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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:

image0.png

To make things easier, you can work with eigenvectors and eigenvalues (eigen is German for “innate” or “natural”). For example,

image1.png

is an eigenvector of the operator A if

  • The number a is a complex constant

    image2.png

Note what’s happening here: Applying A to one of its eigenvectors,

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

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

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

would look like:

image7.png

And here’s what the second eigenvector,

image8.png

would look like:

image9.png

And so on, up to

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.

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