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:
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:
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" (
), the others will be empty (
).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
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: