Possible energy values given Hamiltonian

[Rayan]

Homework Statement

The Hamiltonian for a two level system with the orthonormal states |1⟩ and |2⟩ is given by:

Relevant Equations

H = a|1⟩⟨1| + b|1⟩⟨2| + b|2⟩⟨1| + c|2⟩⟨2| ,

where a,b and c are real constants with energy unit.

So first I rewrote H as a matrix:

$$ H =
\begin{pmatrix}
a & b \\
b & c
\end{pmatrix} $$

And tried to find the eigenvalues/energies of H, so I solved

$$ det (H - \lambda I ) =
\begin{vmatrix}
a-\lambda & b \\
b & c-\lambda
\end{vmatrix} = (a-\lambda)(c-\lambda) - b^2 = ac - a\lambda - c\lambda + \lambda^2 - b^2 = 0
$$

but got a complicated solution

$$ \lambda = \frac{a+c}{2} \pm \sqrt{ ( \frac{a+c}{2} )^2 - (ac-b^2) } $$

What am I doing wrong here?

 

[TSny]

So first I rewrote H as a matrix:

$$ H =
\begin{pmatrix}
a & b \\
b & c
\end{pmatrix} $$

Ok.

And tried to find the eigenvalues/energies of

$$ H - \lambda I $$

Terminology: You want the eigenvalues of $H$, not $H - \lambda I$.

but got a complicated solution
$$ \lambda = \frac{a-c}{2} \pm \sqrt{ ( \frac{a-c}{2} )^2 - (ac-b^2) } $$
What am I doing wrong here?

It's hard to tell where you made the mistakes. Please show the steps in getting the quadratic equation for $\lambda$ and then the steps in solving for $\lambda$.

 

[Rayan]

Ok.Terminology: You want the eigenvalues of $H$, not $H - \lambda I$.It's hard to tell where you made the mistakes. Please show the steps in getting the quadratic equation for $\lambda$ and then the steps in solving for $\lambda$.

You're right! I just updated my question with the steps!:)

 

[TSny]

$$ det (H - \lambda I ) =
\begin{vmatrix}
a-\lambda & b \\
b & c-\lambda
\end{vmatrix} = (a-\lambda)(c-\lambda) - b^2 = ac - a\lambda - c\lambda + \lambda^2 - b^2 = 0
$$

This looks good.

but got a complicated solution

$$ \lambda = \frac{a+c}{2} \pm \sqrt{ ( \frac{a+c}{2} )^2 - (ac-b^2) } $$

This looks correct.

My preference would be to write what you have as $$ \lambda = \frac{a+c}{2} \pm \frac1 2 \sqrt{ ((a+c)^2 - 4(ac-b^2) } $$
You should be able to simplify the expression inside the square root a little. (Work with the terms involving $a$ and $c$.)