Ψ1,ψ2 same E, p, are ⊥; find Ωψ1=ω1ψ1, Ωψ2=ω2ψ2; ω1≠ω2?

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Spinnor
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I think I have two orthogonal solutions, ψ1 and ψ2, to the 1+1 dimensional Dirac equation with the same energy and momentum. How might you proceed to try and find some operator, Ω, if it exists, such that,

Ωψ1 = ω1ψ1 and
Ωψ2 = ω2ψ2 where ω1 ≠ ω2.

Must Ω necessarily commute with the Hamiltonian operator?

Thanks for any help!
 

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Oops, after a little more care looks like ψ1 and ψ2 are not ⊥. Fugawezt!