The negative muon has a charge equal to that of an electron...

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the $n=2$ level to the $n=1$ level?

Key Concepts

Reduced Mass

The concept of reduced mass is used in two-body problems to simplify the dynamics by replacing the two interacting masses with a single effective mass. It is defined as ? = (m1*m2)/(m1 + m2) and allows one to treat the motion as if a single particle of mass ? is moving in the potential. This is particularly important in atomic systems where the masses of the nucleus and orbiting particle can differ greatly.

Bohr Model of the Atom

The Bohr model introduces the idea that the electron (or any orbiting charged particle) in a hydrogen-like atom occupies discrete energy levels. It postulates that angular momentum is quantized and leads to the derivation of energy levels that depend inversely on the square of the principal quantum number (n). This model forms the basis for understanding atomic spectra and transitions between energy levels.

Energy Quantization in Hydrogen-like Atoms

In hydrogen-like atoms, the energy levels are quantized due to the discrete allowed orbits governed by quantum mechanics. The energy of each level is determined by the reduced mass of the system and the Coulomb interaction, and is typically expressed by formulas like E? = - (?e?)/(2(4???)²?²)(1/n²). This quantization explains the specific energy differences observed in spectral lines.

Quantum Transitions and Photon Emission

When an atom transitions between quantized energy levels, the energy difference is released or absorbed as a photon. The wavelength of the emitted or absorbed radiation is directly related to the energy difference through the relationship ? = hc/?E. This concept is central to understanding and analyzing spectral lines and the electromagnetic radiation emitted or absorbed by atoms.

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