Hamiltonian for identical particles

When identical particles are concerned it is intuitive that no physical property can be modified when the roles of the particles are permuted:any physical observable must be invariant under all permutations, that is: if is a physical observable then . This also means that the spaces and are invariant under the action of a physical observable : if . The hamiltonian of a system of identical particles is a physical observable (it corresponds to the energy) and must commute with all . This restricts the possible forms of the interacting potential.

Let us investigate deeply this fact. Consider a system of 2 non-interacting identical particles. The hamiltonian of this system is:

with because of the indistinguishability of the particles. Now suppose that there is an interaction between the particles. In order to preserve the invariance of under permutations of the particles () , that is in order to keep the hamiltonian a physical observable, the interaction potential has to be of the form:
.

Energy[edit | edit source]

For the sake of simplicity we are going to consider a system of N non-interacting identical particles. Then

Suppose we are dealing with particles in the state , particles in the state and so on.The total energy of each state is:
and the total energy of the system is:
For bosons there are no restrictions on the numbers . It follows that the ground state is the one with all the N bosons on the lowest energy level (we suppose it's ), with energy
The situation for fermions is different, since we have learned that can only be 0 or 1. Thus if we have N fermions the ground state is the one with one fermion per quantum state. N quantum states will be "full" ( for ), the others will be empty ( for ).The situation is the following:
The highest individual energy , corresponding to the last full quantum state in the ground state is called Fermi energy

Example (Two identical non-interacting particles in an infinite potential well)

First consider the particles to be spinless. Since we're dealing with bosons. It's known from quantum mechanics that the wave function of a particle in such a system is given by

where defines the width of the well. Energy levels are given by
These expressions define the individual states and the individual energies. Suppose we have a particle on the state and a particle on the state . Taking into account the nature of the particles we can immediately write down the wave function of the system:
Now suppose that the particles have spin .

  • Individual states must take into account the spin degrees of freedom
  • we're now dealing with fermions

We know that a particle is on the level with its spin up and the other on the level with its spin down . Individual states are then given by:

Remembering that a permutation operator acts not only on the orbital variables but also on spin ones, antisymmetrization gives us the following wave function:

 
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