Planck's constant (denoted as h) is a fundamental constant in quantum mechanics and plays a crucial role in the context of atomic energy levels and the behaviour of particles at the quantum level. It is the proportionality constant in the formula E = hf, where E is the energy of a photon and f is its frequency. This relationship illustrates the quantisation of energy, as it shows that the energy of a photon (and thus the energy exchanged in atomic transitions) is quantised and directly proportional to its frequency. Planck's constant essentially sets the scale for these energy quantisations and is fundamental to the understanding of the discrete nature of energy exchange in atomic and subatomic processes. Its value dictates the scale at which quantum mechanical effects become significant and provides a bridge between the macroscopic classical world and the microscopic quantum world. It is also a cornerstone in the development of other quantum theories and principles, such as the Heisenberg Uncertainty Principle.

Line spectra are considered 'fingerprints' for elements due to their unique and specific patterns for each element. When an element is heated or excited, its electrons transition between different energy levels, emitting or absorbing photons of specific energies. These energies correspond to specific wavelengths of light, resulting in distinct spectral lines at those wavelengths. Since each element has a unique set of energy levels due to its electron configuration, the pattern of these spectral lines is unique to each element. This uniqueness allows scientists to identify elements by their line spectra. For instance, by analysing the light from a distant star and matching its spectral lines with known elements' spectra, astronomers can determine the star's composition. This distinctiveness of line spectra as elemental 'fingerprints' is a powerful tool in analytical chemistry, astrophysics, and various other fields, enabling precise identification of elements even in complex mixtures or remote celestial bodies.

Electrons cannot exist between quantised energy levels in an atom due to the fundamental principles of quantum mechanics. In quantum theory, energy levels are seen as discrete states with specific energy values. These discrete states arise from the wave-like nature of electrons, as described by the Schrödinger equation. The solutions to this equation, which represent the probability distribution of finding an electron, only yield certain allowed energy values, leading to quantised energy levels. Any intermediate energy value does not correspond to a solution of the Schrödinger equation and thus is not a permissible state for an electron. This quantisation is a direct consequence of the wave properties of particles at the atomic and subatomic level and is a fundamental aspect of quantum physics, distinguishing it markedly from classical physics where energy can vary continuously.

The quantum mechanical model of the atom provides a more comprehensive and accurate explanation of energy levels compared to the Bohr model. While the Bohr model introduced the concept of quantised orbits for electrons, it still treated electrons as particles in fixed orbits around the nucleus, similar to planets orbiting a star. The quantum mechanical model, however, introduces the concept of electron orbitals - regions of space around the nucleus where an electron is likely to be found. Unlike the Bohr model, it does not define exact paths for the electrons but rather probability distributions. This model is based on the principles of wave-particle duality and uncertainty, which state that the exact position and momentum of an electron cannot be simultaneously known. This approach allows for a more accurate prediction of chemical and physical properties of elements, as it considers electrons as waves distributed in space rather than particles in fixed orbits.

The photoelectric effect supports the concept of quantised energy levels in atoms by demonstrating that light interacts with matter in a quantised manner. In the photoelectric effect, photons of light are absorbed by electrons in a material. If the energy of the photon matches or exceeds the binding energy of an electron, the electron is ejected from the material. The key point here is that only photons with a certain minimum energy (threshold frequency) can cause this ejection. This phenomenon aligns with the idea of quantised energy levels, as it shows that electrons can only absorb or emit energy in specific, quantised amounts. The effect directly contradicts the classical wave theory of light, which predicted that light of any frequency, given sufficient time, should be able to impart enough energy to an electron. Instead, it supports the quantum theory where energy is transferred in discrete packets (quanta), thus reinforcing the concept of quantised energy levels in atoms.