Bosons/fermions trapped in a 1 dimensional trap

[zinDo]

Homework Statement

A system of bosons or fermions trapped in a potential.
I'm being asked to write the Hamiltonian in 1st and 2nd quantization and "describe the ground state" of the system.

My main question is: what does it mean to describe the ground state? What should I look for? Should I look for the lowest energy?

Homework Equations

1 dimensional MOT-trap: $V(x)=mw^2x^2/2$

The Attempt at a Solution

My attempt for writing the hamiltonian:
1st quantization $H=\sum_{i=1}^{N}\frac{p_i^2}{2m}+\frac{1}{2}mw^2x_i^2$
2nd quantization: $H=\int dx \psi^{\dagger}(x)\left[-\frac{\hbar^2}{2m}\nabla^2+\frac{1}{2}mw^2x_i^2\right]\psi(x)$
Where $\psi(x)$ isthe field operator.

Do you think this is enough? Are the hamiltonians equal for both bosons or fermions? Am I missing all the important stuff?

As for describing the ground state I don't know, my attempt has been writing down the many-body wavefunction (I don't know why, just to put something) for each case and explain that bosons can all be at the ground state and fermions have the exclusion principle.
$\Psi^{(S)}=N_S\sum_p \phi_1(x_1) \phi_2(x_2)... \phi_N(x_N)$
$\Psi^{(A)}=N_A\sum_p sgn(p)\phi_1(x_1) \phi_2(x_2)... \phi_N(x_N)$

Any advice/guidance?
Thanks

 

[mfb]

My main question is: what does it mean to describe the ground state? What should I look for? Should I look for the lowest energy?

Yes, that is the ground state by definition.

and explain that bosons can all be at the ground state and fermions have the exclusion principle.

That's the important difference between the two. Which states will be filled in the different cases?