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Exercise:
To test the validity of the quantum mechanical description of the hydrogen atom the energy levels of various exotic atoms have been investigated. The experimental results agree with the theoretical predictions. vspacemm Calculate the first energy level for abcliste abc a muonic hydrogen atom i.e. a hydrogen atom where the electron is replaced by a muon m_mummuO. abc a muonic lead atom. abc positronium a configuration of a positron anti-electron and an electron. Since the two masses are identical the reduced mass mum_e/ has to be used instead of the electron mass. abcliste

Solution:
The energy of the ground state of an electron in the potential of a nucleus with charge Ze is given by E_ -fracZ^ e^ m_eepsilon_^ h^ When the electron is replaced by another particle with charge -e the mass has to be adjusted accordingly. abcliste abc The energy can be calculated as E_ -fracZ^ e^ m_muepsilon_^ h^ -fracZ^ e^ m_eepsilon_^ h^fracm_mum_eE_efracm_mum_e where EeEgrO is the energy of the ground state of the regular hydrogen atom. It follows for the myonic hydrogen atom E_mu Egrtimesfracmmume resultEHmuP For a muonic lead atom the proton number is ZZPbO so E_ textrmPb/mu Z^ E_mu ZPb^timesEHmu resultEPbmuP abc For positronium the only change compared to the regular hydrogen atom is that the relevant mass is only half of the electron mass. The predicted energy for the ground state is therefore E_e^+e^- fracE_ resultEposP abcliste
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Exercise:
To test the validity of the quantum mechanical description of the hydrogen atom the energy levels of various exotic atoms have been investigated. The experimental results agree with the theoretical predictions. vspacemm Calculate the first energy level for abcliste abc a muonic hydrogen atom i.e. a hydrogen atom where the electron is replaced by a muon m_mummuO. abc a muonic lead atom. abc positronium a configuration of a positron anti-electron and an electron. Since the two masses are identical the reduced mass mum_e/ has to be used instead of the electron mass. abcliste

Solution:
The energy of the ground state of an electron in the potential of a nucleus with charge Ze is given by E_ -fracZ^ e^ m_eepsilon_^ h^ When the electron is replaced by another particle with charge -e the mass has to be adjusted accordingly. abcliste abc The energy can be calculated as E_ -fracZ^ e^ m_muepsilon_^ h^ -fracZ^ e^ m_eepsilon_^ h^fracm_mum_eE_efracm_mum_e where EeEgrO is the energy of the ground state of the regular hydrogen atom. It follows for the myonic hydrogen atom E_mu Egrtimesfracmmume resultEHmuP For a muonic lead atom the proton number is ZZPbO so E_ textrmPb/mu Z^ E_mu ZPb^timesEHmu resultEPbmuP abc For positronium the only change compared to the regular hydrogen atom is that the relevant mass is only half of the electron mass. The predicted energy for the ground state is therefore E_e^+e^- fracE_ resultEposP abcliste
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quantum physics
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energy level, hydrogen atom
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