle {\hat {A}}\rangle =\int A(x,p)W(x,p)\,dp\,dx.}$

A point of caution, however: despite the similarity in appearance, W(x, p) is not a genuine joint probability distribution, because regions under it do not represent mutually exclusive states, as required in the third axiom of probability theory. Moreover, it can, in general, take negative values even for pure states, with the unique exception of (optionally squeezed) coherent states, in violation of the first axiom. 

Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a few ħ, and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise localization within phase-space regions smaller than ħ, and thus renders such "negative probabilities" less paradoxical. If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context of quantum optics this equation is known as the optical equivalence theorem.

An alternative phase-space approach to quantum mechanics seeks to define a wave function (not just a quasiprobability density) on phase space, typically by means of the Segal–Bargmann transform. To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. Rather, the Segal–Bargmann transform is a holomorphic function of ${\displaystyle x+ip}$. There is a quasiprobability density associated to the phase-space wave function; it is the Husimi Q representation of the position wave function.

Star product

The fundamental noncommutative binary operator in the phase space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★. Each representation of the phase-space distribution has a different characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner-Weyl representation. 

For notational convenience, we introduce the notion of left and right derivatives. For a pair of functions f and g, the left and right derivatives are defined as

${\displaystyle {\begin{aligned}f{\overleftarrow {\partial }}_{x}g&={\frac {\partial f}{\partial x}}\cdot g\\[5pt]f{\overrightarrow {\partial }}_{x}g&=f\cdot {\frac {\partial g}{\partial x}}.\end{aligned}}}$

The differential definition of the star product is 

${\displaystyle f\star g=f\,\exp {\left({\frac {i\hbar }{2}}\left({\overleftarrow {\partial }}_{x}{\overrightarrow {\partial }}_{p}-{\overleftarrow {\partial }}_{p}{\overrightarrow {\partial }}_{x}\right)\right)}\,g}$

where the argument of the exponential function can be interpreted as a power series. Additional differential relations allow this to be written in terms of a change in the arguments of f and g:

${\displaystyle {\begin{aligned}(f\star g)(x,p)&=f\left(x+{\tfrac {i\hbar }{2}}{\overrightarrow {\partial }}_{p},p-{\tfrac {i\hbar }{2}}{\overrightarrow {\partial }}_{x}\right)\cdot g(x,p)\\&=f(x,p)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\overleftarrow {\partial }}_{p},p+{\tfrac {i\hbar }{2}}{\overleftarrow {\partial }}_{x}\right)\\&=f\left(x+{\tfrac {i\hbar }{2}}{\overrightarrow {\partial }}_{p},p\right)\cdot g\left(x-{\tfrac {i\hbar }{2}}{\overleftarrow {\partial }}_{p},p\right)\\&=f\left(x,p-{\tfrac {i\hbar }{2}}{\overrightarrow {\partial }}_{x}\right)\cdot g\left(x,p+{\tfrac {i\hbar }{2}}{\overleftarrow {\partial }}_{x}\right).\end{aligned}}}$

It is also possible to define the ★-product in a convolution integral form, essentially through the Fourier transform:

${\displaystyle (f\star g)(x,p)={\frac {1}{\pi ^{2}\hbar ^{2}}}\,\int f(x+x',p+p')\,g(x+x'',p+p'')\,\exp {\left({\tfrac {2i}{\hbar }}(x'p''-x''p')\right)}\,dx'dp'dx''dp''~.}$

(Thus, e.g., Gaussians compose hyperbolically,

${\displaystyle \exp \left(-{a}(x^{2}+p^{2})\right)~\star ~\exp \left(-{b}(x^{2}+p^{2})\right)={1 \over 1+\hbar ^{2}ab}\exp \left(-{a+b \over 1+\hbar ^{2}ab}(x^{2}+p^{2})\right),}$

or

${\displaystyle \delta (x)~\star ~\delta (p)={2 \over h}\exp \left(2i{xp \over \hbar }\right),}$

etc.)

The energy eigenstate distributions are known as stargenstates, ★-genstates, stargenfunctions, or ★-genfunctions, and the associated energies are known as stargenvalues or ★-genvalues. These are solved, analogously to the time-independent Schrödinger equation, by the ★-genvalue equation,

${\displaystyle H\star W=E\cdot W,}$

where H is the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian.

Time evolution

The time evolution of the phase space distribution is given by a quantum modification of Liouville flow. This formula results from applying the Wigner transformation to the density matrix version of the quantum Liouville equation, the von Neumann equation.

In any representation of the phase space distribution with its associated star product, this is

${\displaystyle {\frac {\partial f}{\partial t}}=-{\frac {1}{i\hbar }}\left(f\star H-H\star f\right),}$

or, for the Wigner function in particular,

${\displaystyle {\frac {\partial W}{\partial t}}=-\{\{W,H\}\}=-{\frac {2}{\hbar }}W\sin \left({{\frac {\hbar }{2}}({\overset {\leftarrow }{\partial _{x}}}{\overset {\rightarrow }{\partial _{p}}}-{\overset {\leftarrow }{\partial _{p}}}{\overset {\rightarrow }{\partial _{x}}})}\right)\ H=-\{W,H\}+O(\hbar ^{2}),}$

where {{ , }} is the Moyal bracket, the Wigner transform of the quantum commutator, while { , } is the classical Poisson bracket.

This yields a concise illustration of the correspondence principle: this equation manifestly reduces to the classical Liouville equation in the limit ħ → 0. In the quantum extension of the flow, however, the density of points in phase space is not conserved; the probability fluid appears "diffusive" and compressible. The concept of quantum trajectory is therefore a delicate issue here. (Given the restrictions placed by the uncertainty principle on localization, Niels Bohr vigorously denied the physical existence of such trajectories on the microscopic scale. By means of formal phase-space trajectories, the time evolution problem of the Wigner function can be rigorously solved using the path-integral method and the method of quantum characteristics, although there are practical obstacles in both cases.) See the movie for the Morse potential, below, to appreciate the nonlocality of quantum phase flow.

Examples

Simple harmonic oscillator

The Wigner quasiprobability distribution F_n(u) for the simple harmonic oscillator with a) n = 0, b) n = 1, and c) n = 5.

 

The Hamiltonian for the simple harmonic oscillator in one spatial dimension in the Wigner-Weyl representation is

${\displaystyle H={\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}.}$

The ★-genvalue equation for the static Wigner function then reads

${\displaystyle {\begin{aligned}H\star W={}&\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\star W\\[4pt]={}&\left({\frac {1}{2}}m\omega ^{2}\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p}\right)^{2}+{\frac {1}{2m}}\left(p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)^{2}\right)~W\\[4pt]={}&\left({\frac {1}{2}}m\omega ^{2}\left(x^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{p}^{2}\right)+{\frac {1}{2m}}\left(p^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{x}^{2}\right)\right)~W\\[4pt]&{}+{\frac {i\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)~W\\[4pt]={}&E\cdot W.\end{aligned}}}$

Time evolution of combined ground and 1st excited state Wigner function for the simple harmonic oscillator. Note the rigid motion in phase space corresponding to the conventional oscillations in coordinate space.

 

Wigner function for the harmonic oscillator ground state, displaced from the origin of phase space, i.e., a coherent state. Note the rigid rotation, identical to classical motion: this is a special feature of the SHO, illustrating the correspondence principle. From the general pedagogy web-site.

 

Consider, first, the imaginary part of the ★-genvalue equation,

${\displaystyle {\frac {\hbar }{2}}\left(m\omega ^{2}x{\overrightarrow {\partial }}_{p}-{\frac {p}{m}}{\overrightarrow {\partial }}_{x}\right)\cdot W=0}$

This implies that one may write the ★-genstates as functions of a single argument,

${\displaystyle W(x,p)=F\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\equiv F(u).}$

With this change of variables, it is possible to write the real part of the ★-genvalue equation in the form of a modified Laguerre equation (not Hermite's equation!), the solution of which involves the Laguerre polynomials as

${\displaystyle F_{n}(u)={\frac {(-1)^{n}}{\pi \hbar }}L_{n}\left(4{\frac {u}{\hbar \omega }}\right)e^{-2u/\hbar \omega }~,}$

introduced by Groenewold in his paper, with associated ★-genvalues

${\displaystyle E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right)~.}$

For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. An initial W(x,p; t = 0) = F(u) evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simply rigidly rotating in phase space,

${\displaystyle W(x,p;t)=W(m\omega x\cos \omega t-p\sin \omega t,~p\cos \omega t+\omega mx\sin \omega t;0)~.}$

Typically, a "bump" (or coherent state) of energy E ≫ ħω can represent a macroscopic quantity and appear like a classical object rotating uniformly in phase space, a plain mechanical oscillator (see the animated figures). Integrating over all phases (starting positions at t = 0) of such objects, a continuous "palisade", yields a time-independent configuration similar to the above static ★-genstates F(u), an intuitive visualization of the classical limit for large action systems.

Free particle angular momentum

Suppose a particle is initially in a minimally uncertain Gaussian state, with the expectation values of position and momentum both centered at the origin in phase space. The Wigner function for such a state propagating freely is

${\displaystyle W(\mathbf {x} ,\mathbf {p} ;t)={\frac {1}{(\pi \hbar )^{3}}}\exp {\left(-\alpha ^{2}r^{2}-{\frac {p^{2}}{\alpha ^{2}\hbar ^{2}}}\left(1+\left({\frac {t}{\tau }}\right)^{2}\right)+{\frac {2t}{\hbar \tau }}\mathbf {x} \cdot \mathbf {p} \right)}~,}$

where α is a parameter describing the initial width of the Gaussian, and τ = m/α²ħ. 

Initially, the position and momenta are uncorrelated. Thus, in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel. 

However, the position and momentum become increasingly correlated as the state evolves, because portions of the distribution farther from the origin in position require a larger momentum to be reached: asymptotically,

${\displaystyle W\longrightarrow {\frac {1}{(\pi \hbar )^{3}}}\exp \left[-\alpha ^{2}\left(\mathbf {x} -{\frac {\mathbf {p} t}{m}}\right)^{2}\right]\,.}$

(This relative "squeezing" reflects the spreading of the free wave packet in coordinate space.) 

Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the standard quantum-mechanical notion of the ground-state nonzero angular momentum specifying orientation independence:

${\displaystyle K_{\text{rad}}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}\left({\frac {3}{2}}-{\frac {1}{1+(t/\tau )^{2}}}\right)}$

${\displaystyle K_{\text{ang}}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}{\frac {1}{1+(t/\tau )^{2}}}~.}$

Morse potential

The Morse potential is used to approximate the vibrational structure of a diatomic molecule. 

The Wigner function time-evolution of the Morse potential U(x) = 20(1 − e^−0.16x)² in atomic units (a.u.). The solid lines represent level set of the Hamiltonian H(x, p) = p²/2 + U(x).

Quantum tunneling

Tunneling is a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics. 

The Wigner function for tunneling through the potential barrier U(x) = 8e^−0.25x2 in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian H(x, p) = p²/2 + U(x).

Quartic potential

 

The Wigner function time evolution for the potential U(x) = 0.1x⁴ in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian H(x, p) = p²/2 + U(x).

Schrödinger cat state

 

Wigner function of two interfering coherent states evolving through the SHO Hamiltonian. The corresponding momentum and coordinate projections are plotted to the right and under the phase space plot.

Quantum number

From Wikipedia, the free encyclopedia

 

Quantum numbers describe values of conserved quantities in the dynamics of a quantum system. In the case of electrons, the quantum numbers can be defined as "the sets of numerical values which give acceptable solutions to the Schrödinger wave equation for the hydrogen atom". An important aspect of quantum mechanics is the quantization of the observable quantities, since quantum numbers are discrete sets of integers or half-integers, although they could approach infinity in some cases. This distinguishes quantum mechanics from classical mechanics where the values that characterize the system such as mass, charge, or momentum, range continuously. Quantum numbers often describe specifically the energy levels of electrons in atoms, but other possibilities include angular momentum, spin, etc. An important family is flavour quantum numbers – internal quantum numbers which determine the type of a particle and its interactions with other particles through the forces. Any quantum system can have one or more quantum numbers; it is thus difficult to list all possible quantum numbers.

How many quantum numbers exist?

The question of how many quantum numbers are needed to describe any given system has no universal answer. Hence for each system one must find the answer for a full analysis of the system. A quantized system requires at least one quantum number. The dynamics of any quantum system are described by a quantum Hamiltonian, H. There is one quantum number of the system corresponding to the energy, i.e., the eigenvalue of the Hamiltonian. There is also one quantum number for each operator O that commutes with the Hamiltonian. These are all the quantum numbers that the system can have. Note that the operators O defining the quantum numbers should be independent of each other. Often, there is more than one way to choose a set of independent operators. Consequently, in different situations different sets of quantum numbers may be used for the description of the same system.

Spatial and angular momentum numbers

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.

 

Four quantum numbers can describe an electron in an atom completely. As per the following model, these nearly-compatible quantum numbers are:

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

The spin-orbital interaction, however, relates these numbers. Thus, a complete description of the system can be given with fewer quantum numbers, if orthogonal choices are made for these basis vectors.

Traditional nomenclatures

Hund-Mulliken molecular orbital theory

Many different models have been proposed throughout the history of quantum mechanics, but the most prominent system of nomenclature spawned from the Hund-Mulliken molecular orbital theory of Friedrich Hund, Robert S. Mulliken, and contributions from Schrödinger, Slater and John Lennard-Jones. This system of nomenclature incorporated Bohr energy levels, Hund-Mulliken orbital theory, and observations on electron spin based on spectroscopy and Hund's rules.

This model describes electrons using four quantum numbers, n, ℓ, m_ℓ, m_s, given below. It is also the common nomenclature in the classical description of nuclear particle states (e.g. protons and neutrons). A quantum description of molecular orbitals require different quantum numbers, because the Hamiltonian and its symmetries are quite different.

1. The principal quantum number (n) describes the electron shell, or energy level, of an electron. The value of n ranges from 1 to the shell containing the outermost electron of that atom, i.e.

   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.
   For particles in a time-independent potential (see Schrödinger equation), it also labels the nth eigenvalue of Hamiltonian (H), i.e. the energy, E with the contribution due to angular momentum (the term involving J²) left out. This number therefore has a dependence only on the distance between the electron and the nucleus (i.e., the radial coordinate, r). The average distance increases with n, and hence quantum states with different principal quantum numbers are said to belong to different shells.
2. The azimuthal quantum number (ℓ) (also known as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation

   L² = ħ² ℓ (ℓ + 1).

   In chemistry and spectroscopy, "ℓ = 0" is called an s orbital, "ℓ = 1" a p orbital, "ℓ = 2" a d orbital, and "ℓ = 3" an 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:

   ℓ = 0, 1, 2,..., n − 1.

   A quantum number beginning in 3, 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.
3. The magnetic quantum number (m_ℓ) describes the specific orbital (or "cloud") within that subshell, and yields the projection of the orbital angular momentum along a specified axis:

   L_z = m_ℓ ħ.

   The values of m_ℓ range from −ℓ to ℓ, with integer intervals: The s subshell (ℓ = 0) contains only one orbital, and therefore the m_ℓ of an electron in an s orbital will always be 0. The p subshell (ℓ = 1) contains three orbitals (in some systems, depicted as three "dumbbell-shaped" clouds), so the m_ℓ of an electron in a p orbital will be −1, 0, or 1. The d subshell (ℓ = 2) contains five orbitals, with m_ℓ values of −2, −1, 0, 1, and 2.
4. The spin projection quantum number (m_s) describes the spin (intrinsic angular momentum) of the electron within that orbital, and gives the projection of the spin angular momentum S along the specified axis:

   S_z = m_s ħ.

   In general, the values of m_s range from −s to s, where s is the spin quantum number, an intrinsic property of particles:

   m_s = −s, −s + 1, −s + 2, ..., s − 2, s − 1, s.

   An electron has spin number s = ½, consequently m_s will be ±½, referring to "spin up" and "spin down" states. Each electron in any individual orbital must have different quantum numbers because of the Pauli exclusion principle, therefore an orbital never contains more than two electrons.

Note that there is no universal fixed value for m_ℓ and m_s values. Rather, the m_ℓ and m_s values are random. The only requirement is that the naming schematic used within a particular set of calculations or descriptions must be consistent (e.g. the orbital occupied by the first electron in a p orbital could be described as m_ℓ = −1 or m_ℓ = 0 or m_ℓ = 1, but the m_ℓ value of the next unpaired electron in that orbital must be different; yet, the m_ℓ assigned to electrons in other orbitals again can be m_ℓ = −1 or m_ℓ = 0, or m_ℓ = 1 ). 

These rules are summarized as follows:

Name	Symbol	Orbital meaning	Range of values	Value examples
principal quantum number	n	shell	1 ≤ n	n = 1, 2, 3, …
azimuthal quantum number (angular momentum)	ℓ	subshell (s orbital is listed as 0, p orbital as 1 etc.)	0 ≤ ℓ ≤ n − 1	for n = 3: ℓ = 0, 1, 2 (s, p, d)
magnetic quantum number, (projection of angular momentum)	mℓ	energy shift (orientation of the subshell's shape)	−ℓ ≤ mℓ ≤ ℓ	for ℓ = 2: mℓ = −2, −1, 0, 1, 2
spin projection quantum number	ms	spin of the electron (−½ = "spin down", ½ = "spin up")	−s ≤ ms ≤ s	for an electron s = ½, so ms = −½, ½

Example: The quantum numbers used to refer to the outermost valence electrons of the Carbon (C) atom, which are located in the 2p atomic orbital, are; n = 2 (2nd electron shell), ℓ = 1 (p orbital subshell), m_ℓ = 1, 0 or −1, m_s = ½ (parallel spins).

Results from spectroscopy indicated that up to two electrons can occupy a single orbital. However two electrons can never have the same exact quantum state nor the same set of quantum numbers according to Hund's rules, which addresses the Pauli exclusion principle. A fourth quantum number with two possible values was added as an ad hoc assumption to resolve the conflict; this supposition could later be explained in detail by relativistic quantum mechanics and from the results of the renowned Stern–Gerlach experiment.

Total angular momenta numbers

Total momentum of a particle

When one takes the spin–orbit interaction into consideration, the L and S operators no longer commute with the Hamiltonian, and their eigenvalues therefore change over time. Thus another set of quantum numbers should be used. This set includes

1. The total angular momentum quantum number:

   j = |ℓ ± s|,

   which gives the total angular momentum through the relation
   

   J² = ħ² j (j + 1).

2. The projection of the total angular momentum along a specified axis:
   m_j = −j, −j + 1, −j + 2, ..., j − 2, j − 1, j
   analogous to the above and satisfies
   

   m_j = m_ℓ + m_s and |m_ℓ + m_s| ≤ j.

3. Parity This is the eigenvalue under reflection: positive (+1) for states which came from even ℓ and negative (−1) for states which came from odd ℓ. The former is also known as even parity and the latter as odd parity, and is given by
   

   P = (−1)^ℓ.

For example, consider the following 8 states, defined by their quantum numbers:

#	n	ℓ	mℓ	ms		ℓ + s	ℓ − s	mℓ + ms
1.	2	1	1	+1/2		3/2	1/2	3/2
2.	2	1	1	−1/2		3/2	1/2	1/2
3.	2	1	0	+1/2		3/2	1/2	1/2
4.	2	1	0	−1/2		3/2	1/2	−1/2
5.	2	1	−1	+1/2		3/2	1/2	−1/2
6.	2	1	−1	−1/2		3/2	1/2	−3/2
7.	2	0	0	+1/2		1/2	−1/2	1/2
8.	2	0	0	−1/2		1/2	−1/2	−1/2

The [quantum state]s in the system can be described as linear combination of these 8 states. However, in the presence of spin–orbit interaction, if one wants to describe the same system by 8 states that are eigenvectors of the Hamiltonian (i.e. each represents a state that does not mix with others over time), we should consider the following 8 states:

j	mj	parity
3/2	3/2	odd	(coming from state (1) above)
3/2	1/2	odd	(coming from states (2) and (3) above)
3/2	−1/2	odd	(coming from states (4) and (5) above)
3/2	−3/2	odd	(coming from state (6) above)
1/2	1/2	odd	(coming from states (2) and (3) above)
1/2	−1/2	odd	(coming from states (4) and (5) above)
1/2	1/2	even	(coming from state (7) above)
1/2	−1/2	even	(coming from state (8) above)

Nuclear angular momentum quantum numbers

In nuclei, the entire assembly of protons and neutrons (nucleons) has a resultant angular momentum due to the angular momenta of each nucleon, usually denoted 'I'. If the total angular momentum of a neutron is j_n = ℓ + s and for a proton is j_p = ℓ + s (where s for protons and neutrons happens to be ½ again) then the nuclear angular momentum quantum numbers I are given by:

I = |j_n − j_p|, |j_n − j_p| + 1, |j_n − j_p| + 2, ...,

(j_n + j_p) − 2, (j_n + j_p) − 1, (j_n + j_p)

Parity with the number I is used to label nuclear angular momentum states, examples for some isotopes of Hydrogen (H), Carbon (C), and Sodium (Na) are;

1 1H	I = (1/2)+		9 6C	I = (3/2)−		20 11Na	I = 2+
2 1H	I = 1+		10 6C	I = 0+		21 11Na	I = (3/2)+
3 1H	I = (1/2)+		11 6C	I = (3/2)−		22 11Na	I = 3+
			12 6C	I = 0+		23 11Na	I = (3/2)+
			13 6C	I = (1/2)−		24 11Na	I = 4+
			14 6C	I = 0+		25 11Na	I = (5/2)+
			15 6C	I = (1/2)+		26 11Na	I = 3+

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

Elementary particles

Elementary particles contain many quantum numbers which are usually said to be intrinsic to them. However, it should be understood that 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 are lepton number and baryon number or the electric charge. (For a full list of quantum numbers of this kind see the article on flavor.)

Multiplicative quantum numbers

A minor but often confusing point is as follows: 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 a parity, 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).