In quantum physics, when you look at the spin eigenstates and operators for particles of spin 1/2 in terms of matrices, there are only two possible states, spin up and spin down.
The eigenvalues of the S2 operator are
![image0.png](https://www.dummies.com/wp-content/uploads/395103.image0.png)
and the eigenvalues of the Sz operator are
![image1.png](https://www.dummies.com/wp-content/uploads/395104.image1.png)
You can represent these two equations graphically as shown in the following figure, where the two spin states have different projections along the z axis.
![Spin magnitude and <i>z</i> projection.](https://www.dummies.com/wp-content/uploads/395105.image2.jpg)
In the case of spin 1/2 matrices, you first represent the eigenstate
![image3.png](https://www.dummies.com/wp-content/uploads/395106.image3.png)
like this:
![image4.png](https://www.dummies.com/wp-content/uploads/395107.image4.png)
And the eigenstate
![image5.png](https://www.dummies.com/wp-content/uploads/395108.image5.png)
looks like this:
![image6.png](https://www.dummies.com/wp-content/uploads/395109.image6.png)
Now what about spin operators like S2? The S2 operator looks like this in matrix terms:
![image7.png](https://www.dummies.com/wp-content/uploads/395110.image7.png)
And this works out to be the following:
![image8.png](https://www.dummies.com/wp-content/uploads/395111.image8.png)
Similarly, you can represent the Sz operator this way:
![image9.png](https://www.dummies.com/wp-content/uploads/395112.image9.png)
This works out to
![image10.png](https://www.dummies.com/wp-content/uploads/395113.image10.png)
Using the matrix version of Sz, for example, you can find the z component of the spin of, say, the eigenstate
![image11.png](https://www.dummies.com/wp-content/uploads/395114.image11.png)
Finding the z component looks like this:
![image12.png](https://www.dummies.com/wp-content/uploads/395115.image12.png)
Putting this in matrix terms gives you this matrix product:
![image13.png](https://www.dummies.com/wp-content/uploads/395116.image13.png)
Here’s what you get by performing the matrix multiplication:
![image14.png](https://www.dummies.com/wp-content/uploads/395117.image14.png)
And putting this back into ket notation, you get the following:
![image15.png](https://www.dummies.com/wp-content/uploads/395118.image15.png)
How about the raising and lowering operators S+ and S–? The S+ operator looks like this:
![image16.png](https://www.dummies.com/wp-content/uploads/395119.image16.png)
And the lowering operator looks like this:
![image17.png](https://www.dummies.com/wp-content/uploads/395120.image17.png)
Here it is in matrix terms:
![image18.png](https://www.dummies.com/wp-content/uploads/395121.image18.png)
Performing the multiplication gives you this:
![image19.png](https://www.dummies.com/wp-content/uploads/395122.image19.png)
Or in ket form, it’s
![image20.png](https://www.dummies.com/wp-content/uploads/395123.image20.png)
Cool.