1.

When Bohr's theory is used to compute the energy levels in a hydrogen atom, only the Coulomb interaction between the electron and the proton is taken into account, the magnetic moments of these particles being ignored. Assess the resulting error. How will the energy level pattern change, if, in addition to the Coulomb interaction, the magnetic interaction between the electron and the proton is also taken into account?

Answer»


Solution :The orientations of the magnetic moments of the proton and the electron may be cither parallel, or anti-parallel lo one another. THEREFORE the TOTAL energy of interaction of the electron and the proton is
`epsi=epsi_("Coul")pmepsi_(mag)=-e^(2)/(4piepsi_(0)r)pm(2mu_(0)mu_(p)mu_(e))/(4pir^(3))`
The relative error is
`delta=|(delta-delta_("Coul"))/delta| ~~|(delta_(mag))/(delta_("Coul"))|=(2mu_(p)mu_(e))/(e^(2)c^(2)r^(2))le(2mu_(p)mu_(e))/(e^(2)c^(2)a_(0)^(2))`
where `a_(0)` is the Bohr radius.
Every energy level is seen to split into two sub-levels: the upper `epsi_(n).=epsi+|epsi_(mag)|`, and the lower `epsi_(n)..=epsi-|epsi_(mag)|`, where n is the number of the level and `|epsi_(mag)|` is the MAGNITUDE of the energy of magnetic interaction. Dashed LINES in Fig. show the first three energy levels stemming from Bohr.s theory, while the solid lines show the sub-levels due to magnetic interaction. The diagram is, of course, not to scale.


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