A Hamiltonian represented by a matrix, find the eigevalues

[Jdraper]

Homework Statement

Been struggling with a particular problem that keeps coming up in one of my modules, so i thought i'd see if anyone here can enlighten me.

A Hamiltonian H₀ is represented by the matrix:

top row: 3 0 -1
Middle row: 0 a 0
Bottom row: -1 0 3

(Unsure how to display matrices)

where is a dimensionless parameter. Show that (1/√2)(1 0 1) is an
eigenstate of the Hamiltonian and derive its eigenvalue. Find the other
two eigenstates and the associated eigenenergies.

Homework Equations

n/a

The Attempt at a Solution

Can find the eigenenergy associated with the eigenstate given to us with relative ease, it has a value of 2eV.

However finding the remaining eigenstates has always puzzled me. Is there easy way to find them? as opposed to learning how to use row operators.

Thanks, John

 

[rwooduk]

I'll try contribute something, it could be wrong but might give you some ideas. for

$$H = \begin{pmatrix} 3 & 0 & -1\\ 0 & a & 0\\ -1 & 0 & 3 \end{pmatrix}$$

we can write

$$\begin{pmatrix} 3 & 0 & -1\\ 0 & a & 0\\ -1 & 0 & 3 \end{pmatrix} \Psi = E\Psi$$

we can introduce the identity operator

$$\begin{pmatrix} 3 & 0 & -1\\ 0 & a & 0\\ -1 & 0 & 3 \end{pmatrix} \Psi = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} E\Psi$$

put E in the matrix and minus from the left hand side so

$$\begin{pmatrix} 3-E & 0 & -1-E\\ 0 & a-E & 0\\ -1-E & 0 & 3-E \end{pmatrix} \Psi = 0$$

we can then insert the matrix form of the eigen state

$$\begin{pmatrix} 3-E & 0 & -1-E\\ 0 & a-E & 0\\ -1-E & 0 & 3-E \end{pmatrix} \begin{pmatrix} a_{1}\\ a_{2} \\ a_{3} \end{pmatrix} = 0$$

then take the determinant and this will give you the possible energy values. for each energy value put it back into the matrix and it will tell you what a₁, a₂ and a₃ are in relation to each other. once you know this then it should be clear where the given wave function comes from. i.e. a₁=a₃ and a₂ = 0 and what the other possible eigenstates are.

 

[Fightfish]

Essentially what you are doing is what is commonly known as diagonalising the matrix. You should look up some basic linear algebra texts or even google to find out more about the details. The general idea in post #2 is correct, although some careless mistakes slipped in from line 4 onwards.
Give it a try and let us know if you face any further problems when trying to work them out.