- #1

paco_uk

- 22

- 0

## Homework Statement

I'm trying to use the transfer matrix method in statistical mechanics but I'm struggling with the algebra so I'd like to know if there is a simpler way to find the eigenvalues and eigenvectors of a matrix.

For example, studying the lattice gas model produces the transfer matrix:

[tex]

T = \left( \begin{array}{cc}

1 & e^{\beta \mu /2} \\

e^{\beta \mu /2} & e^{\beta ( J - \mu)} \end{array} \right)

[/tex]

## Homework Equations

It's not too hard to find the eigenvalues, although they don't look very nice:

[tex]

\lambda = \frac{1 + e^{\beta (J + \mu)}}{2} \pm \sqrt{4e^{\beta \mu} + (1-e^{\beta(J+\mu)})^2}

[/tex]

## The Attempt at a Solution

When it comes to the eigenvectors it seems like a hopeless case:

[tex]

\left( \begin{array}{cc}

1 & e^{\beta \mu /2} \\

e^{\beta \mu /2} & e^{\beta ( J - \mu)} \end{array} \right) \left( \begin{array}{c}

a \\

b \end{array} \right) = \lambda

\left( \begin{array}{c}

a \\

b \end{array} \right)

[/tex]

The algebra involved here seems unmanageable, especially as I'm supposed to be able to do this under exam conditions.

I'm supposed to be able to show that:

[tex]

\langle n_i \rangle = \frac{1}{1+e^{-2 \theta}}

[/tex]

where

[tex]

\sinh (\theta ) = \exp(\frac{1}{2} \beta J) \sinh (\frac{1}{2} \beta [J + \mu])

[/tex]

I think that [tex] \langle n_i \rangle [/tex] is given by:

[tex]

\langle n_i \rangle = \langle 0 | \mathbf{C} | 0 \rangle

[/tex]

where [tex] | 0 \rangle [/tex] is the eigenvector corresponding to the largest eigenvalue and:

[tex]

C = \left( \begin{array}{cc}

0 & 0 \\

0 & 1 \end{array} \right)

[/tex]

In fact the examiners report for this question suggests that it is not even necessary to find the eigenvalues of T which is why I am wondering if there is some clever way to spot the eigenvectors without going through all the algebra?

A similar question on the Potts model said something about guessing the eigenvectors from the symmetry of the matrix but I wouldn't know how to start.