Addition of spin-½ operators ↩

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shorthands: {}
aliases: [Addition of spin-1/2 operators, Addition of spin-half operators]
created: 2022-01-08 08:37:23
modified: 2022-01-11 15:39:10

For the general case, see Addition of angular momenta. This is the simplest case of adding general angular momenta. This is useful for the treatment of the helium atom.

Here, we consider two spin-½ operators and and their sum is the total spin:

The four states

in which the first (second) symbol refers to the first (second) spin, are eigenstates of . It is reasonable to suppose that the total spin assumes the values and . To show this, we compute:

Furthermore

Let us first consider the two maximally aligned states and . For these, we find from the expansion of that:

And

The states and therefore have total spin and .
The as yet missing spin- state with is obtained by application of to :

The resulting state has been normalized to unity by inserting the factor . It has . Thus, we have found all states.

Triplet and singlet states

Using the notation , in which designates the total spin and it's -component, we have the so-called triplet states:

There is an additional state, which is orthogonal to those given above. It is called the singlet state:

For this state, evidently:

And the total spin vanishes as well:

Thus, we have found all four eigenstates of and .