The three dimensional Hamiltonian operator for an -electron atom (excluding spin-orbit and other interactions) is:

where , and

or

where .

It is the total energy operator of the atom. The first term consists of the kinetic energy operators of the atom’s electrons and acts on a function of , and . The second term, which is the electrostatic potential energy for the attractions between the electrons and the protons, acts on a function of just because we have assumed an infinitely heavy nucleus that is reduced to a point at the origin. The third term, which acts on a function of , and , is the electrostatic potential energy of electron repulsions. To illustrate the double summation for the last term, we consider a four-electron system with electrons e1, e2, e3 and e4. The possible interacting potentials, in the form of **e j**–

**e**are:

*k*,which can be represented by the double summation . Note that this double summation can also be represented as , which is illustrated as:

or , which is

To show that eq239 commutes with the components of the total orbital angular momentum operator , we consider a two electron system. As we know, , where and are the individual orbital angular momentum of the system. Using eq76, we have

where , and are components of .

For the uncoupled case, where the orbital angular momenta of the two electrons do not interact, eq239 becomes . From eq103, eq104 and eq105, we have and , where and . This implies that the components of and commutes with , and that and , and hence and . Therefore, when we analyse the commutation relations of the components of and with , we are left with the commutation relations of the components of and with .

###### Question

Show that the components of , but not those of and , commute with .

###### Answer

Using eq74,

Similarly, the -components of the uncoupled operators and do not commute with with . Therefore, all components of and do not commute with the Hamiltonian .

Using the identity and substituting eq241 and eq242 into , we have . Similarly, we find that and . So,

Therefore, the components of the coupled total orbital angular momentum operator commutes with the Hamiltonian . If so, we can evaluate using the identities and and eq243, which gives us

Finally, also commutes with and because and spin angular momentum operators act on different vector spaces.

###### Question

Show that commutes with , where and ..

###### Answer

Using the identities and , and noting that every component of commutes with every component of because they act on different vector spaces, and that the components of commutes with because they act on different vector spaces, we have