(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The Hamiltonian of a quantum-mechanical system has only two energy eigenstates, namely |1> and |2>. The system has three other properties, denoted by the observables A, B, and C, respectively. The normalized eigenstates |1> and |2> may or may not be eigenstates of A, B, or C. Find, as many as possible, the eigenvalues of the observables A, B, and C by making use of the following known matrix elements:

(a) [itex]<2|A|2> = 1/3, <2|A^{2}|2> = 1/9 [/itex]

(b) [itex]<1|B|1> = 1/2, <1|B^{2}|1> = 5/4, <2|B|2> = 0 [/itex]

(c) [itex]<1|C|1> = 1 , <1|C^{2}|1> = 5/4, <1|C^{3}|1> = 7/4 [/itex]

Comment briefly on whether the observables commute with the Hamiltonian.

2. The attempt at a solution

If a state is an eigenstate I can write it as [itex] a|a> [/itex]. So |2> is an eigenstate of A with eigenvalue 1/3. The states |1> and |2> are not eigenstates of B and C. Is this part correct?

From this, we know that B and C don't commute with the Hamiltonian but A might (depending on whether |1> is also an eigenstate).

That's about all I can get from this problem. I get the feeling that I'm not using a lot of the information so can someone help me figure out more eigenvalues for A, B and C and whether A commutes with H. Thank you

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# Eigenstate problem in QM

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