Deriving Allowed Radii Using the Bohr Model | Modern Physics, Video 2

Hi everyone! In today's video, I will be deriving the relationship between the "allowed" orbital radii of the hydrogen atom and the principal quantum number, n, using the Bohr model. I deliberately put the word "allowed" in quotation marks because, although the Bohr model successfully predicts the quantized energy levels of hydrogen, its description of electron behavior is incomplete. Bohr proposed that electrons travel in specific circular orbits around the nucleus, each corresponding to a particular radius and energy. By introducing the quantization of angular momentum, he was able to derive discrete orbital radii and energy levels. Remarkably, his predictions for hydrogen's energy levels were correct, even though his picture of electrons traveling along fixed circular paths was not. In modern quantum mechanics, electrons are instead described by wavefunctions, which give us the probability of finding an electron at different locations around the nucleus. Although the energy levels of a bound electron are quantized, its distance from the nucleus is not restricted to specific values. Interestingly, the Bohr radius still holds physical significance! For hydrogen in its ground state, it represents the most probable radial distance between the electron and the nucleus. However, this relationship does not necessarily hold for the other energy levels, where the behavior of electrons becomes more complicated. In this video, we will focus on deriving the allowed orbital radii using Coulomb's law, centripetal force, and Bohr's angular momentum quantization condition. While the Bohr model is not a complete description of atomic behavior, it provides an excellent introduction to the ideas that eventually led to modern quantum mechanics. I hope you find this derivation interesting and insightful! This video is part of a series in which I document and share my understanding of important concepts in modern physics as I continue studying them myself. As always, I welcome any questions, corrections, or suggestions in the comments. I appreciate any opportunity to improve my explanations and understanding of these fascinating topics. Thank you for watching! #modernphysics #quantummechanics #bohrmodel