Which of the following is discreted in Bohr's theory? (a) Potential energy (b) Kinetic energy (c) Velocity (d) Angular momentum

In the mysterious quantum world, certain physical properties take on only specific, 'quantized' values, much like rungs on a ladder. In Bohr's theory, one such property is the angular momentum of an electron orbiting the nucleus. Angular momentum is a measure of the amount of rotation an object has, taking into account its mass, shape, and speed.

According to Bohr, instead of smoothly varying, an electron can only possess angular momentum that fits the formula: \( L = n\frac{h}{2\pi} \), where \( L \) is the angular momentum, \( n \) is a positive integer (known as the quantum number), and \( h \) is Planck's constant. This means that electrons can't just orbit at any distance or with any energy; they must 'choose' orbits where the angular momentum is an exact multiple of this specific quantum of action.

This concept was groundbreaking because it helped explain why electrons don't spiral into the nucleus and why atoms emit or absorb energy at certain fixed wavelengths. The idea of quantization is fundamental to quantum mechanics and appears throughout the theory, manifesting in phenomena such as 'leaping' from one energy level to another in the form of photons.

The Bohr model depicts the hydrogen atom as a small, positively charged nucleus surrounded by electrons in circular orbits, akin to planets revolving around the sun. This model brought the atomic world a step closer to understanding by applying classical physics concepts to quantum situations, albeit with a twist. The twist is the quantization rules that govern the permitted orbits.

Bohr postulated that electrons can only travel in orbits where their angular momentum is quantized. Additionally, when jumping between these orbits, electrons absorb or emit energy in discrete amounts, which correspond to the difference in energy levels. The permitted orbits are those where the electron's wavelength forms a standing wave, which doesn't interfere destructively with itself—in layman's terms, the orbit loop must 'fit' a whole number of wavelengths.

The model was successful in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen and marks a pivotal breakthrough for quantum theory. However, later advancements in quantum mechanics, incorporating wave-particle duality and the Heisenberg uncertainty principle, expanded and eventually superseded the Bohr model.

Serving as one of the cornerstones of quantum mechanics, Planck's constant (\( h \) is a physical constant reflecting the scales at which quantum effects become significant. This minuscule constant has the value of approximately \( 6.62607015 \times 10^{-34} \) joule-seconds.

Discovered by Max Planck, this constant originated from his work on black-body radiation, for which he hypothesized that energy is not continuous, but comes in discrete packets called 'quanta'. Planck's constant essentially determines the size of these quanta. It appears in various fundamental equations, including the energy of a photon ( \( E = hv \), where \( v \) is the frequency) and the previously mentioned formula for quantized angular momentum.

Understanding Planck's constant offers insight into the workings of devices like LEDs and solar cells, and principles like photoelectric effect, which all rely on the interaction of energy and matter at the quantum level. Its fundamental role in the fabric of physics is so critical that it was used to redefine the kilogram in the International System of Units (SI), anchoring mass to a constant of nature rather than a physical object.