Energy Levels of Hydrogenlike Atom

Derive the energy level of a hydrogen-like atom.

Deriving the energy levels of a hydrogen-like atom is a foundational problem in quantum mechanics. This task involves understanding how quantum mechanics describes electrons bound in an atom, which is a central potential problem. By solving this, you are addressing a specific quantum mechanical system with spherical symmetry. The solution involves solving the time-independent Schrödinger equation in three dimensions, particularly under a central potential due to a nucleus with a positive charge. This central potential problem reflects a system where the potential energy depends only on the distance from the center, a crucial feature in atomic systems.

The solution process relies on using separation of variables to solve the Schrödinger equation, which effectively breaks down this problem into more manageable parts: radial and angular components. It critically shows how quantum numbers arise from boundary conditions and quantization rules. Through the derived energy expression, we understand that energy levels are quantized and inversely proportional to the square of the principal quantum number, a fundamental insight into atomic structure and spectroscopy.

Moreover, this problem exemplifies how quantum mechanics reveals the discrete nature of energy levels in atoms, significantly different from classical physics' predictions. Understanding the derivation and its results help learners appreciate the fundamental principles of quantum theory and atomic physics, forming a basis for studying more complex atoms and molecules.