In quantum physics, when working with kets, it is useful to know how to use eigenvectors and eigenvalues. Applying an operator to a ket can result in a new ket:
![image0.png](https://www.dummies.com/wp-content/uploads/397731.image0.png)
To make things easier, you can work with eigenvectors and eigenvalues (eigen is German for “innate” or “natural”). For example,
![image1.png](https://www.dummies.com/wp-content/uploads/397732.image1.png)
is an eigenvector of the operator A if
The number a is a complex constant
Note what’s happening here: Applying A to one of its eigenvectors,
![image3.png](https://www.dummies.com/wp-content/uploads/397734.image3.png)
multiplied by that eigenvector’s eigenvalue, a.
Although a can be a complex constant, the eigenvalues of Hermitian operators are real numbers, and their eigenvectors are orthogonal
![image4.png](https://www.dummies.com/wp-content/uploads/397735.image4.png)
Casting a problem in terms of eigenvectors and eigenvalues can make life a lot easier because applying the operator to its eigenvectors merely gives you the same eigenvector back again, multiplied by its eigenvalue — there’s no pesky change of state, so you don’t have to deal with a different state vector.
Take a look at this idea, using the R operator from rolling the dice, which is expressed this way in matrix form:
![image5.png](https://www.dummies.com/wp-content/uploads/397736.image5.png)
The R operator works in 11-dimensional space and is Hermitian, so there’ll be 11 orthogonal eigenvectors and 11 corresponding eigenvalues.
Because R is a diagonal matrix, finding the eigenvectors is easy. You can take unit vectors in the 11 different directions as the eigenvectors. Here’s what the first eigenvector,
![image6.png](https://www.dummies.com/wp-content/uploads/397737.image6.png)
would look like:
![image7.png](https://www.dummies.com/wp-content/uploads/397738.image7.png)
And here’s what the second eigenvector,
![image8.png](https://www.dummies.com/wp-content/uploads/397739.image8.png)
would look like:
![image9.png](https://www.dummies.com/wp-content/uploads/397740.image9.png)
And so on, up to
![image10.png](https://www.dummies.com/wp-content/uploads/397741.image10.png)
Note that all the eigenvectors are orthogonal.
And the eigenvalues? They’re the numbers you get when you apply the R operator to an eigenvector. Because the eigenvectors are just unit vectors in all 11 dimensions, the eigenvalues are the numbers on the diagonal of the R matrix: 2, 3, 4, and so on, up to 12.