Bohr's Model of the Hydrogen atom.
First consider the energies of the electron orbiting around the proton.

> K:=unapply((1/2)*m*v*v,m,v); U:=unapply(-k*e*e/r,k,e,r);

Add Bohr's hypothesis that the angular momentum mvr is quantized in units of hb (Plank's constant divided by 2*Pi). Recall that this is the same as the condition that the electron wave interferes constructively with itself, i.e., deBroglie said that if waves can act like particles, why not particles act like waves? with wavelength h/(mv).

> solve( {k*e*e/r/r=m*v*v/r, m*r*v=n*hb},{v,r});

Compute total energy:

> E:=subs(%,K(m,v)+U(k,e,r));

And write in electron-volts (eV):

> EeV := E/e;

Substitute values for e, hb=h/(2*Pi), m_electron and k_Coulomb in the n=1, or ground state, energy level, E1:

> E1:=subs(n=1.0, hb=1.0546e-34, e=1.6022E-19, k=8.9875E9, m=9.1095e-31, EeV);

Compute the energy levels for n=2, 3, 4, 5, and 6:

> E2:=E1/2/2; E3:=E1/3/3; E4:=E1/4/4; E5:=E1/5/5; E6:=E1/6/6;

Apply the photon concept, E_photon = h*f and wavelength = c/f.
Photon wavelenth for n=3 to n=2 transition (in nm) - red:

> 299792458/((E3-E2)*1.6022E-19/6.6262E-34)*1e9;

Photon wavelenth for n=3 to n=2 transition (in nm) - cyan:

> 299792458/((E4-E2)*1.6022E-19/6.6262E-34)*1e9;

Photon wavelenth for n=5 to n=2 transition (in nm) - blue:

> 299792458/((E5-E2)*1.6022E-19/6.6262E-34)*1e9;

Photon wavelenth for n=6 to n=2 transition (in nm) - violet:

> 299792458/((E6-E2)*1.6022E-19/6.6262E-34)*1e9;

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