Angular momentum in quantum mechanics

Angular momentum in quantum mechanics differs in many profound respects from angular momentum in classical mechanics. In relativistic quantum mechanics, it differs even more, in which the above relativistic definition becomes a tensorial operator.

Spin, orbital, and total angular momentumedit

The classical definition of angular momentum as ${\displaystyle \mathbf{L}=\mathbf{r}\times <!--\; \times \; -->\mathbf{p}}$ can be carried over to quantum mechanics, by reinterpreting r as the quantum position operator and p as the quantum momentum operator. L is then an operator, specifically called the orbital angular momentum operator. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum Hilbert space. (See also the discussion below of the angular momentum operators as the generators of rotations.)

However, in quantum physics, there is another type of angular momentum, called spin angular momentum, represented by the spin operator S. Almost all elementary particles have nonvanishing spin. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All elementary particles have a characteristic spin (possibly zero), for example electrons have "spin 1/2" (this actually means "spin ħ/2"), photons have "spin 1" (this actually means "spin ħ"), and pi-mesons have spin 0.

Finally, there is total angular momentum J, which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, J = L + S.) Conservation of angular momentum applies to J, but not to L or S; for example, the spin–orbit interaction allows angular momentum to transfer back and forth between L and S, with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have fractional values.

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

Quantizationedit

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 ${\displaystyle \mathrm{\hslash <!--\; \hslash \; -->}}$ is the reduced Planck constant and ${\displaystyle \stackrel{^<!--\; ^\; -->}{n}}$ is any Euclidean vector such as x, y, or z:

If you measure...	The result can be...
${\displaystyle L_{\stackrel{^<!--\; ^\; -->}{n}}}$	${\displaystyle \dots <!--\; \dots \; -->,-<!--\; -\; -->2\mathrm{\hslash <!--\; \hslash \; -->},-<!--\; -\; -->\mathrm{\hslash <!--\; \hslash \; -->},0,\mathrm{\hslash <!--\; \hslash \; -->},2\mathrm{\hslash <!--\; \hslash \; -->},\dots <!--\; \dots \; -->}$
${\displaystyle S_{\stackrel{^<!--\; ^\; -->}{n}}}$ or ${\displaystyle J_{\stackrel{^<!--\; ^\; -->}{n}}}$	${\displaystyle \dots <!--\; \dots \; -->,-<!--\; -\; -->\frac{3}{2}\mathrm{\hslash <!--\; \hslash \; -->},-<!--\; -\; -->\mathrm{\hslash <!--\; \hslash \; -->},-<!--\; -\; -->\frac{1}{2}\mathrm{\hslash <!--\; \hslash \; -->},0,\frac{1}{2}\mathrm{\hslash <!--\; \hslash \; -->},\mathrm{\hslash <!--\; \hslash \; -->},\frac{3}{2}\mathrm{\hslash <!--\; \hslash \; -->},\dots <!--\; \dots \; -->}$
	${\displaystyle {\mathrm{\hslash <!--\; \hslash \; -->}}^{2}n(n+1)}$, where ${\displaystyle n=0,1,2,\dots <!--\; \dots \; -->}$
${\displaystyle S^2}$ or ${\displaystyle J^2}$	${\displaystyle {\mathrm{\hslash <!--\; \hslash \; -->}}^{2}n(n+1)}$, where ${\displaystyle n=0,\frac{1}{2},1,\frac{3}{2},\dots <!--\; \dots \; -->}$

(There are additional restrictions as well, see angular momentum operator for details.)

The reduced Planck constant ${\displaystyle \mathrm{\hslash <!--\; \hslash \; -->}}$ is tiny by everyday standards, about 10−34 J s, and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of electron shells and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by Niels Bohr in his Bohr model of the atom and was later predicted by Erwin Schrödinger in his Schrödinger equation.

Uncertaintyedit

In the definition ${\displaystyle \mathbf{L}=\mathbf{r}\times <!--\; \times \; -->\mathbf{p}}$, six operators are involved: The position operators ${\displaystyle r_x}$, ${\displaystyle r_y}$, ${\displaystyle r_z}$, and the momentum operators ${\displaystyle p_x}$, ${\displaystyle p_y}$, ${\displaystyle p_z}$. However, the Heisenberg uncertainty principle tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's magnitude and its component along one axis.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not commute, for example ${\displaystyle L_x{L}_y\ne <!--\; \ne \; -->L_y{L}_x}$. (For the precise commutation relations, see angular momentum operator.)

Total angular momentum as generator of rotationsedit

As mentioned above, orbital angular momentum L is defined as in classical mechanics: ${\displaystyle \mathbf{L}=\mathbf{r}\times <!--\; \times \; -->\mathbf{p}}$, but total angular momentum J is defined in a different, more basic way: J is defined as the "generator of rotations". More specifically, J is defined so that the operator

${\displaystyle R(\stackrel{^<!--\; ^\; -->}{n},\mathrm{\varphi <!--\; \varphi \; -->})\equiv <!--\; \equiv \; -->\exp <!--\; \; -->\left(-<!--\; -\; -->\frac{i}{\mathrm{\hslash <!--\; \hslash \; -->}}\mathrm{\varphi <!--\; \varphi \; -->}\mathbf{J}\cdot <!--\; \cdot \; -->\stackrel{^<!--\; ^\; -->}{\mathbf{n}}\right)}$

is the rotation operator that takes any system and rotates it by angle ${\displaystyle \mathrm{\varphi <!--\; \varphi \; -->}}$ about the axis ${\displaystyle \stackrel{^<!--\; ^\; -->}{\mathbf{n}}}$. (The "exp" in the formula refers to operator exponential) To put this the other way around, whatever our quantum Hilbert space is, we expect that the rotation group SO(3) will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between Lie algebras and Lie groups in mathematics. The close relationship between angular momentum and rotations is reflected in Noether's theorem that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.