Quantum orbital motion involves the quantum mechanical motion of rigid particles (such as electrons) about some other mass, or about themselves. In classical mechanics, an object's orbital motion is characterized by its orbital angular momentum (the angular momentum about the axis of rotation) and spin angular momentum, which is the object's angular momentum about its own center of mass. In quantum mechanics there are analogous orbital and spin angular momenta which describe the orbital motion of a particle, represented as quantum mechanical operators instead of vectors.

The paradox of Heisenberg's Uncertainty Principle and the wavelike nature of subatomic particles make the exact motion of a particle impossible to represent using classical mechanics. The orbit of an electron about a nucleus is a prime example of quantum orbital motion. While the Bohr model describes the electron's motion as uniform circular motion, in reality its location in space is described by probability functions. Each probability function has a different average energy level, and corresponds to the likelihood of finding the electron in a specific atomic orbital, which are functions representing 3 dimensional regions around the nucleus. The description of orbital motion as probability functions for wavelike particles rather than the specific paths of orbiting bodies is the essential difference between quantum mechanical and classical orbital motion.

Orbital Angular Momentum

In quantum mechanics, the position of an electron in space is represented by its spatial wave function, and specified by three variables (as with x, y, and z Cartesian coordinates). The square of an electron's wave function at a given point in space is proportional to the probability of finding it at that point, and each wave function is associated with a particular energy. There are limited allowed wave functions, and thus limited allowed energies of particles in a quantum mechanical system; wave functions are solutions to Schrödinger's Equation.

For Hydrogen-like atoms, spatial wave function has the following representation:

${\displaystyle \psi _{nlm}(r,\theta ,\phi )=R_{nl}(r)Y_{l}^{m}(\theta ,\phi )}$

Electrons do not "orbit" the nucleus in the classical sense of angular momentum, however there is a quantum mechanical analog to the mathematical representation of L = r × p in classical mechanics. In quantum mechanics, these vectors are replaced by operators; the angular momentum operator is defined as the cross product of the position operator and the momentum operator, ${\displaystyle {\hat {p))=-i\hbar \nabla }$

Just as in classical mechanics, the law of conservation of angular momentum still holds.^[1]

Spin

An electron is considered to be a point charge.^[2] The motion of this charge about the atomic nucleus produces a magnetic dipole moment that can be oriented in an external magnetic field, as with magnetic resonance. The classical analog to this phenomena would be a charged particle moving around a circular loop, which constitutes a magnetic dipole. The magnetic moment and angular momentum of this particle would be proportional to each other by the constant ${\displaystyle \gamma _{0))$, the gyromagnetic ratio. However, unlike bodies in classical mechanics an electron carries an intrinsic property called spin, which creates an additional (spin) magnetic moment.

The total angular momentum of a particle is the sum of both its orbital angular momentum and spin angular momentum.^[3]

A particle's spin is generally represented in terms of spin operators. It turns out for particles that make up ordinary matter (protons, neutrons, electrons, quarks, etc.) particles are of spin 1/2,^[4] meaning that only two eigenvectors of the Hamiltonian exist for a spin 1/2 state, implying that there are only two values of energy that can be measured. Thus showing that the inherent quantum property of energy quantization is a direct result of electron spin.

Atomic Orbitals

For a particle to remain in orbit, it must be bound to its center of rotation by some radial potential. Electrons orbiting an atomic nucleus are bound to the nucleus via the Coulomb Potential, given by ${\displaystyle V(r)={\frac {e^{2)){4\pi \epsilon _{0))}{\frac {1}{r))}$.

Using the formalisms of wave mechanics developed by physicist Erwin Schrödinger in 1926, each electron's distribution is described by a 3-dimensional standing wave. This was motivated by the work of 18th century mathematician Adrien Legendre.

The spatial distribution of an electron about a nucleus is represented by three quantum numbers: ${\displaystyle n,l}$ and ${\displaystyle m_{l))$. These three numbers describe the electron's atomic orbital, which is the region of space occupied by the electron. Each set of these numbers constitutes a principal shell with a specific number of sub shells, each with a specific number of orbitals. Roughly speaking, the principle quantum number ${\displaystyle n}$ describes the average distance of an electron from the nucleus. The azimuthal quantum number ${\displaystyle l}$ describes the relative shape of the region of space (orbital) occupied by the electron. Finally, The magnetic quantum number ${\displaystyle m_{l))$ describes the relative orientation of the orbital with respect to an applied magnetic field. The allowed values of ${\displaystyle l}$ and ${\displaystyle m_{l))$ depend on the value of ${\displaystyle n}$.

Applications

The Einstein-de Haas effect describes the phenomena in which a change in this magnetic moment causes the electron to rotate.

The Barnett effect describes the magnetization of the electron resulting from being spun on its axis.