Linearity is a fundamental property of quantum mechanics. If I’m trying to just describe it in wave mechanics terms, I would say that the linearity of quantum mechanics derives from the fact that the wave equation describes a linear system and thus solutions to it must obey the (general, mathematical) principle of superposition.
It’d be fine if it were linear in general, but it’s not for combinations that aren’t orthogonal. Suppose a is drawn from R^2. P(sqrt(2))=P(|(1,1)|)=P(1,1)=P(1,0)+P(0,1)=2*P(|(1,0)|)=2*P(1) which agrees with your analysis, but P(sqrt(5))=P(|(2,1)|)=P(2,1)/=P(1,0)+P(1,1)=3*P(1) doesn’t add up.
It’s been a while since I’ve done any wave mechanics, but I’ll try to take a crack at this. The Schrodinger equation describes a linear PDE such that the sum of any two solutions is also a solution and any constant multiple of a solution is also a solution. Furthermore, the Schrodinger equation just takes the form ^HΨ=EΨ, thus “solutions to the Schrodinger equation” is equivalent to “eigenfunctions of the Hamiltonian.” Thus, if ϕi are eigenfunctions of the Hamiltonian with eigenvalues Ei, then ∑iaiϕi must also be an eigenfunction of the Hamiltonian. This raises a problem for any theory with P nonlinear across a sum of eigenfunctions, however, because it lets me change bases into an equivalent form with a potentially different result.
If a1 is 2 and phi1 has eigenvalue 3, and a2 is 4 and phi2 has eigenvalue 5, then 2*phi1+3*phi2 is mapped to 6*phi1+20*phi2 and therefore not an eigenfunction.
Linearity is a fundamental property of quantum mechanics. If I’m trying to just describe it in wave mechanics terms, I would say that the linearity of quantum mechanics derives from the fact that the wave equation describes a linear system and thus solutions to it must obey the (general, mathematical) principle of superposition.
It’d be fine if it were linear in general, but it’s not for combinations that aren’t orthogonal. Suppose a is drawn from R^2. P(sqrt(2))=P(|(1,1)|)=P(1,1)=P(1,0)+P(0,1)=2*P(|(1,0)|)=2*P(1) which agrees with your analysis, but P(sqrt(5))=P(|(2,1)|)=P(2,1)/=P(1,0)+P(1,1)=3*P(1) doesn’t add up.
It’s been a while since I’ve done any wave mechanics, but I’ll try to take a crack at this. The Schrodinger equation describes a linear PDE such that the sum of any two solutions is also a solution and any constant multiple of a solution is also a solution. Furthermore, the Schrodinger equation just takes the form ^HΨ=EΨ, thus “solutions to the Schrodinger equation” is equivalent to “eigenfunctions of the Hamiltonian.” Thus, if ϕi are eigenfunctions of the Hamiltonian with eigenvalues Ei, then ∑iaiϕi must also be an eigenfunction of the Hamiltonian. This raises a problem for any theory with P nonlinear across a sum of eigenfunctions, however, because it lets me change bases into an equivalent form with a potentially different result.
If a1 is 2 and phi1 has eigenvalue 3, and a2 is 4 and phi2 has eigenvalue 5, then 2*phi1+3*phi2 is mapped to 6*phi1+20*phi2 and therefore not an eigenfunction.
Ah, I see the confusion. Since we’re in a wave mechanics setting, I should have written ^HΨ=i¯h∂∂tΨ rather than ^HΨ=EΨ.