Angular motion of electrons: from classical physics to quantum physics

Angular momentum is a fundamental concept in physics and plays a key role in describing the behavior of rotating objects and subatomic particles. At the microscopic level, electrons not only orbit the atomic nucleus but also possess angular momentum that determines their quantum properties. Understanding electron angular momentum is essential for understanding atomic structure, spectra, and many quantum phenomena. This article explores the concept of electron angular momentum from the perspectives of classical and quantum physics.

Definition of angular motion

Angular momentum is a vector quantity that describes the rotational speed of an object around a given point In classical physics, the angular momentum for a particle of mass $,$ velocity $v,$ and distance $defined as follows:$

$L⃗=r⃗×p⃗=r⃗×(mv⃗)\vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m \vec{v})$

The direction $is determined by the right-hand rule$ . The value of this quantity represents the “angle of rotation”, and its direction is the “axis of rotation”.

In classical physics, if the force acting on a particle has no momentum (i.e., $τ = r \times F = 0$ ), then its angular momentum remains constant. This is the law of conservation of angular momentum .

Angular motion of electrons in the Bohr atomic model

Niels Bohr was the first to use the concept of angular momentum to explain the stability of the atom. In his hydrogen atom model , he postulated that electrons move in specific circular orbits around the atomic nucleus and that only orbits with quantized angular momentum can exist.

$\hbar = n \frac{h}{2\pi}$

In this case, an integer (the principal quantum number) $constant$ .
This simple assumption enabled Bohr to successfully predict the spectral lines of the light emitted by the hydrogen atom. The larger the value of $n, the farther$ $is$ from the atomic nucleus and the higher its energy.

However, Bohr’s atomic model is only applicable to hydrogen atoms (with one electron) and cannot explain the more complex structures of atoms with multiple electrons. To solve this problem, quantum mechanics was developed.

Angular momentum in quantum mechanics

In quantum mechanics, angular momentum is no longer simply the product $r$ $\times$ but a factor that influences the electron’s wavefunction . In general, there are two types of electron angular momentum $:$

The orbital angular momentum
associated with the motion of electrons around an atomic nucleus is characterized by $the quantum number$

$\sqrt{l(l+1)}\hbar$

can an integer from $0$ $to$ $n$ For example, for $n$ $=$ $possible$ values of $l are$ $0, 1 and 2$ $, which$ $the s, p$ $and$ $d$ $sublevels$ in the periodic table $, respectively .$
Intrinsic angular momentum (spin angular momentum)
is a special type of angular momentum caused by an intrinsic property of the electron and independent of its actual spin in space. The electron possesses an intrinsic spin whose magnitude is constant, as follows:

$\sqrt{s(s+1)}\hbar$

Where $s = 2 1.$ This property allows the electron two permissible spin states (+1/2 and -1/2)

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Unification of orbital angular momentum and rotational angular momentum

In an atom, the total angular momentum of an electron is a vector combination of orbital angular momentum and spin angular momentum:

$J⃗=L⃗+S⃗\vec{J} = \vec{L} + \vec{S}$

The total angular momentum is defined as follows:

$\sqrt{j(j+1)}\hbar$

where is a common subnumber and its admissible value is $:$

$\, l + s – 1, \, \ldots, \, |l – s|$

This combination leads to a reduction in the fine structure of the atomic spectra , as the interaction of rotational and orbital motion leads to the splitting of the spectral lines into highly compressed components.

Directional and spatial quantification

In quantum mechanics, angular momentum cannot have an angle to the z-axis. its z-component (i.e., $L_z$ takes on discrete values:

$Lz=mlℏL_z = m_l\$

Here, $mlm_l$ the magnetic quantum number $values from -l$ to $l$ $.$ This phenomenon $_{is}$ called spatial quantization. The Stern-Glass experiment confirmed this property by splitting $a beam of silver atoms in a magnetic field into two different paths due to the two possible spin directions of the electrons.$

The role of angular motion in spectroscopy and atomic chemistry

$The angular motion of the electrons determines the structure$ and $_{mlm_l}$ represents a specific orbital $with different geometry$ $.$ For example:

The lower shell s ( $l$ $0)$ has a spherical shape.
The substrate p ( $)$ has three spatial directions (pₓ, pᵧ, p_z $)$ $.$
Layers d and f have a more complex shape.

Differences in angular momentum lead to differences in the electron configuration of elements, which in turn cause differences in their chemical properties.
Furthermore, changes in angular momentum during transitions between energy levels must obey certain selection rules, such as:

$_{Δl = \pm1 and Δml = 0, \pm1} (where Δl = \pm 1 and Δml = 0, \pm 1$

Determination of permissible spectral lines.

Conservation of angular momentum

In atomic and subatomic processes, angular momentum is a conserved quantity . This principle finds widespread application in particle interactions, decay, and the excitation of atoms.
For example, when an excited electron emits a photon, the sum of the angular momenta before and after the process must remain constant. Since the photon has a spin of 1, this determines the transitions allowed in quantum mechanics.

New models and applications

In modern physics, the concept of electron angular momentum has expanded to a wide range of applications. In computational chemistry, the distribution of angular momentum across orbitals plays a crucial role in chemical bonding and reactivity.
Emerging technologies such as spin electronics utilize the electron’s spin, rather than its charge, to store and transmit information. This field of science is directly based on the intrinsic properties of angular momentum.

In particle physics, the study of spin and angular momentum interactions contributes to our understanding of the fundamental forces of nature. In quantum field theory, angular momentum is studied in more complex forms, for example as the orbital angular momentum of a photon or as the spin of a gluon.

In conclusion

The angular momentum of electrons is a cornerstone of understanding atomic structure and quantum phenomena . From the classical Bohr model of the atom to modern quantum theory, this concept plays a key role in explaining the behavior of elementary particles.
Orbital angular momentum, spin, and their combinations not only determine the energy levels and spectral structure of atoms, but also have important practical applications in modern technologies such as spin electronics and nanoelectronics.

A deeper understanding of this concept brings us closer to new scientific horizons, from atoms to the universe: rotation and rotation form the hidden language of the universe.