2.3 Identical particles

2.3.1 Symmetries

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.

Figure 2.1: Indistinguishable particles in quantum mechanics: (left) initially there are two particles at A and B, later on two particles are found at C and D; (middle) but we cannot be certain whether the particles travelled from A to D and B to C or (right) from A to C and B to D, because they are identical.

Consider a system of two identical particles represented by the wave-function $\Psi({\bf r}_1,{\bf r}_2)$ and a particle-exchange operator ${\hat P}_{12}$ which swaps the particles i.e.

$${\hat P}_{12} \Psi({\bf r}_1,{\bf r}_2) = \Psi({\bf r}_2,{\bf r}_1) .$$ (2.24)

However, since the system must be unchanged by such an exchange of identical particles, the two states appearing in equation 2.25 must be the same and hence differ only by a multiplicative complex constant;

$$\Psi({\bf r}_2,{\bf r}_1) = c \Psi({\bf r}_1,{\bf r}_2) ,$$ (2.25)

so that many-body wave-functions of identical particles must be eigenstates of the particle interchange operator. Performing the exchange twice clearly returns the system precisely to its original state and so leads to

$${\hat P}_{12}^2 \Psi({\bf r}_1,{\bf r}_2) = c^2 \Psi({\bf r}_1,{\bf r}_2) = \Psi({\bf r}_1,{\bf r}_2)$$ (2.26)

i.e. $c^2 = 1$ so $c = \pm 1$, and the many-body wave-function at most changes sign under particle exchange. This result is readily extended to systems of more than two identical particles, so that the wave-functions are either symmetric or antisymmetric under exchange of any two identical particles.

2.3.2 Spin and statistics

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 express^2.8. A result of the relativistic theory is that particles may possess intrinsic angular momentum known as spin, which is quantised in units of ${1 \over 2}$. 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 [14]). For a system of free non-interacting particles, the single-particle states are characterised by linear momentum ${\bf p}$ and spin $\sigma$. We denote the occupation numbers of these states $N_{{\bf p}\sigma}$, 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 $\vert 1_{{\bf p}\sigma} , 1_{{\bf p'}\sigma'} \rangle$ denotes a state in which a particle was put in state $({\bf p'}\sigma')$ and then a second particle was added in state $({\bf p}\sigma)$.

Annihilation and creation operators^2.9 ${\hat a}_{{\bf p}\sigma}$, ${\hat a}_{{\bf p}\sigma} ^{\dag }$ are introduced for each state which act in the following manner:

$${\hat a}_{{\bf p}\sigma} \vert 1_{{\bf p}\sigma} \rangle = \vert 0 \rangle ,$$ (2.27)

$${\hat a}_{{\bf p}\sigma}^{\dag } \vert 0 \rangle = \vert 1_{{\bf p}\sigma} \rangle .$$ (2.28)

In order that the sign of the state is unambiguously defined in this notation, it is necessary for consistency that creation operators act on the right-most void and annihilation operators act on the left-most appropriately-filled slot.

We now consider the state $\vert 1_{{\bf p}\sigma} , 1_{{\bf p'}\sigma'} \rangle$ and use the exchange operator ${\hat a}_{{\bf p'}\sigma'}^{\dag } {\hat a}_{{\bf p}\sigma}^{\dag } {\hat a}_{{\bf p}\sigma} {\hat a}_{{\bf p'}\sigma'}$ to obtain the state $\vert 1_{{\bf p'}\sigma'} , 1 _{{\bf p}\sigma} \rangle$, in which particles have been exchanged between state $({\bf p}\sigma)$ and $({\bf p'}\sigma')$:

$${\hat a}_{{\bf p'}\sigma'}^{\dag }{\hat a}_{{\bf p}\sigma}^{\dag ... ...\hat a}_{{\bf p'}\sigma'}\vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle = {\hat a}_{{\bf p'}\sigma'}^{\dag } {\hat a}_{{\bf p}\sigma}^{\dag... ..._{{\bf p'}\sigma'}^{\dag } {\hat a}_{{\bf p}\sigma}^{\dag } \vert 0 , 0 \rangle$$

$$= {\hat a}_{{\bf p'}\sigma'}^{\dag } \vert 0 , 1_{{\bf p}\sigma} \rangle = \vert 1 _{{\bf p'}\sigma'} , 1_{{\bf p}\sigma} \rangle .$$ (2.29)

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:

$$\left[ {\hat a}_{{\bf p}\sigma}^{\dag } , {\hat a}_{{\bf p'}\sigma'} \right] = \delta_{\bf p p'} \delta_{\sigma \sigma'} ,$$ (2.30)

$$\left[ {\hat a}_{{\bf p}\sigma} , {\hat a}_{{\bf p'}\sigma'} \right] = 0 ,$$ (2.31)

$${\mathrm{where~}} \left[ {\hat p} , {\hat q} \right] = {\hat p}{\hat q} - {\hat q}{\hat p} .$$ (2.32)

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

$${\hat a}_{{\bf p'}\sigma'}^{\dag }{\hat a}_{{\bf p}\sigma}^{\dag ... ...\hat a}_{{\bf p'}\sigma'}\vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle = \vert 1 _{{\bf p'}\sigma'} , 1_{{\bf p}\sigma} \rangle$$

$$= {\hat a}_{{\bf p}\sigma}^{\dag }{\hat a}_{{\bf p'}\sigma'}^{\da... ...\hat a}_{{\bf p'}\sigma'}\vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle = \vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle$$ (2.33)

i.e. states describing particles whose creation and annihilation operators obey the Bose commutation rules ( bosons) must have symmetric wave-functions.

The second set of commutation rules, which is due to Fermi, describes fermions:

$$\left\{ {\hat a}_{{\bf p}\sigma}^{\dag } , {\hat a}_{{\bf p'}\sigma'} \right\} = \delta_{\bf p p'} \delta_{\sigma \sigma'} ,$$ (2.34)

$$\left\{ {\hat a}_{{\bf p}\sigma} , {\hat a}_{{\bf p'}\sigma'} \right\} = 0 ,$$ (2.35)

$${\mathrm{where~}} \left\{ {\hat p} , {\hat q} \right\} = {\hat p}{\hat q} + {\hat q}{\hat p} ,$$ (2.36)

and gives rise to antisymmetric wave-functions:

$${\hat a}_{{\bf p'}\sigma'}^{\dag }{\hat a}_{{\bf p}\sigma}^{\dag ... ...\hat a}_{{\bf p'}\sigma'}\vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle = \vert 1 _{{\bf p'}\sigma'} , 1_{{\bf p}\sigma} \rangle$$

$$= -{\hat a}_{{\bf p}\sigma}^{\dag }{\hat a}_{{\bf p'}\sigma'}^{\d... ...\hat a}_{{\bf p'}\sigma'}\vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle = -\vert 1_{{\bf p}\sigma} , 1 _{{\bf p'}\sigma'} \rangle .$$ (2.37)

In particular, note that the Fermi rules (equation 2.36 for $({\bf p}\sigma) = ({\bf p'}\sigma')$) require that

$${\hat a}_{{\bf p}\sigma} {\hat a}_{{\bf p}\sigma} = {\hat a}_{{\bf p}\sigma}^{\dag } {\hat a}_{{\bf p}\sigma}^{\dag } = 0$$ (2.38)

i.e. it is impossible to put more than one fermion in any single-particle state. This result is known as the Pauli exclusion principle and it is ultimately responsible for the stability of matter. In an atom, for instance, the Pauli exclusion principle prevents all of the electrons from falling into the lowest-lying energy level.

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 ${\hat b}_{{\bf p}\sigma}^{\dag }$ and ${\hat b}_{{\bf p}\sigma}$ 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, ${\hat N}_{{\bf p}\sigma} = {\hat a}_{{\bf p}\sigma}^{\dag } {\hat a}_{{\bf p}\sigma}$, so-called because its action is simply to return the number of particles in state $({{\bf p}\sigma})$. For antiparticles, ${\hat{\bar N}}_{{\bf p}\sigma} = {\hat b}_{{\bf p}\sigma}^{\dag } {\hat b}_{{\bf p}\sigma}$.

Using the method of second quantisation, the Hamiltonian can be written as (see [15]):

$${\hat H} = \sum_{\bf p} \sum_{\sigma} \varepsilon({\bf p}) \... ...at b}_{{\bf p}\sigma} {\hat b}_{{\bf p}\sigma}^{\dag } \right)$$ (2.39)

where the $\varepsilon({\bf p}) = \sqrt{{\bf p}^2 + m^2}$ are the energies of the single-particle states, and the plus sign occurs for particles of integral spin and the minus sign for particles with half-integral spin. We note that the particle creation and annihilation operators occur in the correct order to be rewritten as the particle number operator, whereas the antiparticle operators are in the wrong order, so we can use the appropriate set of commutation rules to reverse this order. The Hamiltonian for free particles must be positive-definite, and therefore turns out to be of the form

$${\hat H} = \sum_{\bf p} \sum_{\sigma} \varepsilon({\bf p}) \... ...{\bf p}\sigma} + {\hat {\bar N}}_{{\bf p}\sigma} + 1 \right) .$$ (2.40)

The constant $\sum_{\bf p} \sum_{\sigma} \varepsilon({\bf p})$ in equation 2.41 represents the energy of the vacuum and is usually ignored. In order to obtain the Hamiltonian in this form, particles with half-integral spin (minus sign in 2.40) must have creation and annihilation operators which anticommute according to the Fermi rules, whereas particles with integral spin (plus sign in 2.40) must have operators which commute according to the Bose rules.

We thus come to the following conclusions:

• particles with half-integral spin are fermions and have antisymmetric wave-functions,
• particles with integral spin are bosons and have symmetric wave-functions.

In particular, electrons (which have spin ${1 \over 2}$) are fermions with antisymmetric wave-functions and obey the Pauli exclusion principle. These consequences of relativistic quantum mechanics must be carried over by hand into the non-relativistic theory if we are to correctly describe nature.

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'').