Explaining the Stability of Revolving Electrons: Bohr’s Model and Quantum Mechanics

For physics enthusiasts, the question of how the Bohr model of an atom accounts for the stability of a revolving electron, which is accelerating, is a fascinating topic. This article delves into this concept, providing a detailed explanation based on both the Bohr model and the principles of quantum mechanics.

Bohr’s Model: Stationary Orbits

According to Niels Bohr, some orbits in atoms are stationary, meaning that electrons in these orbits accelerate but do not radiate energy. This solution was a pivotal concept in the early development of quantum mechanics, as it resolved the problem of stationary solutions for electrons while adhering to the quantum theory's principle that energy can only be emitted in discrete quanta.

Stability and Quantum Numbers

The stability of the electron's motion is central to the Bohr model, which posits that electrons exist in stable orbits. However, a more comprehensive understanding of electron stability and motion is achieved through the quantum mechanical framework. Modern quantum mechanical calculations of the hydrogen atom, using the Schr?dinger equation, provide a deeper insight into electron behavior.

The Schr?dinger equation offers a more detailed explanation of electron behavior, introducing three quantum numbers: the principal quantum number (n), the orbital quantum number (l), and the magnetic quantum number (m). These numbers describe the electron's energy, angular momentum, and space orientation, respectively. In contrast, the Bohr model only introduces the principal quantum number (n) to explain the quantized energy levels of electrons.

Wave Properties and Stability

The stability of the electron is not just a postulate but is fundamentally tied to its wave properties. Albert Einstein provided the foundational equation for the photoelectric effect, which showed that light can be described as a particle (photon) that carries energy in discrete quanta. Louis de Broglie further advanced this concept by proposing that matter, including electrons, exhibits wave-like properties. This wave-particle duality is central to the understanding of electron stability within the quantum mechanical framework.

The Heisenberg Uncertainty Principle, which states that the more precisely the position of some particle is determined, the less precisely its momentum can be predicted, and vice versa, contributes to the stability of electrons. This principle explains why electrons do not lose energy in their orbits, which would otherwise lead to their collapse into the nucleus, as suggested by classical physics.

Matrix Representation and Equivalence

The matrix representation of the Heisenberg Uncertainty Principle and the quantized solutions of the Schr?dinger equation show their equivalence. The wave equation and the matrix representation of quantum mechanics often employ operators, such as the Hamiltonian (which represents the total energy of the system), which can be expressed in a matrix form. This equivalence further supports the stability of electrons in their orbits, as it ensures that the energy of an electron remains quantized and does not emit continuous radiation.

The Ground State and Electronic Jiggling

According to the Bohr model, the electron in the hydrogen atom's ground state has a specific distance from the nucleus, roughly half an angstrom. In this state, the electron cannot lower its energy further and indeed cannot fall into the nucleus, as there are no lower energy states allowed by quantum mechanics. This ground state can be described as the electron's "resting state," where it is not actually orbiting but rather in a spatially extended quantum field, "jiggling" randomly.

Quantum mechanics also explains that all quantum objects, including electrons, are inherently probabilistic. An electron in the ground state of a hydrogen atom is not confined to a discrete orbit; instead, it is a spatially extended bundle of energy. This bundle is constantly fluctuating, giving rise to its "quantum jiggling."

The ground state of an electron is characterized by minimal angular momentum, indicating that the electron is not rotating around the nucleus but rather has a spread of energy that avoids the zero-point energy of rotating systems. This non-rotational state further supports the stability of the electron, as it does not radiate energy according to classical physics.

For further insights into these topics, the non-technical book Tales of the Quantum, published by Oxford University Press in 2017, offers a detailed explanation of these concepts. Other key literature includes the paper "There are no particles; there are only fields," published in Am. J. Phys., Vol. 81, 211-233, March 2013, and Am. J. of Phys., Vol. 73, 630, July 2005.

Keywords: Bohr’s model, quantum mechanics, electron stability