System of 2 particles: why is the wavefunction a product?

I am trying to solve for the energy of 2 non-interacting identical particles in a 1D infinite potential well. I want to do it as much "from scratch" as possible, making sure I fully understand every step.

H = -ħ2/2m * (∂2/∂x12 + ∂2/∂x22)

Hψ=Eψ

2ψ/∂x12 + ∂2ψ/∂x22 = kψ, where k=-2mE/ħ2

I got up to here, where I need to write down a form for ψ to take. Both textbooks I referred to (Griffiths and Shankar) use the example of a wavefunction for 2 distinguishable particles before discussing the 2-term form for a wavefunction of 2 indistinguishable particles. Apparently for 2 distinguishable particles, ψ(x1,x2)=ψa(x1b(x2), but I don't understand the reasoning behind writing it as a product of the individual wavefunctions. Griffiths discusses it a little bit in footnote 2, but I don't follow it. I do understand the later part about writing ψ for indistinguishable particles as a sum of the two products, but I don't understand why each term is written as a product in the first place.

atyy
It can be taken as an additional axiom that the Hilbert space for two particles is the tensor product of the individual Hilbert spaces, eg. http://www.theory.caltech.edu/~preskill/ph219/chap2_13.pdf (Axiom 5). Then the basis functions of the two-particle Hilbert space are products of basis functions of one-particle Hilbert space.

bhobba
Mentor
It can be taken as an additional axiom that the Hilbert space for two particles is the tensor product of the individual Hilbert spaces, eg. http://www.theory.caltech.edu/~preskill/ph219/chap2_13.pdf (Axiom 5). Then the basis functions of the two-particle Hilbert space are products of basis functions of one-particle Hilbert space.

Atty is correct.

Its an axiom, but its so natural and obvious many textbooks don't state it.

Thanks
Bill