(p.345) Appendix E Second Quantization of Fermions
(p.345) Appendix E Second Quantization of Fermions
With few exceptions, the systems considered in examples and problems of this book are assumed to be bosonic in character. That is, they are assumed not to have occupation probabilities dominated by spin-related considerations. Their spectra generally have large numbers of energy levels and their statistics are governed by the Bose–Einstein distribution, unlike fermionic systems that have a very small number of energy levels and obey Fermi–Dirac statistics (to describe the latter we shall follow the discussion in Ref. [E.1]). This of course ignores the important role played by the Pauli exclusion principle in many systems of practical importance, such as semiconductors, as well as the possibility of converting bosonic systems into fermionic systems and vice versa by altering experimental conditions. At the end of Chapter 3, an example of the conversion of a multi-level atom into a two-level atom was considered, and other examples can be found in the literature on Bose–Einstein condensates near Feshbach resonances [E.2]. Here we briefly consider some of the implications of the exclusion principle that were omitted from earlier discussions.
Fermions obey population statistics that are different from those of bosons, and this difference is reflected in the commutation relations for fermionic operators. Fermionic operators anti-commute. For example, the annihilation and creation operators of a fermionic state labeled with index k, namely and , obey the commutation rules:
where the symbol implies the sum . Other than electrons, what systems are fermionic, and do they exhibit familiar features? On the basis of Eq. (E.1) and our earlier representation of two-level atoms by anti-commuting Pauli matrices, we may infer that two-level atoms are fermionic as one example. On the other hand, electromagnetic fields and simple harmonic oscillators, both of which are characterized by an infinite number of equally spaced energy levels, are bosonic. In this appendix we discuss differences in the way wavefunctions are constructed for fermionic and bosonic systems, while emphasizing that their system energies share a simple feature in the limit of weak interactions.
Define a fermionic number operator and note that according to Eq. (E.1)
If has eigenvalues nk, then obviously
For a given state k, the operator can be represented by the matrix
(p.346) when there is no energy level degeneracy. Suitable matrices for which are consistent with the commutation relations are
The two possible states of the system are
and it is easily verified that and play the roles of destruction and creation operators respectively. That is,
To describe a complete system of fermions we must introduce the exclusion principle by permitting the sign of the wavefunction to be altered by the action of . Order the states of the system in an arbitrary but definite way, for example, . Then permit to act as before, except that the sign will be determined by whether the state is preceded in the assumed order by an even or odd number of occupied states.
is the number of states, preceding state k, which are occupied. For a particular single particle state k of the system, it is easy to show that
(p.347) From these relations it is easily seen that the commutation relations are satisfied for . For example,
Hence . Remaining commutation relations are satisfied in the same way.
Notice from this that the result of applying fermi operators depends not only on occupation number k but also on the occupation numbers of all preceding states. Thus and are not entirely independent. This means that the order of processes in fermionic systems is even more important to the outcome of interactions than it is for bosonic systems. It should therefore not be taken for granted in a fermionic system that the application of a sum of single particle operators to the system wavefunction yields a sum of single particle energies. Let us test whether this is true in the perturbative limit of a fermionic system that has extremely weak interactions.
To the extent that particles can be viewed as being distinct, the total system Hamiltonian in the configuration representation can be written
where gives the spatial and spin coordinates of each of the N fermions. Here is the single particle Hamiltonian for the particle at coordinate . Single particle states and their eigenvalues are determined by
To see what the form of the total system Hamiltonian is in the occupation number representation, we introduce field operators defined by
where . The single particle states are assumed to form a complete set. Hence the total Hamiltonian operator may be changed into the occupation representation using the transformation (compare Eq. (2.4.34) for discrete states)
(p.348) where the integration over coordinates implicitly includes a sum over spin variables . At a fixed time t it may readily be shown that
Substituting these expressions into Eq. (E.15), one finds
This result shows that the energy of a system of non-interacting fermions is just the sum of the individual particle energies, a familiar result that holds for purely bosonic systems also. In interacting systems of fermions, a similar decomposition of the system into a simple sum of (renormalized) individual particle energies can be performed only for extremely weak interactions that can be taken into account via an effective field. For this to be true, the fermions have to be widely spaced. For close particles and strong interactions, the concept of single particle states and single particle energies breaks down completely. Equations (E.12) and (E.19) are no longer valid in this limit. In order to ensure that no two fermions have the same wavefunction (occupy the same state) in the strong-coupling limit, many-body techniques are required for analysis.