The radius of the stationary state which is also called Bohr radius is given by the expression \(r_{n}=n^{2} a_{0}\) where the value of \(a_{0}\) is (a) \(52.9 \mathrm{pm}\) (b) \(5.29 \mathrm{pm}\) (c) \(529 \mathrm{pm}\) (d) \(0.529 \mathrm{pm}\)

The Bohr model is a fundamental concept in understanding atomic structure. Introduced by Niels Bohr in 1913, this model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus, similar to the structure of the Solar System. Bohr's revolutionary idea was to apply quantum theory to these orbits. According to the Bohr model, electrons can only exist in certain discrete orbits, each with its energy level. As electrons jump from one orbit to another, they emit or absorb energy, explaining the spectral lines observed from atoms.

One notable aspect of the Bohr model is its calculation of the Bohr radius (denoted as \(a_{0}\)), which represents the size of the hydrogen atom in its lowest energy state, also known as the ground state. The Bohr model was a significant milestone in the development of the quantum mechanical model and offered a simple explanation for the atomic spectrum of hydrogen.

Understanding atomic structure is crucial in the fields of chemistry and physics. The atom, the smallest unit of an element that retains the properties of that element, consists of three main particles: protons, neutrons, and electrons. Protons carry a positive charge and reside in the nucleus, while neutrons are electrically neutral and also located within the nucleus. Electrons, which are negatively charged, orbit the nucleus in various energy levels or shells.

The behavior and arrangement of these subatomic particles give an atom its characteristics. For instance, the number of protons (atomic number) defines the element itself, and the number of neutrons can result in different isotopes of an element. Electrons are significant when it comes to bonding and the reactivity of elements. The Bohr model helps visualize an atom's structure with defined circular paths for electron orbits, although modern quantum mechanics depicts more complex and accurate representations like electron clouds or orbitals.

Converting units of length is a common task in physics and chemistry, especially when dealing with atomic and molecular scales. The picometer (pm) and angstrom (\(\mathring{A}\)) are two units often used for such small distances. To convert from picometers to angstroms, it's important to know that one angstrom is equivalent to 100 picometers. This conversion factor simplifies the process of moving between these two units of measurement.

Example Conversion:

Consider you have a distance of 200 picometers that you want to convert into angstroms. Using the conversion factor, you would divide by 100, giving you 2 angstroms. On the other hand, to convert from angstroms to picometers, you would multiply by 100. This type of conversion is essential when comparing distances in different notations in scientific work, such as interpreting spectroscopy data or sizing up molecules in nanotechnology.

Quantum states are the key to understanding the behavior of particles at the microscopic level. In quantum mechanics, opposed to classical physics, particles such as electrons don't have defined trajectories. Instead, they are described by wave functions that give the probabilities of finding the particles in various locations and conditions. The 'state' of a quantum system refers to the set of all properties, such as energy, position, momentum, and spin, that fully describe a particle at any given time.

Electrons in atoms exist in certain quantum states, each corresponding to an energy level. They can transition between these states by absorbing or emitting energy in discrete amounts called quanta. Quantum states are critical not only in explaining the atomic emissions and absorption spectra but also in the development of modern technologies such as lasers, semiconductors, and quantum computing.