Embarassing question about eigenvectors

[Ed Quanta]

Ok, so let us suppose we have a spinor which is a spin 1/2 state vector (a)
b

Now spinors exist in 2 dimensional complex space. How do I find the eigenvalues which correspond to the eigenvector
(a)
b


I am confused because we are dealing with eigenvalues for a matrix which is not a square matrix. I know for a square matrix we just find the eigenvalues such that the determinant of the matrix becomes zero. I am not sure how to deal with determinants of non-square matrices however. Helo anybody?

 

[Dr Transport]

in spin 1/2 space, the eigen vectors are usually found using the z component of the angular momentum. Use the Pauli matricies for $$S_{z}$$ to find the eigenvalues and vectors.

 

[M98Ranger]

Hi there,

First of all, there is no need to feel embarrassed about asking this question. Understanding eigenvectors and eigenvalues can be tricky, and it is completely normal to have questions about them.

To answer your question, in order to find the eigenvalues for a spinor, we need to first convert it into a square matrix. This can be done by taking the outer product of the spinor with its conjugate transpose. For your spinor (a, b), the corresponding square matrix would be:

|a|^2 a*b
a*b |b|^2

Now, to find the eigenvalues of this matrix, we can use the same method as with a regular square matrix - set the determinant equal to zero and solve for the eigenvalues. In this case, the determinant would be:

|a|^2 * |b|^2 - |a*b|^2 = 0

Solving this equation will give you the eigenvalues for your spinor. Keep in mind that since spinors are complex numbers, the eigenvalues will also be complex numbers.

I hope this helps clarify things for you. Don't hesitate to ask for further clarification if needed.