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Bosons/fermions trapped in a 1 dimensional trap

  1. Mar 28, 2015 #1
    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)##

    Any advice/guidance?
    Thanks
     
  2. jcsd
  3. Mar 28, 2015 #2

    mfb

    User Avatar
    2016 Award

    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?
     
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