Angular Momentum of Electron

The angular momentum of an electron is a fundamental property that describes its rotational motion. It has two components: orbital angular momentum, arising from the electron’s motion around the nucleus, and spin angular momentum, which is an intrinsic property of the electron. Angular momentum is crucial in determining atomic structure and behavior, and it has important applications in explaining spectral lines, magnetic properties of materials, chemical bonding, electron configurations in atoms, and technologies like magnetic resonance imaging (MRI) and electron spin-based devices.

1.0Definition of Angular Momentum

• Angular momentum describes the rotational motion of an object.The measure of rotational motion equal to the product of the moment of inertia and angular velocity.

It depends on:

• how massive the object is,
• how fast it rotates,
• and how far it is from the center of rotation.

2.0Momentum of an Electron

When discussing atoms, Niels Bohr proposed that electrons revolve around the nucleus in fixed circular paths. He suggested that the angular momentum of an electron is not continuous, but restricted to specific values.

This idea was revolutionary because:

• It explained why electrons do not spiral into the nucleus,
• and why atoms emit or absorb radiation at specific wavelengths.

3.0Bohr’s Quantization of Angular Momentum

• Bohr states that angular momentum of an electron is given as $L=mvr=n\frac{h}{2\pi }$
• This means electrons can occupy only certain stable orbits,each with a fixed angular momentum value.

4.0De Broglie’s Explanation of Quantized Angular Momentum

 If an electron is a wave, only those orbits are stable where the electron’s wave can “fit” exactly along the circumference of the orbit.

$2\pi r=n\lambda$

$\lambda =\frac{h}{mv}$

$2\pi r=n\frac{h}{mv}$

$L=mvr=n\frac{h}{2\pi }$ This is the Bohr quantisation condition.

General form,$2\pi r_n=n\lambda$ where

 n=order of the orbit

$r_n$= radius of the nth Bohr orbit

$r_n=(0.259\mathring{A})\frac{n^2}{z}(Z=\text{ Atomic Number })$

5.0Orbital Angular Momentum And Spin Angular Momentum

Orbital Angular Momentum

Orbital angular momentum refers to the rotational motion of an electron around the nucleus. In quantum mechanics, the electron does not follow a fixed circular path; instead, it occupies regions of space called orbitals. The shape and orientation of these orbitals determine the electron’s orbital angular momentum. It arises because the electron behaves both like a particle and a wave, and its motion around the nucleus creates a form of rotation. Orbital angular momentum also influences the energy levels of atoms and helps determine the arrangement of electrons into shells and subshells. It is responsible for the formation of different orbital shapes such as s, p, d, and f. Additionally, orbital angular momentum plays a key role in explaining atomic spectra, chemical bonding, and magnetic effects in atoms.

$L=\sqrt{l(l+1)\hslash }$ where $\hslash =\frac{h}{2\pi }$

Angular Momentum of a p-Electron,l=1

$L=\sqrt{l(l+1)h}=\sqrt{1(1+1)h}=\sqrt{2}h$

Angular Momentum of a d-Electron,l=2

$L=\sqrt{l(l+1)h}=\sqrt{2(2+1)h}=\sqrt{6}h$

6.0Spin Angular Momentum 

• Spin angular momentum is an intrinsic property of the electron, meaning it is a built-in form of angular momentum that does not depend on any physical rotation or orbital motion. Even if the electron were motionless, it would still possess spin. This property gives rise to two possible spin states, often referred to as “spin-up” and “spin-down.” Spin angular momentum is responsible for the electron’s magnetic moment, making electrons behave like tiny magnets. This property is essential in understanding the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum characteristics. Spin also influences fine structure in atomic spectra and is critical in many modern technologies, including magnetic materials, data storage devices, spintronics, and medical imaging techniques like MRI.

$S=\sqrt{s(s+1)\hslash }$ where $\hslash =\frac{h}{2\pi }$

For electrons: $s=\frac{1}{2}$Behaviour

Spin determines:

• Magnetic Behaviour,
• Energy Level Splitting,
• Chemical Bonding Patterns.

Illustration-1. Calculate the de-broglie wavelength of an electron revolving in the first excited state in hydrogen atom.

Solution:Given n=2 (first excited states); Z=1 for hydrogen

$2\pi r_2=2\lambda \Rightarrow \lambda =\pi r_2$

$=\pi (0.53\mathring{A})\frac{(2{)}^2}{(1)}=2.116\mathring{A}$