Fermions in a one-dimensional harmonic potential

MeiLai
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Two identical spin-1/2 fermions are placed in the one-dimensional harmonic potential

V(x)=(1/2) m w^2 x^2,

where m is the mass of the fermion and w its angular frequency.



(1) Find the energies of the ground and first excited states of this two-fermion system. Express the eigenstates corresponding to these two energy levels in terms of harmonic oscillator wavefunctions and spin states.



(2) Calculate the square of the separation of the two fermions,

<(x1-x2)^2>=<(x1^2+x2^2-2x1x2)>

for the lowest energy state of the two-fermion system.



[(3) Repeat the calculations for the first excited states.]




~~So I'm unsure how to add the spatial components to that or if it's a good start at all. I've only done this for parrallel spin stuff
 

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Quickie hint: what is the difference between a system of bosons and one of fermions regarding particles occuping the same ground state energy level? You've stated you know how to do it for parallel spins... so how will it change slightly for this system?
 

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