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Appendix A Second Quantization An exact description of an interacting many-body system requires the solving of the corresponding many-body Schrodinger¨ equations. The formalism of second quantization leads to a substantial simplification in the description of such a many- body system, but it should be noted that it is only a reformulation of the original Schrodinger¨ equation but not yet a solution. The essential step in the second quanti- zation is the introduction of so-called creation and annihilation operators. By doing this, we eliminate the need for the laborious construction, respectively, of the sym- metrized or the anti-symmetrized N-particle wavefunctions from the single-particle wavefunctions. The entire statistics is contained in fundamental commutation rela- tions of these operators. Forces and interactions are expressed in terms of these “creation” and “annihilation” operators. How does one handle an N-particle system? In case the particles are distinguish- able, that is, if they are enumerable, then the method of description follows directly the general postulates of quantum mechanics: H(i) {|φ(i) } 1 : Hilbert space of the ith particle with the orthonormal basis α : (i) (i) φα |φβ =δαβ (A.1) HN : Hilbert space of the N-particle system H = H(1) ⊗ H(2) ⊗···⊗H(N) N 1 1 1 (A.2) with the basis {|φN }: |φ =|φ(1)φ(2) ···φ(N) =|φ(1) |φ(2) ···|φ(N) (A.3) N α1 α2 αN α1 α2 αN An arbitrary N-particle state |ψN , |ψ = c(α ···α )|φ(1)φ(2) ···φ(N) (A.4) N 1 N α1 α2 αN α1···αN underlies the same statistical interpretation as in the case of a 1-particle system. The dynamics of the N-particle system results from the formally unchanged Schrodinger¨ equation: W. Nolting, A. Ramakanth, Quantum Theory of Magnetism, 491 DOI 10.1007/978-3-540-85416-6 BM2, C Springer-Verlag Berlin Heidelberg 2009 492 A Second Quantization ∂|ψ i N = H |ψ (A.5) ∂t N The handling of the many-body problem in quantum mechanics, in the case of distinguishable particles, confronts exactly the same difficulties as in the classical physics, simply because of the large number of degrees of freedom. There are no extra, typically quantum mechanical complications. A.1 Identical Particles What are “identical particles”? To avoid misunderstandings let us strictly separate “particle properties” from “measured quantities of particle observables”. “Parti- cle properties” as e.g. mass, spin, charge, magnetic moment, etc. are in principle unchangeable intrinsic characteristics of the particle. The “measured values of par- ticle observables” as e.g. position, momentum, angular momentum, spin projection, etc., on the other hand, can always change with time. We define “identical particles” in the quantum mechanical sense as particles which agree in all their particle prop- erties. Identical particles are therefore particles, which behave exactly in the same manner under the same physical conditions, i.e. no measurement can differentiate one from the other. Identical particles also exist in classical mechanics. However, if the initial conditions are known, the state of the system for all times is determined by the Hamilton’s equations of motion. Thus the particles are always identifiable! In quantum mechanics, on the contrary, there is the principle of indistinguishability, which says that, identical particles are intrinsically indistinguishable. In quantum mechanics, in some sense, identical particles lose their individuality. This originates from the fact that there do not exist sharp particle trajectories (only “spreading” wave packets!). The regions, where the probability of finding the particle is unequal to zero, overlap for different particles. Any question whose answer requires the observation of one single particle is physically meaningless. We face now the problem, that, essentially for calculational purposes, we can- not avoid, to number the particles. Then, however, the numbering must be so that all physically relevant statements, that are made, are absolutely invariant under a change of this particle labelling. How to manage this is the subject of the following considerations. We introduce the permutation operator P, which, in the N-particle state, inter- changes the particle indices: P|φ(1)φ(2) ···φ(N) =|φ(i1)φ(i2) ···φ(iN ) (A.6) α1 α2 αN α1 α2 αN Every permutation operator can be written as a product of transposition operators Pij. On application of Pij to a state of identical particles, results, according to the principle of indistinguishability, in a state, which, at the most, is different from the initial state by an unimportant phase factor λ = exp(iη): A.1 Identical Particles 493 P |···φ(i) ···φ( j) ··· = |···φ( j) ···φ(i) ··· ij αi α j αi α j = λ |···φ(i) ···φ( j) ··· (A.7) αi α j 2 = λ =± Since Pij 1, it is necessary that 1. Therefore, a system of identical particles must be either symmetric or antisymmetric against interchange of particles! This defines two different Hilbert spaces: H(+) |ψ (+) N : the space of symmetric states N (+) (+) Pij |ψN =|ψN (A.8) H(−) |ψ (−) N : the space of antisymmetric states N (−) (−) Pij |ψN =−|ψN (A.9) In these spaces, Pij are Hermitian and unitary! What are the properties the observables must have for a system of identical par- ticles? They must necessarily depend on the coordinates of all the particles A = A(1, 2, ··· , N) (A.10) and must commute with all the transpositions (permutations) , = Pij A − 0 (A.11) This is valid specially for the Hamiltonian H and therefore also for the time evolu- tion operator i U t, t = exp − H t − t H = H t ( 0) ( 0) ;( ( )) (A.12) , = Pij U − 0 (A.13) That means the symmetry character of an N-particle state remains unchanged for all times! Which Hilbert space out of H(+) and H(−) is applicable for which type of particles is established in relativistic quantum field theory. We will take over the spin-statistics theorem from there, without any proof. H(+) N : Space of symmetric states of N identical particles with integral spin. These particles are called Bosons. H(−) N : Space of antisymmetric states of N identical particles with half-integral spin. These particles are called Fermions. 494 A Second Quantization A.2 Continuous Fock Representation A.2.1 Symmetrized Many-Particle States H(ε) Let N be the Hilbert space of a system of N identical particles. Here + : Bosons ε = (A.14) − : Fermions Let φ be a 1-particle observable (or a set of 1-particle observables) with a continuous spectrum, φα being a particular eigenvalue corresponding to the 1-particle eigenstate |φα : φ |φα =φα |φα (A.15) φα|φβ =δ(φα − φβ )(≡ δ(α − β)) (A.16) dφα|φα φα|=1l i n H1 (A.17) H(ε) A basis of N is constructed from the following (anti-) symmetrized N-particle states: ) * (ε) 1 p (1) (2) (N) |φα ···φα = ε P |φ |φ ···|φ (A.18) 1 N α1 α2 αN N! P p is the number of transpositions in P. The summation runs over all the possible permutations P. The sequence of the 1-particle states in the ket on the left-hand side of (A.18) is called the standard ordering. It is arbitrary, but has to be fixed right at the beginning. Interchange of two 1-particle symbols on the left means only a constant factor ε on the right. One can easily prove the following relations for the above introduced N-particle states: Scalar product: (ε) φ ···|φ ··· (ε) β1 α1 ) * 1 pα = ε Pα δ(φβ − φα ) ···δ(φβ − φα ) (A.19) N! 1 1 N N Pα The index α shall indicate that Pα permutes only the φα’s. Completeness relation: ··· φ ··· φ |φ ··· (ε)(ε) φ ···|= d β1 d βN β1 β1 1l (A.20) A.2 Continuous Fock Representation 495 This leads to a formal representation of an observable of the N-particle system, which will be used later a few times: = ··· φ ··· φ φ ··· φ ∗ A d α1 d αN d β1 d βN ∗|φ ··· (ε)(ε) φ ···| |φ ··· (ε)(ε) φ ···| α1 α1 A β1 β1 (A.21) A.2.2 Construction Operators H(ε) | We want to build up the basis states of N , step by step, from the vacuum state 0 ( 0|0 =1) with the help of the operator † ≡ † cφα cα These operators are defined by their action on the states: √ † (ε) (ε) c |0 = 1|φα ∈ H (A.22) α1 1 1 √ † (ε) (ε) (ε) c |φα = 2|φα φα ∈ H (A.23) α2 1 2 1 2 Or, in general √ † (ε) (ε) c |φα ···φα = N + 1 |φβ φα ···φα (A.24) β 1 N 1 N ∈H(ε) ∈H(ε) N N+1 † cβ is called the creation operator.Itcreates an extra particle in the N-particle state. The relation (A.24) is obviously reversible: (ε) 1 † † † |φα φα φα ···φα = √ cα cα ···cα |0 (A.25) 1 2 3 N N! 1 2 N |φ ··· (ε) | The N-particle state αi can be built up from the vacuum state 0 by apply- ing a sequence of N creation operators, where the order of the operators is to be strictly obeyed. For a product of two creation operators, it follows from (A.24) † † (ε) (ε) c c |φα ···φα = N(N − 1)|φα φα φα ···φα (A.26) α1 α2 3 N 1 2 3 N If the sequence of the operators is reversed, then we have † † (ε) c c |φα ···φα α2 α1 3 N = − |φ φ φ ···φ (ε) N(N 1) α2 α1 α3 αN = ε − |φ φ φ ···φ (ε) N(N 1) α1 α2 α3 αN (A.27) 496 A Second Quantization The last step follows because of (A.18).
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  • Second Quantization∗

    Second Quantization∗

    Second Quantization∗ Jörg Schmalian May 19, 2016 1 The harmonic oscillator: raising and lowering operators Lets first reanalyze the harmonic oscillator with potential m!2 V (x) = x2 (1) 2 where ! is the frequency of the oscillator. One of the numerous approaches we use to solve this problem is based on the following representation of the momentum and position operators: r x = ~ ay + a b 2m! b b r m ! p = i ~ ay − a : (2) b 2 b b From the canonical commutation relation [x;b pb] = i~ (3) follows y ba; ba = 1 y y [ba; ba] = ba ; ba = 0: (4) Inverting the above expression yields rm! i ba = xb + pb 2~ m! r y m! i ba = xb − pb (5) 2~ m! ∗Copyright Jörg Schmalian, 2016 1 y demonstrating that ba is indeed the operator adjoined to ba. We also defined the operator y Nb = ba ba (6) which is Hermitian and thus represents a physical observable. It holds m! i i Nb = xb − pb xb + pb 2~ m! m! m! 2 1 2 i = xb + pb − [p;b xb] 2~ 2m~! 2~ 2 2 1 pb m! 2 1 = + xb − : (7) ~! 2m 2 2 We therefore obtain 1 Hb = ! Nb + : (8) ~ 2 1 Since the eigenvalues of Hb are given as En = ~! n + 2 we conclude that the eigenvalues of the operator Nb are the integers n that determine the eigenstates of the harmonic oscillator. Nb jni = n jni : (9) y Using the above commutation relation ba; ba = 1 we were able to show that p a jni = n jn − 1i b p y ba jni = n + 1 jn + 1i (10) y The operator ba and ba raise and lower the quantum number (i.e.