from math import sqrt
n = 2
Ms = 1.
M = 1.
omega = 1.
u = 3.*Ms**3/27-2*M*omega**2*(n+1./2)**2
E = (-u/2+1./2*sqrt(u**2-4*(Ms/3)**6))**(1./3) + Ms**3/9*(-u/2+1./2*sqrt(u**2-4*(Ms/3)**6))**(-1./3) - Ms/3
E
2.0285764580832724
n = 2.
c = 137.03599907
M = 1.
En = 1.49999501
En = 3.49989517
En = 2.49997504
En = 2.49993510
sqrt((En+2*M*c**2)/(2*M*c**2))*En - (n+1./2)*1.
1.8299537026589263e-05