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

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In summary, Ψ1 and ψ2 represent wave functions, E represents energy, and p represents momentum in this equation. If Ψ1 and ψ2 have the same energy and momentum, it means that they are identical except for a possible difference in their quantum states, known as degeneracy. To find Ωψ1 and Ωψ2, an operator Ω is used on the wave functions Ψ1 and ψ2, resulting in Ωψ1 and Ωψ2 having the same energies ω1ψ1 and ω2ψ2. If ω1 and ω2 are unequal, it means that Ωψ1 and Ω
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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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  • #2
Oops, after a little more care looks like ψ1 and ψ2 are not ⊥. Fugawezt!
 

1. What does Ψ1, ψ2, E, and p represent in this equation?

Ψ1 and ψ2 represent wave functions, E represents energy, and p represents momentum. These are all important variables in quantum mechanics.

2. What does it mean for Ψ1 and ψ2 to have the same E and p?

If Ψ1 and ψ2 have the same energy and momentum, it means that they are identical except for a possible difference in their quantum states. This is known as degeneracy.

3. How do you find Ωψ1 and Ωψ2 in this equation?

To find Ωψ1 and Ωψ2, you would need to use an operator Ω on the wave functions Ψ1 and ψ2. This operator would result in Ωψ1 having the same energy ω1ψ1 and Ωψ2 having the same energy ω2ψ2.

4. What does it mean for ω1 and ω2 to be unequal?

If ω1 and ω2 are unequal, it means that Ωψ1 and Ωψ2 will have different energies. This could indicate a difference in the quantum states of Ψ1 and ψ2.

5. Is this equation always true, or are there exceptions?

This equation is generally true in quantum mechanics, but there may be exceptions depending on the specific system being studied. In some cases, the wave functions Ψ1 and ψ2 may not have the same energy and momentum, or the operator Ω may not result in the same energies for Ωψ1 and Ωψ2.

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