The orbital angular momentum of an electron is \(\sqrt{3} \frac{h}{\pi}\). Which of the following may be the permissible value of angular momentum of this electron revolving in unknown Bohr orbit? (a) \(\frac{h}{\pi}\) (b) \(\frac{h}{2 \pi}\) (c) \(\frac{3 h}{2 \pi}\) (d) \(\frac{2 h}{\pi}\)

When we discuss atoms and their structure, an essential concept is the Bohr orbit. Named after Niels Bohr, it revolutionized our understanding of how electrons move around the nucleus. Bohr proposed that electrons travel in specific paths or orbits, and each Bohr orbit corresponds to a particular energy level.

Unlike the paths planets take around the sun, Bohr orbits are quantized, meaning electrons can only occupy certain orbits with fixed sizes and energies. This idea is akin to climbing a ladder; you can stand on a rung but not in between rungs. For electrons, this means a jump between orbits involves absorbing or emitting distinct amounts of energy, visible as light in the form of photons.

In the context of the exercise, identifying the correct Bohr orbit for an electron involves understanding the quantized nature of its angular momentum. This feature is fundamental to the atomic structure and is a testament to the marriage of classical physics and quantum theory in the early 20th century.

The Planck constant (symbolized as h) is one of the fundamental constants in physics, pivotal to quantum mechanics. Max Planck first introduced the concept to explain the discrete nature of energy emission from black bodies, which eventually laid the foundation for quantum theory.

The Planck constant defines the size of the 'quantum' of action, linking the energy (E) of particles such as photons to their frequency (f) through the simple equation, E = hf. This relationship has significant consequences, for example, in the emission spectra of atoms and the operation of lasers.

In simple terms, think of the Planck constant as the fundamental 'currency' of quantum transactions. It's akin to the smallest denomination of a currency that can't be broken down further, reflecting the granular nature of energy at microscopic scales. In our exercise, the Planck constant is a crucial factor to determine the quantized angular momentum of an electron. Thanks to its role, we are able to calculate and predict physical quantities that would otherwise seem random and chaotic.

Quantum numbers are akin to an address system for electrons in an atom, specifying their location and movement characteristics. There are four quantum numbers, but in the context of the exercise, we focus specifically on the principal quantum number, which tells us the size and energy level of the Bohr orbit an electron occupies.

The principal quantum number, symbolized by n, is a positive integer (1, 2, 3,...). It's also directly linked to the quantization of angular momentum, as shown in the given equation for angular momentum, L = n(h/2π), where n must be an integer. This ensures that angular momentum is not arbitrary but comes in discrete 'chunks'.

Understanding this digitized nature of motion inside atoms is eye-opening because it suggests a structured and orderly universe at the quantum level. This organization allows physicists to predict the behavior of atoms under various circumstances, such as in the step-by-step solution we provided for the exercise. The principal quantum number is key to the permissible values of angular momentum and thus to the structure of atoms themselves.