Understanding Muonium Energy Levels and Relativistic Velocities

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SUMMARY

The discussion centers on the energy levels of muonium, an atom composed of a proton and a muon, which is approximately 200 times more massive than an electron. The energy of the first orbit of muonium is more negative than that of hydrogen due to the greater reduced mass, which directly influences the Rydberg constant (R). The expression for energy levels, E = -Rhc/n², indicates that a larger reduced mass results in a more negative energy value. Additionally, the muon's higher mass leads to a higher velocity, approximately 14 times that of an electron, raising questions about relativistic effects in such systems.

PREREQUISITES
  • Understanding of atomic structure and energy levels
  • Familiarity with the concept of reduced mass
  • Knowledge of the Rydberg formula for energy levels
  • Basic principles of relativistic physics
NEXT STEPS
  • Explore the implications of reduced mass in quantum mechanics
  • Study the Rydberg formula and its applications to different atomic systems
  • Investigate the behavior of particles at relativistic speeds
  • Examine the quantization of angular momentum in muonium and hydrogen
USEFUL FOR

Physicists, students of quantum mechanics, and anyone interested in the properties of exotic atoms and relativistic particle behavior.

physgirl
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A muonium is an atom consisting of a proton and a muon. A muon has the same charge as an electron but about 200x the mass of an electron. Conceptually, how does it make sense that the energy of the first orbit of a muonium is more negative than that of a hydrogen?
 
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physgirl said:
A muonium is an atom consisting of a proton and a muon. A muon has the same charge as an electron but about 200x the mass of an electron. Conceptually, how does it make sense that the energy of the first orbit of a muonium is more negative than that of a hydrogen?

Compare the the reduced mass of an electron in muonium to the reduced mass of an electron in hydrogen.
 
How does the expression for energy level of the hydrogen atom depend on the mass of the electron? How would you modify it for muonium?
 
Reduced mass for muonium is much greater than reduced mass for hydrogen...

And expression for energy level that I used was: E=-Rhc/n^2... but R depends directly on reduced mass, so the greater the reduced mass,the greater the R, and therefore the more negative the E...

But I guess I'm wondering how I can picture this in my head. Is it that since muons are heavier than electrons, the proton has to pull on the muon with greater energy to keep it orbiting there and therefore it has more energy? Can I think of it in those terms?
 


so does this mean the de broglie wavelength is 1/200th of that of an electron?

and the velocity is approx 14 times higher than that of an electron - and if so are we anywhere near relativistic speeds yet?

If two positive muons were bound to one negative muon in the same manner as a h2+ ion (basically an electron cloud separating two protons) what would the result be? would the velocities be relativistic?
 

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