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# Fermions in harmonic potential.

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Two identical, non-interacting spin-1/2 fermions are placed in the 1-D harmonic potential

V(x) = (1/2)m ω2x2,

Where m is the mass of the fermion and ω is its angular frequency.

a. 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 wave functions and the singlet and triplet spin states.
b. Calculate the square of the separation of the two fermions,
<(x1-x2)2>=<(x12+x22-2x1x2)>
for the lowest energy state of the two-fermion system.
c. Repeat the calculations for the first excited states.

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#### Solution Preview

Please see the attached file.
You might want to take a look at:
http://electron6.phys.utk.edu/phys594/archives/quantum/Identical/identical3.htm

Part A.

Ground state:

We have two particles at the ground state, each with energy of so the total energy is

First excited states:

...

#### Solution Summary

The energies of the found and first excited states of this two-fermion systems are determined. The square of the separation of the two fermions is determined.

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