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.(adsbygoogle = window.adsbygoogle || []).push({});

H = -ħ^{2}/2m * (∂^{2}/∂x_{1}^{2}+ ∂^{2}/∂x_{2}^{2})

Hψ=Eψ

∂^{2}ψ/∂x_{1}^{2}+ ∂^{2}ψ/∂x_{2}^{2}= kψ, where k=-2mE/ħ^{2}

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(x_{1},x_{2})=ψ_{a}(x_{1})ψ_{b}(x_{2}), 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.

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

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