It is a consequence of quantum mechanics, usually expressed in the terms of the Heisenberg uncertainty principle that, in contrast to Newtonian mechanics, the trajectory of a particle is undefined. When dealing with identical particles this leads to complications, as illustrated in figure 2.1.
Consider a system of two identical particles represented by the wave-function
and a particle-exchange operator
which swaps the particles i.e.
It is necessary to go to the full relativistic theory of quantum mechanics in order to ascertain which sign is appropriate to a system of particles, although the result itself is simple enough to express2.8. A result of the relativistic theory is that particles may possess intrinsic angular momentum known as spin, which is quantised in units of . A brief outline of the relationship between spin and statistics follows.
We use the method of second quantisation of fields of particles with spin (see ). For a system of free non-interacting particles, the single-particle states are characterised by linear momentum and spin . We denote the occupation numbers of these states , but for now we will only consider situations in which every single-particle state is either empty or singly occupied, and in addition will consider a system of at most two particles and focus on just two single-particle states. The state-vector is represented by a series of ``slots'', whose order is important at this stage, each containing an occupation number. Thus denotes a state in which a particle was put in state and then a second particle was added in state .
Annihilation and creation
are introduced for each state which act in the following manner:
We now consider the state
and use the exchange operator
to obtain the state
, in which particles have
been exchanged between state
The initial discussion in this section showed that under this exchange, the
wave-function at most changes sign. This requires that the creation and
annihilation operators obey one of two sets of commutation rules, as we will
now show. The first set is due to Bose and can be summarised by:
Under the Bose commutation rules, the two creation operators in the
exchange operator, which refer to
different states, commute and so can be swapped, and the result is that
The second set of commutation rules, which is due to Fermi, describes fermions:
In particular, note that the Fermi rules (equation 2.36 for
) require that
Thus all systems of identical particles must subscribe to one of the sets of rules above: bosons have symmetric wave-functions and fermions antisymmetric wave-functions.
The second result of the relativistic theory which needs to be considered is the existence of antiparticles, which have the same mass but opposite charge to their corresponding particles. The antiparticles are assigned their own set of creation and annihilation operators, denoted and respectively, which obey the same commutation rules as the particle operators.
The creation and annihilation operators can be combined to form a Hermitian product, the number operator, , so-called because its action is simply to return the number of particles in state . For antiparticles, .
Using the method of second quantisation, the Hamiltonian can be written as
We thus come to the following conclusions:
In this dissertation we will not address the issues which arise in spin-polarised systems, in which the numbers of electrons in different spin states differ. In our case, it is only necessary to ensure that the many-body electronic wave-function is antisymmetric under exchange and that each single-particle state is never more than doubly-occupied (with one spin ``up'' electron and one spin ``down'').