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

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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 goal is to determine if there exists an operator such that Ωψ1 equals ω1ψ1 and Ωψ2 equals ω2ψ2, with ω1 not equal to ω2. A key question raised is whether Ω must commute with the Hamiltonian operator. The initial assumption of orthogonality between ψ1 and ψ2 is later reconsidered, suggesting they may not be orthogonal after all. The inquiry seeks clarification on the properties of the operator in relation to the solutions.
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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!
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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