Eigenvalues and eigenkets of a two level system

[White_M]

Homework Statement

The Hamiltonian for a two level system is given:

H=a(|1><1|-|2><2|+|1><2|+|2><1|)

where 'a' is a number with the dimentions of energy.

Find the energy eigenvalues and the corresponding eigenkets (as a combination of |1> and |2>).

Homework Equations

H|ψ>=E|ψ>

The Attempt at a Solution

Using |a>=∑ci |a'>

I wrote |ψ> as a combunation of the two system kets |ψ>=c1|1>+c2|2> (c1,c2 are complex numbers).

so H|ψ>= a(|1><1|-|2><2|+|1><2|+|2><1|)*(c1|1>+c2|2>)= a(c1|1>-c2|2>+c2|1>c1|2>)=a ((c1+c2)|1>+(c1-c2)|2>)=E|ψ>.

How do I continue?

Thank you :)

 

[hilbert2]

Find out how to write the Hamiltonian as a matrix, and then use standard linear algebra methods to find the eigenvalues and eigenvectors.

An example: if the Hamiltonian were $H=2|1><1|+3|2><2|$, the corresponding matrix would be the diagonal matrix

$H=\left( \begin{array}{cc} 2 & 0 \\ 0 & 3 \\ \end{array} \right)$

Here the kets correspond to column vectors. Do you know how to find the eigenvalues of matrix $A$ from the characteristic equation $det|A-\lambda I|=0$?

 

[White_M]

Hilbert2

Thank you for responding.

I wrote the Hamiltonian as:

H11=a
H12=a
H21=a
H22=-a

Where H11,H12,H21,H22 are the H matrix components (sorry, I could not figure out how to writhe it as a matrix with this Latex Reference).

Solving Det(H-λI)=0 I got that the eigenvalues are +a\sqrt{2} and -a\sqrt{2}

For λ=a\sqrt{2} I got the vector |ψ1>= \frac{1}{4-2\sqrt{2}}(|1>+(\sqrt{2}-1)|2>)
and for λ=-a\sqrt{2} I got the vector |ψ2>= \frac{1}{4+2\sqrt{2}}(|1>-(\sqrt{2}+1)|2>)

Is that correct?

 

[hilbert2]

Just operate on your solution vectors with the hamiltonian to test if they are correct. This kind of eigenvalue problems with small matrices can also be solved with Mathematica or WolframAlpha.

If you want to see how my matrix was written in Latex, right click on it and choose "Show Math As > TeX commands".