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

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SUMMARY

The discussion centers on finding an operator, Ω, that acts on two orthogonal solutions, ψ1 and ψ2, of the 1+1 dimensional Dirac equation, both having the same energy and momentum. The user seeks to determine if Ω can be defined such that Ωψ1 = ω1ψ1 and Ωψ2 = ω2ψ2, with the condition that ω1 ≠ ω2. A critical question raised is whether Ω must commute with the Hamiltonian operator. The user later realizes that ψ1 and ψ2 are not orthogonal, which complicates the analysis.

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This discussion is beneficial for quantum physicists, graduate students studying quantum mechanics, and researchers focusing on operator theory and the Dirac equation.

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!
 

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