# Bosons/fermions trapped in a 1 dimensional trap

1. Mar 28, 2015

### zinDo

1. The problem statement, all variables and given/known data
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?

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

3. 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.
$\Psi^{(S)}=N_S\sum_p \phi_1(x_1) \phi_2(x_2)... \phi_N(x_N)$
$\Psi^{(A)}=N_A\sum_p sgn(p)\phi_1(x_1) \phi_2(x_2)... \phi_N(x_N)$

Thanks

2. Mar 28, 2015

### Staff: Mentor

Yes, that is the ground state by definition.
That's the important difference between the two. Which states will be filled in the different cases?