In Sakurai's Quantum mechanics book, he says the hydrogen atom has $SO(4)$ symmetry by explicitly exhibiting operators $I_i,K_i$ that satisfy the commutation relation of the Lie algebra $so(4)$. Namely, if we restrict our attention to some eigenspace of the Hamiltonian $H$, then it is possible to find operators $I_i,K_i,i=1,2,3$ such that $$[I_i,I_j]=i\hbar\epsilon_{ijk}I_k,$$$$[K_i,K_j]=i\hbar\epsilon_{ijk}K_k,$$$$[I_i,K_j]=0.$$ Then he claims $SO(4)$ symmetry follows because these form the Lie algebra $so(4)=su(2)\oplus su(2)$. My question is there are many groups with the same Lie algebra as $SO(4)$, its universal cover $SU(2)\times SU(2)$ for instance. So what tells us the symmetry group is $SO(4)$ and not those other groups?
Asked
Active
Viewed 306 times
