How to Work with Eigenvectors and Eingenvalues
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:
To make things easier, you can work with eigenvectors and eigenvalues (eigen is German for “innate” or “natural”). For example,
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,
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
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:
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,
would look like:
And here’s what the second eigenvector,
would look like:
And so on, up to
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.