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".

 

[HalfLostAndSearching]

To find the energy eigenvalues, you need to set up the equation E|ψ>=H|ψ>, as you have done. Then, you can solve for the values of E that satisfy this equation. This will give you the energy eigenvalues for the system.

To find the corresponding eigenkets, you can use the values of E that you have found and plug them back into the original equation E|ψ>=H|ψ>. This will give you a system of equations with the coefficients c1 and c2. Solving for these coefficients will give you the corresponding eigenkets for each energy eigenvalue.

Alternatively, you can also use the formula for the eigenvalues and eigenvectors of a 2x2 matrix to find the eigenvalues and eigenvectors directly. This formula is given by:

λ1,2 = (Tr(H) ± √(Tr(H)^2 - 4det(H)))/2

and

|ψ1,2> = (1/√(c1^2 + c2^2)) * (c1|1> ± c2|2>)

where Tr(H) is the trace of the Hamiltonian matrix and det(H) is the determinant of the Hamiltonian matrix. This will give you the same result as solving the system of equations mentioned above.

I hope this helps! Let me know if you have any further questions.