Angular momentum operator#Quantization

File:LS coupling (corrected).png between measuring angular momentum components (see below). ]]

In quantum mechanics, angular momentum can refer to one of three different, but related things.

The classical definition of angular momentum is $\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}$. The quantum-mechanical counterparts of these objects share the same relationship:

$$\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}$$

where r is the quantum position operator, p is the quantum momentum operator, × is cross product, and L is the orbital angular momentum operator. L (just like p and r) is a vector operator (a vector whose components are operators), i.e. $\backslash mathbf\{L\}\; =\; \backslash left(L\_x,\; L\_y,\; L\_z\backslash right)$ where L_x, L_y, L_z are three different quantum-mechanical operators.

In the special case of a single particle with no electric charge and no spin, the orbital angular momentum operator can be written in the position basis as:$$\backslash mathbf\{L\}\; =\; -i\backslash hbar(\backslash mathbf\{r\}\; \backslash times\; \backslash nabla)$$

where {{math|∇}} is the vector differential operator, del.

{{main|Spin (physics)}}

There is another type of angular momentum, called spin angular momentum (more often shortened to spin), represented by the spin operator $\backslash mathbf\{S\}\; =\; \backslash left(S\_x,\; S\_y,\; S\_z\backslash right)$. Spin is often depicted as a particle literally spinning around an axis, but this is only a metaphor: the closest classical analog is based on wave circulation.{{Cite journal |last=Ohanian |first=Hans C. |date=1986-06-01 |title=What is spin? |url=https://physics.mcmaster.ca/phys3mm3/notes/whatisspin.pdf |journal=American Journal of Physics |language=en |volume=54 |issue=6 |pages=500–505 |doi=10.1119/1.14580 |bibcode=1986AmJPh..54..500O |issn=0002-9505}} All elementary particles have a characteristic spin (scalar bosons have zero spin). For example, electrons always have "spin 1/2" while photons always have "spin 1" (details below).

Finally, there is total angular momentum $\backslash mathbf\{J\}\; =\; \backslash left(J\_x,\; J\_y,\; J\_z\backslash right)$, which combines both the spin and orbital angular momentum of a particle or system:

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

Conservation of angular momentum states that J for a closed system, or J for the whole universe, is conserved. However, L and S are not generally conserved. For example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total J remaining constant.

The orbital angular momentum operator is a vector operator, meaning it can be written in terms of its vector components $\backslash mathbf\{L\}\; =\; \backslash left(L\_x,\; L\_y,\; L\_z\backslash right)$. The components have the following commutation relations with each other:{{cite book|chapter-url=https://books.google.com/books?id=dRsvmTFpB3wC&pg=PA171|title= Quantum Mechanics|first=G. |last=Aruldhas |page=171|chapter= formula (8.8) | isbn=978-81-203-1962-2 |date=2004-02-01|publisher= Prentice Hall India}}

$$\backslash left[L\_x,\; L\_y\backslash right]\; =\; i\backslash hbar\; L\_z,\; \backslash ;\backslash ;\; \backslash left[L\_y,\; L\_z\backslash right]\; =\; i\backslash hbar\; L\_x,\; \backslash ;\backslash ;\; \backslash left[L\_z,\; L\_x\backslash right]\; =\; i\backslash hbar\; L\_y,$$

where {{math|[ , ]}} denotes the commutator

$$[X,\; Y]\; \backslash equiv\; XY\; -\; YX.$$

This can be written generally as

$$\backslash left[L\_l,\; L\_m\backslash right]\; =\; i\; \backslash hbar\; \backslash sum\_\{n=1\}^\{3\}\; \backslash varepsilon\_\{lmn\}\; L\_n,$$

where l, m, n are the component indices (1 for x, 2 for y, 3 for z), and {{math|ε_lmn}} denotes the Levi-Civita symbol.

A compact expression as one vector equation is also possible:{{cite book |last1=Shankar| first1=R. |title=Principles of quantum mechanics | url=https://archive.org/details/principlesquantu00shan_139 |url-access=limited | date=1994|publisher=Kluwer Academic / Plenum|location=New York | isbn=9780306447907 |page=[https://archive.org/details/principlesquantu00shan_139/page/n338 319]|edition=2nd}}

$$\backslash mathbf\{L\}\; \backslash times\; \backslash mathbf\{L\}\; =\; i\backslash hbar\; \backslash mathbf\{L\}$$

The commutation relations can be proved as a direct consequence of the canonical commutation relations $[x\_l,p\_m]\; =\; i\; \backslash hbar\; \backslash delta\_\{lm\}$, where {{math|δ_lm}} is the Kronecker delta.

There is an analogous relationship in classical physics:H. Goldstein, C. P. Poole and J. Safko, Classical Mechanics, 3rd Edition, Addison-Wesley 2002, pp. 388 ff.

$$\backslash left\backslash \{L\_i,\; L\_j\backslash right\backslash \}\; =\; \backslash varepsilon\_\{ijk\}\; L\_k$$

where L_n is a component of the classical angular momentum operator, and $\backslash \{\; ,\backslash \}$ is the Poisson bracket.

The same commutation relations apply for the other angular momentum operators (spin and total angular momentum):

$$\backslash left[S\_l,\; S\_m\backslash right]\; =\; i\; \backslash hbar\; \backslash sum\_\{n=1\}^\{3\}\; \backslash varepsilon\_\{lmn\}\; S\_n,\; \backslash quad\; \backslash left[J\_l,\; J\_m\backslash right]\; =\; i\; \backslash hbar\; \backslash sum\_\{n=1\}^\{3\}\; \backslash varepsilon\_\{lmn\}\; J\_n.$$

These can be assumed to hold in analogy with L. Alternatively, they can be derived as discussed below.

These commutation relations mean that L has the mathematical structure of a Lie algebra, and the {{math|ε_lmn}} are its structure constants. In this case, the Lie algebra is SU(2) or SO(3) in physics notation ($\backslash operatorname\{su\}(2)$ or $\backslash operatorname\{so\}(3)$ respectively in mathematics notation), i.e. Lie algebra associated with rotations in three dimensions. The same is true of J and S. The reason is discussed below. These commutation relations are relevant for measurement and uncertainty, as discussed further below.

In molecules the total angular momentum F is the sum of the rovibronic (orbital) angular momentum N, the electron spin angular momentum S, and the nuclear spin angular momentum I. For electronic singlet states the rovibronic angular momentum is denoted J rather than N. As explained by Van Vleck,{{cite journal

| author1=J. H. Van Vleck

| title = The Coupling of Angular Momentum Vectors in Molecules

| journal = Reviews of Modern Physics

| volume = 23 | page= 213 | year = 1951 | issue = 3

| doi = 10.1103/RevModPhys.23.213| bibcode = 1951RvMP...23..213V

}}

the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those given above which are for the components about space-fixed axes.

Like any vector, the square of a magnitude can be defined for the orbital angular momentum operator,

$$L^2\; \backslash equiv\; L\_x^2\; +\; L\_y^2\; +\; L\_z^2.$$

$L^2$ is another quantum operator. It commutes with the components of $\backslash mathbf\{L\}$,

$$\backslash left[L^2,\; L\_x\backslash right]\; =\; \backslash left[L^2,\; L\_y\backslash right]\; =\; \backslash left[L^2,\; L\_z\backslash right]\; =\; 0\; .$$

One way to prove that these operators commute is to start from the [L_ℓ, L_m] commutation relations in the previous section:

{{math proof|title=Proof of [L², L_x] = 0, starting from the [L_ℓ, L_m] commutation relations{{cite book | last=Griffiths | first = David J. | title=Introduction to Quantum Mechanics | url=https://archive.org/details/introductiontoqu00grif_200 | url-access=limited | publisher=Prentice Hall | year=1995 | page=[https://archive.org/details/introductiontoqu00grif_200/page/n159 146] }}

|proof=

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

\left[L^2, L_x\right]

&= \left[L_x^2, L_x\right] + \left[L_y^2, L_x\right] + \left[L_z^2, L_x\right] \\

&= L_y \left[L_y, L_x\right] + \left[L_y, L_x\right] L_y + L_z \left[L_z, L_x\right] + \left[L_z, L_x\right] L_z \\

&= L_y \left(-i \hbar L_z\right) + \left(-i \hbar L_z\right) L_y + L_z \left(i \hbar L_y\right) + \left(i \hbar L_y\right) L_z \\

&= 0

\end{align}

}}

Mathematically, $L^2$ is a Casimir invariant of the Lie algebra SO(3) spanned by $\backslash mathbf\{L\}$.

As above, there is an analogous relationship in classical physics:

$$\backslash left\backslash \{L^2,\; L\_x\backslash right\backslash \}\; =\; \backslash left\backslash \{L^2,\; L\_y\backslash right\backslash \}\; =\; \backslash left\backslash \{L^2,\; L\_z\backslash right\backslash \}\; =\; 0$$

where $L\_i$ is a component of the classical angular momentum operator, and $\backslash \{\; ,\backslash \}$ is the Poisson bracket.Goldstein et al, p. 410

Returning to the quantum case, the same commutation relations apply to the other angular momentum operators (spin and total angular momentum), as well,

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

\left[ S^2, S_i \right] &= 0, \\

\left[ J^2, J_i \right] &= 0.

\end{align}

{{main|Uncertainty principle|Uncertainty principle derivations}}

In general, in quantum mechanics, when two observable operators do not commute, they are called complementary observables. Two complementary observables cannot be measured simultaneously; instead they satisfy an uncertainty principle. The more accurately one observable is known, the less accurately the other one can be known. Just as there is an uncertainty principle relating position and momentum, there are uncertainty principles for angular momentum.

The Robertson–Schrödinger relation gives the following uncertainty principle:

$$\backslash sigma\_\{L\_x\}\; \backslash sigma\_\{L\_y\}\; \backslash geq\; \backslash frac\{\backslash hbar\}\{2\}\; \backslash left|\; \backslash langle\; L\_z\; \backslash rangle\; \backslash right|.$$

where $\backslash sigma\_X$ is the standard deviation in the measured values of X and $\backslash langle\; X\; \backslash rangle$ denotes the expectation value of X. This inequality is also true if x, y, z are rearranged, or if L is replaced by J or S.

Therefore, two orthogonal components of angular momentum (for example L_x and L_y) are complementary and cannot be simultaneously known or measured, except in special cases such as $L\_x\; =\; L\_y\; =\; L\_z\; =\; 0$.

It is, however, possible to simultaneously measure or specify L² and any one component of L; for example, L² and L_z. This is often useful, and the values are characterized by the azimuthal quantum number (l) and the magnetic quantum number (m). In this case the quantum state of the system is a simultaneous eigenstate of the operators L² and L_z, but not of L_x or L_y. The eigenvalues are related to l and m, as shown in the table below.

{{see also|Azimuthal quantum number|Magnetic quantum number}}

In quantum mechanics, angular momentum is quantized – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where $\backslash hbar$ is reduced Planck constant:{{cite book |last1=Condon |first1=E. U. |author-link1= Edward Condon |last2=Shortley |first2=G. H. |title = Quantum Theory of Atomic Spectra |url=https://books.google.com/books?id=hPyD-Nc_YmgC |publisher=Cambridge University Press |year=1935 |chapter=Chapter III: Angular Momentum |chapter-url= https://books.google.com/books?id=hPyD-Nc_YmgC&pg=PA45 |isbn=9780521092098}}

{{main|Ladder operator#Angular momentum}}

A common way to derive the quantization rules above is the method of ladder operators.{{cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | url=https://archive.org/details/introductiontoqu00grif_200 | url-access=limited | publisher=Prentice Hall | year=1995 | pages=[https://archive.org/details/introductiontoqu00grif_200/page/n160 147]–149}} The ladder operators for the total angular momentum $\backslash mathbf\{J\}\; =\; \backslash left(J\_x,\; J\_y,\; J\_z\backslash right)$ are defined as:

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

J_+ &\equiv J_x + i J_y, \\

J_- &\equiv J_x - i J_y

\end{align}

Suppose $|\backslash psi\backslash rangle$ is a simultaneous eigenstate of $J^2$ and $J\_z$ (i.e., a state with a definite value for $J^2$ and a definite value for $J\_z$). Then using the commutation relations for the components of $\backslash mathbf\{J\}$, one can prove that each of the states $J\_+\; |\backslash psi\backslash rangle$ and $J\_-|\backslash psi\backslash rangle$ is either zero or a simultaneous eigenstate of $J^2$ and $J\_z$, with the same value as $|\backslash psi\backslash rangle$ for $J^2$ but with values for $J\_z$ that are increased or decreased by $\backslash hbar$ respectively. The result is zero when the use of a ladder operator would otherwise result in a state with a value for $J\_z$ that is outside the allowable range. Using the ladder operators in this way, the possible values and quantum numbers for $J^2$ and $J\_z$ can be found.

{{math proof

| title = Derivation of the possible values and quantum numbers for $J\_z$ and $J^2$.{{harvnb|Condon|Shortley|1935|pp=[https://books.google.com/books?id=hPyD-Nc_YmgC&pg=PA46 46–47]}}

| proof =

Let $\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\; )$ be a state function for the system with eigenvalue $\{J^2\}\text{'}$ for $J^2$ and eigenvalue $J\_z\text{'}$ for $J\_z$.{{NoteTag|In the derivation of Condon and Shortley that the current derivation is based on, a set of observables $\backslash Gamma$ along with $J^2$ and $J\_z$ form a complete set of commuting observables. Additionally they required that $\backslash Gamma$ commutes with $J\_x$ and $J\_y$. The present derivation is simplified by not including the set $\backslash Gamma$ or its corresponding set of eigenvalues $\backslash gamma$.}}

From $J^2\; =\; J\_x^2\; +J\_y^2\; +\; J\_z^2$ is obtained,

$$J\_x^2\; +J\_y^2\; =\; J^2\; -\; J\_z^2\; .$$

Applying both sides of the above equation to $\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\; )$,

$$(J\_x^2\; +J\_y^2)\; \backslash ;\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\; )\; =\; (\{J^2\}\text{'}\; -\; J\_z\text{'}^2)\; \backslash ;\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\; ).$$

Since $J\_x$ and $J\_y$ are real observables, $\{J^2\}\text{'}-J\_z\text{'}^2$ is not negative and $|J\_z\text{'}|\; \backslash le\; \backslash sqrt\{\; \{J^2\}\text{'}\}$. Thus $J\_z\text{'}$ has an upper and lower bound.

Two of the commutation relations for the components of $\backslash mathbf\{J\}$ are,

$$[J\_y,\; J\_z]\; =\; i\backslash hbar\; J\_x,\; \backslash ;\backslash ;\; [J\_z,\; J\_x]\; =\; i\backslash hbar\; J\_y.$$

They can be combined to obtain two equations, which are written together using $\backslash pm$ signs in the following,

$$J\_z(J\_x\backslash pm\; iJ\_y)\; =\; (J\_x\backslash pm\; iJ\_y)(J\_z\backslash pm\; \backslash hbar)\; ,$$

where one of the equations uses the $+$ signs and the other uses the $-$ signs.

Applying both sides of the above to $\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\; )$,

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

J_z(J_x\pm iJ_y) \;\psi ({J^2}' J_{z}' )

& = (J_x\pm iJ_y)(J_z\pm \hbar) \;\psi ({J^2}' J_z' ) \\

& = (J_z'\pm \hbar)(J_x\pm iJ_y) \;\psi ({J^2}' J_z' )\;. \\

\end{align}

The above shows that $(J\_x\backslash pm\; iJ\_y)\; \backslash ;\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'})$ are two eigenfunctions of $J\_z$ with respective eigenvalues $\{J\_z\}\text{'}\backslash pm\; \backslash hbar$ , unless one of the functions is zero, in which case it is not an eigenfunction. For the functions that are not zero,

$$\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\backslash pm\backslash hbar\; )\; =\; (J\_x\backslash pm\; iJ\_y)\; \backslash ;\backslash psi\; (\{J^2\}\text{'}\; J\_z\text{'}\; )\; .$$

Further eigenfunctions of $J\_z$ and corresponding eigenvalues can be found by repeatedly applying $J\_x\backslash pm\; iJ\_y$ as long as the magnitude of the resulting eigenvalue is $\backslash le\; \backslash sqrt\{\{J^2\}\text{'}\}$.

Since the eigenvalues of $J\_z$ are bounded, let $J\_z^0$ be the lowest eigenvalue and $J\_z^1$ be the highest. Then

$$(J\_x-iJ\_y)\; \backslash ;\backslash psi\; (\{J^2\}\text{'}\; J\_z^0\; )\; =\; 0$$ and

$$(J\_x+iJ\_y)\; \backslash ;\backslash psi\; (\{J^2\}\text{'}\; J\_z^1\; )\; =\; 0\; ,$$

since there are no states where the eigenvalue of $J\_z$ is

$$\{J^2\}\text{'}-(J\_z^0)^2+\backslash hbar\; J\_z^0\; =\; 0$$ and

$$\{J^2\}\text{'}-(J\_z^1)^2-\backslash hbar\; J\_z^1\; =\; 0\; .$$

Subtracting the first equation from the second and rearranging,

$$(J\_z^1+J\_z^0)(J\_z^0-J\_z^1-\backslash hbar)\; =\; 0\; .$$

Since $J\_z^1\; \backslash ge\; J\_z^0$, the second factor is negative. Then the first factor must be zero and thus $J\_z^0\; =\; -J\_z^1$.

The difference $J\_z^1-J\_z^0$ comes from successive application of $J\_x-iJ\_y$ or $J\_x+iJ\_y$ which lower or raise the eigenvalue of $J\_z$ by $\backslash hbar$ so that,

$$J\_z^1-J\_z^0\; =\; 0,\; \backslash hbar,\; 2\backslash hbar,\; \backslash dots$$

Let

$$J\_z^1-J\_z^0\; =\; 2j\backslash hbar,$$ where $$j\; =\; 0,\; \backslash tfrac\{1\}\{2\},\; 1,\; \backslash tfrac\{3\}\{2\},\; \backslash dots\; \backslash ;.$$

Then using $J\_z^0\; =\; -J\_z^1$ and the above,

$$J\_z^0\; =\; -j\backslash hbar$$ and $$J\_z^1\; =\; j\backslash hbar\; ,$$

and the allowable eigenvalues of $J\_z$ are

$$J\_z\text{'}\; =\; -j\backslash hbar,\; -j\backslash hbar+\backslash hbar,\; -j\backslash hbar+2\backslash hbar,\; \backslash dots,\; j\backslash hbar\; .$$

Expressing $J\_z\text{'}$ in terms of a quantum number $m\_j\; \backslash ;$, and substituting $J\_z^0=-j\backslash hbar$ into $\{J^2\}\text{'}-(J\_z^0)^2+\backslash hbar\; J\_z^0=0$ from above,

{{equation box 1

|align=left

|fontsize=100%

|border=2px

|equation=$\backslash begin\{align\}$

J_z' &= m_j\hbar &

m_j &= -j, -j + 1, -j + 2, \dots, j \\

{J^2}' &= j(j+1)\hbar^2 &

j &= 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \dots \;.

\end{align}}}

}}

Since $\backslash mathbf\{S\}$ and $\backslash mathbf\{L\}$ have the same commutation relations as $\backslash mathbf\{J\}$, the same ladder analysis can be applied to them, except that for $\backslash mathbf\{L\}$ there is a further restriction on the quantum numbers that they must be integers.

{{math proof

| title = Derivation of the restriction to integer quantum numbers for $L\_z$ and $L^2$.{{harvnb|Condon|Shortley|1935|pages=[https://books.google.com/books?id=hPyD-Nc_YmgC&pg=PA50 50–51]}}

| proof =

In the Schroedinger representation, the z component of the orbital angular momentum operator can be expressed in spherical coordinates as,{{harvnb|Condon|Shortley|1935|p=50, Eq 1}}

$$L\_z\; =\; -i\backslash hbar\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash phi\}.$$

For $L\_z$ and eigenfunction $\backslash psi$ with eigenvalue $L\_z\text{'}$,

$$-i\backslash hbar\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; \backslash phi\}\backslash psi\; =\; L\_z\text{'}\; \backslash psi.$$

Solving for $\backslash psi$,

$$\backslash psi\; =\; A\; e^\{iL\_z\text{'}\backslash phi/\backslash hbar\},$$

where $A$ is independent of $\backslash phi$. Since $\backslash psi$ is required to be single valued, and adding $2\backslash pi$ to $\backslash phi$ results in a coordinate for the same point in space,

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

A e^{iL_z'(\phi + 2\pi)/\hbar} &= A e^{iL_z'\phi/\hbar}, \\

e^{iL_z'2\pi/\hbar} &= 1 .

\end{align}

Solving for the eigenvalue $L\_z\text{'}$,

$$L\_z\text{'}\; =\; m\_l\backslash hbar\; \backslash ;,$$

where $m\_l$ is an integer.{{harvnb|Condon|Shortley|1935|p=50, Eq 3}}

From the above and the relation $m\_\backslash ell\; =\; -\backslash ell,\; (-\backslash ell\; +\; 1),\; \backslash ldots,\; (\backslash ell\; -\; 1),\; \backslash ell\backslash \; \backslash$, it follows that $\backslash ell$ is also an integer. This shows that the quantum numbers $m\_\backslash ell$ and $\backslash ell$ for the orbital angular momentum $\backslash mathbf\{L\}$ are restricted to integers, unlike the quantum numbers for the total angular momentum $\backslash mathbf\{J\}$ and spin $\backslash mathbf\{S\}$, which can have half-integer values.{{harvnb|Condon|Shortley|1935|p=51}}

}}

File:Vector model of orbital angular momentum.svg

Since the angular momenta are quantum operators, they cannot be drawn as vectors like in classical mechanics. Nevertheless, it is common to depict them heuristically in this way. Depicted on the right is a set of states with quantum numbers $\backslash ell\; =\; 2$, and $m\_\backslash ell\; =\; -2,\; -1,\; 0,\; 1,\; 2$ for the five cones from bottom to top. Since $|L|\; =\; \backslash sqrt\{L^2\}\; =\; \backslash hbar\; \backslash sqrt\{6\}$, the vectors are all shown with length $\backslash hbar\; \backslash sqrt\{6\}$. The rings represent the fact that $L\_z$ is known with certainty, but $L\_x$ and $L\_y$ are unknown; therefore every classical vector with the appropriate length and z-component is drawn, forming a cone. The expected value of the angular momentum for a given ensemble of systems in the quantum state characterized by $\backslash ell$ and $m\_\backslash ell$ could be somewhere on this cone while it cannot be defined for a single system (since the components of $L$ do not commute with each other).

The quantization rules are widely thought to be true even for macroscopic systems, like the angular momentum L of a spinning tire. However they have no observable effect so this has not been tested. For example, if $L\_z/\backslash hbar$ is roughly 100000000, it makes essentially no difference whether the precise value is an integer like 100000000 or 100000001, or a non-integer like 100000000.2—the discrete steps are currently too small to measure. For most intents and purposes, the assortment of all the possible values of angular momentum is effectively continuous at macroscopic scales.{{cite web |last1=Downes |first1=Sean |title=Spin Angular Momentum |url=https://pasayteninstitute.substack.com/p/spin-angular-momentum |website=Physics! |date=29 July 2022}}

{{see also|Total angular momentum quantum number}}

The most general and fundamental definition of angular momentum is as the generator of rotations.{{cite web|url=http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf|title=Lecture notes on rotations in quantum mechanics|first=Robert|last=Littlejohn|author-link1=Robert Grayson Littlejohn|access-date=13 Jan 2012|work=Physics 221B Spring 2011|year=2011|archive-date=26 August 2014|archive-url=https://web.archive.org/web/20140826003155/http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf|url-status=dead}} More specifically, let $R(\backslash hat\{n\},\backslash phi)$ be a rotation operator, which rotates any quantum state about axis $\backslash hat\{n\}$ by angle $\backslash phi$. As $\backslash phi\backslash rightarrow\; 0$, the operator $R(\backslash hat\{n\},\backslash phi)$ approaches the identity operator, because a rotation of 0° maps all states to themselves. Then the angular momentum operator $J\_\{\backslash hat\{n\}\}$ about axis $\backslash hat\{n\}$ is defined as:

$$J\_\backslash hat\{n\}\; \backslash equiv\; i\backslash hbar\; \backslash lim\_\{\backslash phi\; \backslash rightarrow\; 0\}\; \backslash frac\{R\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; -\; 1\}\{\backslash phi\}\; =\; \backslash left.\; i\backslash hbar\; \backslash frac\{\backslash partial\; R\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\}\{\backslash partial\backslash phi\}\; \backslash right|\_\{\backslash phi\; =\; 0\}$$

where 1 is the identity operator. Also notice that R is an additive morphism : $R\backslash left(\backslash hat\{n\},\; \backslash phi\_1\; +\; \backslash phi\_2\backslash right)\; =\; R\backslash left(\backslash hat\{n\},\; \backslash phi\_1\backslash right)R\backslash left(\backslash hat\{n\},\; \backslash phi\_2\backslash right)$ ; as a consequence

$$R\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; =\; \backslash exp\backslash left(-\backslash frac\{i\; \backslash phi\; J\_\backslash hat\{n\}\}\{\backslash hbar\}\backslash right)$$

where exp is matrix exponential. The existence of the generator is guaranteed by the Stone's theorem on one-parameter unitary groups.

In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics, as discussed further below.

File:RotationOperators.svg. The top box shows two particles, with spin states indicated schematically by the arrows.

{{ordered list

| list-style-type = upper-alpha

| The operator R, related to J, rotates the entire system.

| The operator R_spatial, related to L, rotates the particle positions without altering their internal spin states.

| The operator R_internal, related to S, rotates the particles' internal spin states without changing their positions.

}}]]

Just as J is the generator for rotation operators, L and S are generators for modified partial rotation operators. The operator

$$R\_\backslash text\{spatial\}\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; =\; \backslash exp\backslash left(-\backslash frac\{i\; \backslash phi\; L\_\backslash hat\{n\}\}\{\backslash hbar\}\backslash right),$$

rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator

$$R\_\backslash text\{internal\}\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; =\; \backslash exp\backslash left(-\backslash frac\{i\; \backslash phi\; S\_\backslash hat\{n\}\}\{\backslash hbar\}\backslash right),$$

rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation J = L + S comes from:

$$R\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; =\; R\_\backslash text\{internal\}\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; R\_\backslash text\{spatial\}\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)$$

i.e. if the positions are rotated, and then the internal states are rotated, then altogether the complete system has been rotated.

{{main|Spin (physics)}}

Although one might expect $R\backslash left(\backslash hat\{n\},\; 360^\backslash circ\backslash right)\; =\; 1$ (a rotation of 360° is the identity operator), this is not assumed in quantum mechanics, and it turns out it is often not true: When the total angular momentum quantum number is a half-integer (1/2, 3/2, etc.), $R\backslash left(\backslash hat\{n\},\; 360^\backslash circ\backslash right)\; =\; -1$, and when it is an integer, $R\backslash left(\backslash hat\{n\},\; 360^\backslash circ\backslash right)\; =\; +1$. Mathematically, the structure of rotations in the universe is not SO(3), the group of three-dimensional rotations in classical mechanics. Instead, it is SU(2), which is identical to SO(3) for small rotations, but where a 360° rotation is mathematically distinguished from a rotation of 0°. (A rotation of 720° is, however, the same as a rotation of 0°.)

On the other hand, $R\_\backslash text\{spatial\}\backslash left(\backslash hat\{n\},\; 360^\backslash circ\backslash right)\; =\; +1$ in all circumstances, because a 360° rotation of a spatial configuration is the same as no rotation at all. (This is different from a 360° rotation of the internal (spin) state of the particle, which might or might not be the same as no rotation at all.) In other words, the $R\_\backslash text\{spatial\}$ operators carry the structure of SO(3), while $R$ and $R\_\backslash text\{internal\}$ carry the structure of SU(2).

From the equation $+1\; =\; R\_\backslash text\{spatial\}\backslash left(\backslash hat\{z\},\; 360^\backslash circ\backslash right)\; =\; \backslash exp\backslash left(-2\backslash pi\; i\; L\_z\; /\; \backslash hbar\backslash right)$, one picks an eigenstate $L\_z\; |\backslash psi\backslash rangle\; =\; m\backslash hbar\; |\backslash psi\backslash rangle$ and draws

$$e^\{-2\backslash pi\; i\; m\}\; =\; 1$$

which is to say that the orbital angular momentum quantum numbers can only be integers, not half-integers.

{{main|Particle physics and representation theory|Representation theory of SU(2)|Rotation group SO(3)#A note on Lie algebras }}

Starting with a certain quantum state $|\backslash psi\_0\backslash rangle$, consider the set of states $R\backslash left(\backslash hat\{n\},\; \backslash phi\backslash right)\; \backslash left|\backslash psi\_0\backslash right\backslash rangle$ for all possible $\backslash hat\{n\}$ and $\backslash phi$, i.e. the set of states that come about from rotating the starting state in every possible way. The linear span of that set is a vector space, and therefore the manner in which the rotation operators map one state onto another is a representation of the group of rotation operators.

{{block indent | text = When rotation operators act on quantum states, it forms a representation of the Lie group SU(2) (for R and R_internal), or SO(3) (for R_spatial).}}

From the relation between J and rotation operators,

{{block indent | text = When angular momentum operators act on quantum states, it forms a representation of the Lie algebra $\backslash mathfrak\{su\}(2)$ or $\backslash mathfrak\{so\}(3)$.}}

(The Lie algebras of SU(2) and SO(3) are identical.)

The ladder operator derivation above is a method for classifying the representations of the Lie algebra SU(2).

Classical rotations do not commute with each other: For example, rotating 1° about the x-axis then 1° about the y-axis gives a slightly different overall rotation than rotating 1° about the y-axis then 1° about the x-axis. By carefully analyzing this noncommutativity, the commutation relations of the angular momentum operators can be derived.

(This same calculational procedure is one way to answer the mathematical question "What is the Lie algebra of the Lie groups SO(3) or SU(2)?")

The Hamiltonian H represents the energy and dynamics of the system. In a spherically symmetric situation, the Hamiltonian is invariant under rotations:

$$RHR^\{-1\}\; =\; H$$

where R is a rotation operator. As a consequence, $[H,\; R]\; =\; 0$, and then $[H,\backslash mathbf\{J\}]=\backslash mathbf\; 0$ due to the relationship between J and R. By the Ehrenfest theorem, it follows that J is conserved.

To summarize, if H is rotationally-invariant (The Hamiltonian function defined on an inner product space is said to have rotational invariance if its value does not change when arbitrary rotations are applied to its coordinates.), then total angular momentum J is conserved. This is an example of Noether's theorem.

If H is just the Hamiltonian for one particle, the total angular momentum of that one particle is conserved when the particle is in a central potential (i.e., when the potential energy function depends only on $\backslash left|\backslash mathbf\{r\}\backslash right|$). Alternatively, H may be the Hamiltonian of all particles and fields in the universe, and then H is always rotationally-invariant, as the fundamental laws of physics of the universe are the same regardless of orientation. This is the basis for saying conservation of angular momentum is a general principle of physics.

For a particle without spin, J = L, so orbital angular momentum is conserved in the same circumstances. When the spin is nonzero, the spin–orbit interaction allows angular momentum to transfer from L to S or back. Therefore, L is not, on its own, conserved.

{{main|Angular momentum coupling|Clebsch–Gordan coefficients}}

Often, two or more sorts of angular momentum interact with each other, so that angular momentum can transfer from one to the other. For example, in spin–orbit coupling, angular momentum can transfer between L and S, but only the total J = L + S is conserved. In another example, in an atom with two electrons, each has its own angular momentum J₁ and J₂, but only the total J = J₁ + J₂ is conserved.

In these situations, it is often useful to know the relationship between, on the one hand, states where $\backslash left(J\_1\backslash right)\_z,\; \backslash left(J\_1\backslash right)^2,\; \backslash left(J\_2\backslash right)\_z,\; \backslash left(J\_2\backslash right)^2$ all have definite values, and on the other hand, states where $\backslash left(J\_1\backslash right)^2,\; \backslash left(J\_2\backslash right)^2,\; J^2,\; J\_z$ all have definite values, as the latter four are usually conserved (constants of motion). The procedure to go back and forth between these bases is to use Clebsch–Gordan coefficients.

One important result in this field is that a relationship between the quantum numbers for $\backslash left(J\_1\backslash right)^2,\; \backslash left(J\_2\backslash right)^2,\; J^2$:

$$j\; \backslash in\; \backslash left\backslash \{\; \backslash left|j\_1\; -\; j\_2\backslash right|,\; \backslash left(\backslash left|j\_1\; -\; j\_2\backslash right|\; +\; 1\backslash right),\; \backslash ldots,\; \backslash left(j\_1\; +\; j\_2\backslash right)\; \backslash right\backslash \}\; .$$

For an atom or molecule with J = L + S, the term symbol gives the quantum numbers associated with the operators $L^2,\; S^2,\; J^2$.

Angular momentum operators usually occur when solving a problem with spherical symmetry in spherical coordinates. The angular momentum in the spatial representation is{{Cite book

| publisher = Springer Berlin Heidelberg

| last = Bes

| first = Daniel R.

| isbn = 978-3-540-46215-6

| title = Quantum Mechanics

| series = Advanced Texts in Physics

| location = Berlin, Heidelberg

| year = 2007

| page= 70

| doi = 10.1007/978-3-540-46216-3

| bibcode = 2007qume.book.....B

}}Compare and contrast with the contragredient classical L.

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

\mathbf L &= i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right) \\

&= i\hbar\left(

\hat{\mathbf{x}} \left(\sin(\phi) \frac{\partial}{\partial\theta} + \cot(\theta)\cos(\phi) \frac{\partial}{\partial\phi}

\right)

+ \hat{\mathbf{y}} \left(-\cos(\phi)\frac{\partial}{\partial\theta} + \cot(\theta)\sin(\phi) \frac{\partial}{\partial\phi}\right)

- \hat{\mathbf z} \frac{\partial}{\partial\phi}

\right) \\

L_+ &= \hbar e^{i\phi} \left( \frac{\partial}{\partial\theta} + i\cot(\theta) \frac{\partial}{\partial\phi} \right), \\

L_- &= \hbar e^{-i\phi} \left( -\frac{\partial}{\partial \theta} + i\cot(\theta) \frac{\partial}{\partial\phi} \right), \\

L^2 &= -\hbar^2 \left(\frac{1}{\sin(\theta)} \frac{\partial}{\partial\theta} \left(\sin(\theta) \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}\right), \\

L_z &= -i \hbar \frac{\partial}{\partial\phi}.

\end{align}

In spherical coordinates the angular part of the Laplace operator can be expressed by the angular momentum. This leads to the relation

$$\backslash Delta\; =\; \backslash frac\{1\}\{r^2\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; \backslash left(r^2\backslash ,\; \backslash frac\{\backslash partial\}\{\backslash partial\; r\}\backslash right)\; -\; \backslash frac\{L^2\}\{\backslash hbar^\{2\}\; r^2\}.$$

When solving to find eigenstates of the operator $L^\{2\}$, we obtain the following

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

L^2 \left| \ell, m \right\rangle &= \hbar^2 \ell(\ell + 1) \left| \ell, m \right\rangle \\

L_z \left| \ell, m \right\rangle &= \hbar m \left| \ell, m \right\rangle

\end{align}

where

$$\backslash left\backslash langle\; \backslash theta,\; \backslash phi\; |\; \backslash ell,\; m\; \backslash right\backslash rangle\; =\; Y\_\{\backslash ell,m\}(\backslash theta,\; \backslash phi)$$

are the spherical harmonics.Sakurai, JJ & Napolitano, J (2010), Modern Quantum Mechanics (2nd edition) (Pearson) {{isbn|978-0805382914}}

{{colbegin}}

• Runge–Lenz vector (used to describe the shape and orientation of bodies in orbit)
• Holstein–Primakoff transformation
• Jordan map (Schwinger's bosonic model of angular momentum{{cite book |last=Schwinger |first=Julian |date=1952 |title=On Angular Momentum |url=http://www.ifi.unicamp.br/~cabrera/teaching/paper_schwinger.pdf |publisher=U.S. Atomic Energy Commission}})
• Pauli–Lubanski pseudovector
• Angular momentum diagrams (quantum mechanics)
• Spherical basis
• Tensor operator
• Orbital magnetization
• Orbital angular momentum of free electrons
• Orbital angular momentum of light

{{colend}}

{{NoteFoot}}

• {{Cite book|title=Quantum Mechanics|first=E. |last=Abers|publisher=Addison Wesley, Prentice Hall Inc|year=2004|isbn=978-0-13-146100-0}}
• {{Cite book |last1=Biedenharn |first1=L. C. |url=https://www.cambridge.org/core/books/angular-momentum-in-quantum-physics/53AFDEE1D64D0256AD874534F084C402 |title=Angular Momentum in Quantum Physics: Theory and Application |last2=Louck |first2=James D. |date=1984 |publisher=Cambridge University Press |isbn=978-0-521-30228-9 |series=Encyclopedia of Mathematics and its Applications |location=Cambridge |doi=10.1017/cbo9780511759888|bibcode=1984amqp.book.....B |author-link=Lawrence Biedenharn}}
• {{Cite book|title=Physics of Atoms and Molecules|first1=B.H.|last1=Bransden|first2=C.J.|last2=Joachain|publisher=Longman|year=1983|isbn=0-582-44401-2}}
• {{Cite book|chapter-url=https://www.feynmanlectures.caltech.edu/III_18.html|title=The Feynman Lectures on Physics Vol. III|edition=The New Millennium|chapter=Ch. 18: Angular Momentum|first1=Richard P.|last1=Feynman|first2=Robert B.|last2=Leighton|first3=Matthew|last3=Sands}}
• {{Cite book|title=Quantum Mechanics Demystified|first=D.|last=McMahon|publisher=Mc Graw Hill (USA)|year=2006|isbn=0-07-145546 9}}
• {{Cite book |last=Zare|first=R.N.|title=Angular Momentum. Understanding Spatial Aspects in Chemistry and Physics|publisher=Wiley-Interscience|year=1991|isbn=978-0-47-1858928}}

{{Physics operator}}

Category:Angular momentum

Category:Quantum operators

Category:Rotational symmetry