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.
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.