Oops! May be not SHM

There is no SHM about the nucleus.  The electron will simply be in circular motion around the nucleus at a radius $r_o$ and a velocity of $c$, as given by.

$r_o=\cfrac{q^2}{4\pi \varepsilon_o{m_ec^2}}$

However if the electron does not approach close enough ie $> r_o$ then it may be possible that the electron over shoot the nucleus only to reverse direction at some distance $x$ then,

$\cfrac { 1 }{ 2 } m_{ e }c^{ 2 }=\cfrac { q^{ 2 } }{ 4\pi \varepsilon _{ o }x } =\cfrac { 1 }{ 2 } kx^{ 2 }$

As a first approximation, the resonance is liken to be from Hooke's spring formulation. Speed in the perpendicular direction does not change and so does not concern us here.

$\cfrac { k }{ { m }_{ e } } =\cfrac { c^{ 2 } }{ x^{ 2 } } =c^{ 2 }\left\{ \cfrac { 2\pi \varepsilon _{ o } }{ q^{ 2 } } m_{ e }c^{ 2 } \right\} ^{ 2 }$

 $f=\cfrac { 1 }{ 2\pi  } \sqrt { \cfrac { k }{ { m }_{ e } }  }=\cfrac { 1 }{ 2\pi  } c\left\{ \cfrac { 2\pi \varepsilon _{ o } }{ q^{ 2 } } m_{ e }c^{ 2 } \right\}$

$f=\cfrac{m_{ e }c^{ 3 }\varepsilon _{ o }}{q^2}$

$f$=9.10938291e-31*(299792458)^3*8.854187817e-12/(1.602176565e-19)^2=8.4660e21 Hz

Was this resonance from hydrogen ever detected?  The sun might have this resonance.