Consider a hydrogen-like ion where an electron is orbiting a...

Consider a hydrogen-like ion where an electron is orbiting a nucleus that has charge $q=+Z e$. Derive the formulas for the energy $E_{n}$ of the electron in $n$ th orbit and the orbital radius $r_{n}$.

Key Concepts

Orbital Radius Determination

The orbital radius in the Bohr model is derived by equating the centripetal force, necessary for circular motion, to the Coulomb attraction between the nucleus and the electron. This derivation leads to an expression for the radius r_n in terms of the principal quantum number n, the nuclear charge Z, and fundamental constants. It demonstrates that orbital radii increase with n and adjust inversely with Z, reflecting how a larger nuclear charge draws the electron into a tighter orbit.

Energy Quantization

The quantization of energy in the Bohr model arises from the interplay between the electron’s kinetic energy and the potential energy due to the Coulomb interaction. By using the quantized angular momentum condition and the force balance between the Coulomb force and the centripetal force required for circular motion, one can derive a formula for the allowed energy levels, which shows that the energy is inversely proportional to the square of the principal quantum number n.

Angular Momentum Quantization

One of the fundamental postulates of the Bohr model is that the angular momentum of the electron is quantized, meaning that it can only take on values that are integer multiples of ?. This quantization condition, expressed as m_e v_n r_n = n?, is crucial for deriving both the discrete energy levels and the orbital radii of the electron.

Bohr Model

The Bohr model is a semi-classical description in which the electron orbits the nucleus in discrete circular orbits. In this model, the electron's angular momentum is quantized in integer multiples of ?, which restricts the allowed orbits and directly leads to discrete radii and energy levels. This concept is key to understanding the structure of hydrogen-like systems.

Hydrogen-like Ion

A hydrogen?like ion is an atomic system consisting of a nucleus with positive charge Ze and a single electron orbiting it. This makes the system similar to a hydrogen atom, allowing it to be analyzed with simplified models that capture essential quantum mechanical effects of electron motion around a nucleus, despite the increased nuclear charge.

Coulomb Interaction

Coulomb's law governs the force between the positively charged nucleus and the negatively charged electron. In the context of the Bohr model, the electron is treated as moving in the Coulomb potential created by the nucleus, and the balance between this attractive force and the necessary centripetal force for circular motion determines the structure of the orbits.

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