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Understanding atomic structure is fundamental in the study of chemistry and physics. An atom is composed of three primary particles: protons, neutrons, and electrons. Protons and neutrons form the nucleus, the dense core at the center of an atom, while electrons are tiny particles that orbit the nucleus in various energy levels.

The idea that electrons move in fixed orbits or 'shells' around the nucleus comes from early atomic models, including that proposed by Niels Bohr. This concept was built upon Rutherford's model, which suggested that atoms have a central nucleus surrounded by electrons, similar to planets orbiting the sun. However, this classical understanding was later refined as scientists discovered that electrons behave more like waves on a string rather than particles in simple orbits.

Classical electrodynamics is a branch of physics that deals with the interactions between electric charges and currents. It is often referenced in terms of Maxwell's equations, which are the foundation for classical theories of electromagnetic radiation. According to these theories, a charged particle that accelerates, which includes moving in a curved path like an orbit, should emit electromagnetic radiation, thus losing energy.

In Rutherford's atomic model, this posed a problem. If electrons were truly orbiting the nucleus like planets around the sun, they would be accelerating due to the centripetal force required to maintain their circular paths. As a result, they should radiate away their energy and eventually spiral into the nucleus, leading to the collapse of the atom. This inconsistency with observed stable atoms was a significant limitation in the model that prompted further advances in atomic theory.

The quantum mechanical model of the atom provides a more accurate description of atomic structure. It introduces the concept of quantum states and the principle of wave-particle duality. Rather than having electrons in fixed orbits, electrons are described by wave functions that predict a probability distribution of where an electron might be around the nucleus—these are known as atomic orbitals.

One of the fundamental tenets of quantum mechanics is the Heisenberg uncertainty principle, which states that we cannot simultaneously know the exact position and momentum of an electron. This principle signified a departure from the deterministic nature of classical physics and explained why, contrary to classical predictions, atoms do not collapse. This new model successfully accounted for the stability of atoms and explained the spectra of light emitted and absorbed by them, which was a significant achievement of 20th-century physics.

The stability of atoms is a concept that puzzled many scientists during the transition from classical to quantum physics. Rutherford's model couldn't explain why atoms didn't collapse, as was predicted by classical electrodynamics. Quantum mechanics, however, resolved this issue by introducing quantized energy levels.

According to quantum mechanics, electrons can only exist in certain energy states, and any energy they radiate must be released in specific amounts, or 'quanta.' Moreover, the lowest energy state available to an electron—known as the ground state—prevents it from spiraling into the nucleus. This quantization of energy levels means that electrons cannot simply lose energy and collapse into the nucleus; they are stabilized by the rules of quantum mechanics, which govern the microscopic world. It's this revolutionary approach that explains the persistent and robust existence of the diverse array of atoms that make up our universe.