Appendix A Second Quantization The most remarkable fact about a system of fermions is that no more than one fermion can occupy a quantum particle state (Pauli’s exclusion principle). For bosons no such restriction applies. That is, any number of bosons can occupy the same state. We shall discuss the second quantization formalism in which creation and annihilation operators associated with each quantum state are used. This formalism is extremely useful in treating of many-boson and/or many-fermion system. A.1 Boson Creation and Annihilation Oper- ators The quantum state for a system of bosons (or fermions) can most conve- niently be represented by a set of occupation numbers where are the numbers of bosons (or fermions) occupying the quantum particle-states This representation is called the occupation number representation or simply the number representation. For bosons, the possible values for are zero, one, two, or any positive integers: The many-boson state can best be represented by the distribution of particles (balls) in the states (boxes) as shown in Fig. A.1. Let us introduce operators (without prime) whose eigenvalues are given by 0, 1, 2, ... Since Eq. (A.1) is meant for each and every state indepen- 333 334 APPENDIX A. SECOND QUANTIZATION dently, we assume that It is convenient to introduce complex dynamic variables and instead of directly dealing with the number operators We attach labels to the dynamic variables and associated with the states and assume that and satisfy the following Bose commutation rules: Let us set which is Hermitean. Clearly, satisfy Eqs. (A.2). We shall show that has as eigenvalues all non-negative integers. Let be an eigenvalue of (dropping the suffix a) and an eigenket belonging to it. By definition Now is the squared length of the ket and hence A.1. BOSON CREATION AND ANNIHILATION OPERATORS 335 Also by definition hence from Eqs. (A.5) and (A.6), we obtain the case of equality occurring only if Consider now We may use the following identities: and obtain Hence Now if then is, according to Eq. (A.11), an eigenket of belonging to the eigenvalue Hence for non-zero is another eigenvalue. We can repeat the argument and deduce that, if is another eigenvalue of Continuing in this way, we obtain a series of eigenvalues which can terminate only with the value 0 because of inequality (A.7). By a similar process, we can show from the Hermitean conjugate of Eq. (A.10): that the eigenvalue of has no upper limit [Problem A.1.1]. Hence, the eigenvalues of are non- negative integers: 0,1,2,... (q.e.d) Let be a normalized eigenket of belonging to the eigenvalue 0 so that By multiplying all these kets together, we construct a normalized eigen- ket: which is a simultaneous eigenket of all belonging to the eigenvalues zero. This ket is called the vacuum ket. It has the following property: 336 APPENDIX A. SECOND QUANTIZATION Using the commutation rules (A.3) we obtain a relation (dropping suffix a) which may be proved by induction (Problem A.1.2). Multiply Eq. (A.15) by from the left and operate the result to Using Eq. (A.14) we obtain indicating that is an eigenket belonging to the eigenvalue The square length of is We see from Eq. (A.11) that is an eigenket of belonging to the eigenvalue Similarly, we can show from that is an eigenket of belonging to the eigenvalue Thus operator acting on the number eigenket, annihilates a particle while operator creates a particle. Therefore, and are called annihilation and creation operators, respectively. From Eqs. (A.16) and (A.17) we infer that if are any non-negative integers, is a normalized simultaneous eigenket of all the belonging to the eigenvalues Various kets obtained by taking different form a complete set of kets all orthogonal to each other. Following Dirac [1], we postulate that the quantum states for N bosons can be represented by a symmetric ket where S is the symmetrizing operator: and P are permutation operators for the particle-indices (1,2,..., N). The ket in Eq. (A.19) is not normalized but A.2. OBSERVABLES 337 is a normalized ket representing the same state. Comparing Eqs. (A.21) and (A.18), we obtain That is, unnormalized symmetric kets for the system can be constructed by applying N creation operators to the vacuum ket So far we have tacitly assumed that the total number of bosons is fixed at N' . If this number is not fixed but is variable, we can easily extend the theory to this case. Let us introduce a Hermitean operator N defined by the summation extending over the whole set of boson states. Clearly, the operator N has eigenvalues 0,1,2,..., and the ket is an eigenket of N belonging to the eigenvalue N'. We may arrange kets in the order of N', i.e., zero-particle state, one-particle states, two-particle states, ... : These kets are all orthogonal to each other, two kets referring to the same number of bosons are orthogonal as before, and two referring to different numbers of bosons are orthogonal because they have different eigenvalues N'. By normalizing the kets, we get a set of kets like (A.21) with no restriction on These kets form the basic kets in a representation where are diagonal. Problem A.1.1. (a) Show (twice) that by taking the Hermitian-conjugation of Eq. (A.10) and also by using Eqs. (A.9). (b) Use this relation and obtain a series of eigenvalues where is an eigenvalue of Problem A.1.2. Prove Eq. (A.15) by mathematical induction. Hint: use Eqs. (A.9). A. 2 Observables We wish to express observable physical quantities (observables) for the system of identical bosons in terms of and These observables are by postulate symmetric functions of the boson variables. 338 APPENDIX A. SECOND QUANTIZATION An observable may be written in the form: where is a function of the dynamic variables of the boson, that of the dynamic variables of the and bosons, and so on. We take Since acts only on the ket of the boson, we have The matrix element does not depend on the particle index Summing Eq. (A.26) over all and applying operator S to the result, we obtain Since Y is symmetric, we can replace SY by YS for the lhs. After straight- forward calculations, we obtain, from Eq. (A.27), Using the commutation rules and the property (A.14) we can show that A.3. FERMIONS CREATION AND ANNIHILATION OPERATORS 339 (Problem A.2.1). Using this relation, we obtain from Eq. (A.28) Since the kets form a complete set, we obtain In a similar manner Z in Eq. (A.25) can be expressed by [Problem A.2.2] Problem A.2.1. Prove Eq. (A.29). Hint: Start with cases of one- and two- particle-state kets. Problem A.2.2. Prove Eq. (A.32) by following those steps similar to (A.27)- (A.31). A.3 Fermions Creation and Annihilation Op- erators In this section we treat a system of identical fermions in a parallel manner. The quantum states for fermions, by postulate, are represented by anti- symmetric kets: where is the antisymmetrizing operator, with being +1 or – 1 according to whether P is even or odd. Each antisymmetric ket in Eq. (A.34) is charac- terized such that it changes its sign if an odd permutation of particle indices 340 APPENDIX A. SECOND QUANTIZATION is applied to it, and the fermion states are all different. Just as for a boson system, we can introduce observables each with eigenvalues 0 or 1, representing the number of fermions in the states respectively. The many-fermion occupation-number can be represented as in Fig. A.2. We can also introduce a set of linear operators one pair for each state satisfying the Fermi anticommutation rules: The number of fermions in the state is again represented by Using Eqs. (A.36), we obtain If an eigenket of belonging to the eigenvalue is denoted by Eq. (A.38) yields Since we obtain meaning that the eigenvalues are either 0 or 1 as required: A.4. HEISENBERG EQUATION OF MOTION 341 Similarly to the case of bosons, we can show that which is normalized to unity. Observables describing the system of fermions can be expressed in terms of operators and and the results have the same form Eqs. (A.31) and (A.32) as for the case of bosons. In summary both states and observables for a system of identical particles can be expressed in terms of creation and annihilation operators. This formal- ism, called the second quantization formalism, has some notable advantages over the usual Schrödinger formalism. First, the permutation-symmetry property of the quantum particles is represented simply in the form of Bose commutation (or Fermi anticommutation) rules. Second, observables in sec- ond quantization are defined for an arbitrary number of particles so that the formalism may apply to systems in which the number of particles is not fixed, but variable. Third, and most importantly, all relevant quantities (states and observables) can be defined referring only to the single-particle states. This property allows one to describe the motion of the many-body system in the 3D space. This is a natural description since all particles in nature move in 3D. In fact, relativistic quantum field theory can be developed only in second quantization. A.4 Heisenberg Equation of Motion In the Schrödinger Picture (SP), the energy eigenvalue equation is where H is the Hamiltonian and E the eigenvalue.