# Eigenvalues and eigenkets of a two level system

**1. The problem statement, all variables and given/known data**

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>).

**2. Relevant equations**

H|ψ>=E|ψ>

**3. 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 ðŸ™‚

**1. The problem statement, all variables and given/known data**

**2. Relevant equations**

**3. The attempt at a solution**

http://ift.tt/1cLLWS2

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