Are there legal ways to quickly find eigenvalues of an operator?

[cscott]

If I have an operator of the form [tex]1+3\vec{e}\cdot\vec{\sigma}[/tex] where [tex]\vec{e}\cdot\vec{e}=1[/tex].

How can I find the eigenvalues quickly?

 

[malawi_glenn]

you write out the matrix representation of the operator, and then you find the eigenvalues and eigenvectors to that matrix, same as you do in linear algebra.

 

[cscott]

How can I write out the matrix representation without knowing e?

 

[malawi_glenn]

You know this:

[tex]\vec{e}\cdot\vec{e}=1[/tex]

Why not just make the ansatz:
[tex] \vec{e} = (a,b,c) [/tex]
with:
[tex] a^2 + b^2 + c^2 = 1 [/tex]

When you don't have any numbers or explicit expressions, but you have a condition to be fulfilled, you can atleast do an asatz.

 

[cscott]

Thanks

 

[Avodyne]

Or, you could just choose to use a coord system in which e is in the z direction.

 

[nadavgeva]

listen man if you download the book Schaum's Outline of Quantum Mechanics off emule, go to page 54 there's a whole section on how to represent an operator in matrix form. There are also plenty of problems on the subject in 5,6,7.
I guess you can also check these stuff in the Cohen-Tannoudji book, also availabe in emule. Good luck.
And by the way, I find those one line advices to be very unhelpful. that's why I usually turn to the books.

 

[malawi_glenn]

Is it a legal download?

why download a book that costs 12$ ?

 

[nadavgeva]

Is it not a legal download. However, if this book was available on the internet as a scanned and well edited pdf file, I'd be more than glad to pay for it as this price. Just as I used to illegaly download mp3 before the age of iTunes.
Furthermore, as an undergraduate myself, I feel the moral need to help other undergrads regardless of who they are and where they live.