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

[eudyptula]

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)=ψ_a(x₁)ψ_b(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]

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