The correct expression derived for the energy of an electron in the \(n^{\text {th }}\) energy level is for H-atom (a) \(E_{n}=\frac{2 \pi^{2} m e^{4}}{n^{2} h^{2}}\) (b) \(E_{n}=-\frac{\pi^{2} m e^{4}}{2 n^{2} h^{2}}\) (c) \(E_{n}=-\frac{2 \pi^{2} m e^{2}}{n^{2} h^{2}}\) (d) \(E_{n}=-\frac{2 \pi^{2} m e^{4}}{n^{2} h^{2}}\)

The hydrogen atom represents the simplest and most studied system in quantum mechanics. Interestingly, it consists of just a single electron orbiting a single proton, which forms the nucleus. The behavior of the electron in a hydrogen atom is governed by quantum mechanics, and it can only occupy certain energy levels, which are quantized. This means that the energy of the electron is restricted to specific values and it cannot exist in between these levels.

Under the Bohr model, which was an early model explaining the quantized nature of atomic energy levels, these levels are depicted as orbits where the electron can reside without radiating energy. An electron can make transitions between these levels by absorbing or emitting a photon whose energy corresponds to the difference in energy between these levels. The formula provided in the exercise reflects the quantized energy levels that an electron can occupy within a hydrogen atom.

The principal quantum number, represented by the symbol 'n', is one of four quantum numbers used to describe the unique quantum state of an electron. It specifies the electron's energy level or shell in an atom. Values of the principal quantum number start from 1, which is the level closest to the nucleus and also the level with the lowest energy. As 'n' increases, the electron's energy increases and it occupies shells farther from the nucleus.

The principal quantum number also contributes to the determination of the size of the electron orbit and its energy. As we can see from the solution, the energy levels are inversely proportional to the square of the principal quantum number. This means that as electrons occupy higher energy levels (with larger 'n'), their energy increases in the negative scale and they are less tightly bound to the nucleus.

Planck's constant is a fundamental quantity in quantum mechanics denoted by the symbol 'h'. It reveals the quantization of energy and has a value of approximately 6.626 x 10^-34 joule-seconds. It was first introduced by Max Planck, and it connects the energy of a photon to its frequency through the relation E = hf, where E is the energy, h is Planck's constant, and f is the frequency.

Planck's constant is crucial in the Bohr model of the hydrogen atom, as it appears in the energy level formula and determines the scale of quantum effects. Its presence in the equation also underscores the intrinsic quantum mechanical nature of the universe - that energy is not continuous, but rather comes in discrete packets called quanta. Hence, the constant plays a pivotal role in calculating the energy levels any electron can occupy in an atom, including the hydrogen atom as demonstrated in the exercise.