How Do Eigenstates and Eigenvalues Relate to Quantum Observables?

[martyf]

Homework Statement

I have the hamiltonian :

H=C(|2><1|+|1><2|)

where :
C=costant
|1> and |2> are eigenstates of an osservable A.

what are the eigenstate and eigenvalues of the hamiltonian?
what is the probability that the system is in the state |2>?

The Attempt at a Solution

eigenstates :

|1>+|2>, |1> - |2>, -|1> - |2>,-|1>+|2>

eigenvalues (respectively):

C , -C, C, -C

ad es:

H(|1>+|2>)=(C(|2><1|+|1><2|)) (|1>+|2>)=C |2> <1|1> + C |2><1|2> + C |1> <2|1> + C |1> <2|2>= C |2> <1|1> + C |1> <2|2>= C (|1>+|2>)

 

[Redbelly98]

The Attempt at a Solution

eigenstates :

|1>+|2>, |1> - |2>, -|1> - |2>,-|1>+|2>

eigenvalues (respectively):

C , -C, C, -C

A 2x2 matrix should only have 2 eigenstates.