aa r X i v : . [ qu a n t - ph ] F e b Note on a less ad hoc derivation of the second quantization formalism
Ning Wu ∗ Center for Quantum Technology Research, School of Physics,Beijing Institute of Technology, Beijing 100081, China
Second quantization is an essential topic in senior undergraduate and postgraduate level QuantumMechanics course. However, it seems that there is a lack of transparent and natural derivation ofthis formalism from the first-quantization one in most existing textbooks. Without introducing theconcept of a simple harmonic oscillator and taking the case of an assembly of identical fermions asan example, we provide a less ad hoc derivation of the second quantization formalism based on theequivalence of the fully antisymmetric many-fermion states and the Fock states.
I. INTRODUCTION
Second quantization is a useful and important toolfor treating systems of many identical interacting par-ticles. It is also an essential topic for upper-division andpostgraduate physics students when they study advancedquantum mechanics. According to the author’s teach-ing experience, it is usually difficult for the students tofully understand the construction of the second quantiza-tion formalism from the wave-function description of themany-particle system, due to the reasons will be detailedbelow. The aim of this note is to provide a less ad hoc derivation of the second quantization formalism basedmerely on the equivalence of the fully symmetric or an-tisymmetric many-particle states and the correspondingFock states in the occupation number representation.It is well known that the free classical electromagneticfield is described by an infinite set of uncoupled har-monic oscillators, which makes the second quantizationof a multiple boson system natural by directly extendingthe formalism of a single harmonic oscillator to the mul-timode case. This is best illustrated in Dirac’s classicalbook [1] (see §
60 there) and is perhaps one reason whymost textbooks on quantum mechanics often introducethe commutation or anticommutation relations amongthe particle creation and annihilation operators immedi-ately after introducing the concept of a Fock space, see,for example, Refs. [2–5]. We argue that the knowledgeof the algebraic solution of a harmonic oscillator is actu-ally not necessary for the construction. On the contrary,it may even lead to confusions for the beginners on thistopic. For instance, the students may get confused whenconsidering N identical fermions trapped in a harmonicoscillator potential: on the one hand, the creation andannihilation operators for the energy quanta of a singleoscillator satisfy the (single-mode) bosonic commutationrelations; on the other hand, the creation and annihi-lation operators for the real fermions in second quanti-zation have to obey the fermionic anticommutation rela-tions. The only useful idea we borrow from the harmonicoscillator is the concept of “ filling the vacuum ”, which is ∗ Electronic address: [email protected] also the basis for a Fock space.As pointed out in the book by Sakurai et. al. [2],imposing these canonical commutation relations is ac-tually “ very ad hoc ” and “it is not possible to do a fullyself-consistent treatment minimizing ad hoc assumptionswithout developing relativistic quantum field theory”. Inthis note, we will provide a “less ad hoc” derivation of thesecond quantization. Our development is based on i) Thesymmetrization postulate, ii) The concept of “filling thevacuum” in the Fock space, iii) The equivalence betweenthe symmetrized or antisymmetrized many-particle wavefunction and the Fock state. We will focus on the case ofFermi statistics due to the simplicity in the normalizationof the many-particle wave function.
II. EQUIVALENCE BETWEEN FULLYANTISYMMETRIC STATES AND FOCK STATESA. Setup in the first-quantization language
Consider an interacting system composed of N iden-tical fermions. In the “first quantization” language, weare forced to label the single-particle states for particle i as | φ α i i , where i = 1 , , · · · , N and α is the index forthe single particle states (SPSs). We assume there aretotally M SPSs and M can be either a finite (e.g., thetwo spin states for a spin-1/2 particle) or an infinite num-ber (e.g., the eigenstates of a simple harmonic oscillator),depending on the physical system being considered.Assuming that the M SPSs {| φ α i i | α = 1 , , · · · , M } form a complete set for particle i , i.e, P Mα =1 | φ α i i i h φ α | =1 i , one complete set of the N -fermion system is sim-ply the set of all product states, | φ P i ⊗ | φ P i ⊗· · · ⊗ | φ P N i N ≡ | φ P i | φ P i · · · | φ P N i N , where P i ∈{ , , · · · , M } is the index of the SPS occupied by par-ticle i . Since the N fermions are identical, we can alwayschoose 1 ≤ P ≤ P ≤ · · · ≤ P N ≤ M . Below we shallcall this particular product state | φ P i | φ P i · · · | φ P N i N ,in which the SPS indices are ordered as 1 ≤ P ≤ P ≤· · · ≤ P N ≤ M , a reference state .The first-quantization Hamiltonian for an N -fermionsystem can generally be written as H = F (1) + F (2) , F (1) = N X i =1 f (1) i , F (2) = 12 X i = j f (2) ij , (1)where F (1) and F (2) are the one-body and two-body op-erators, respectively. It is apparent that H is, and shouldbe, symmetric under the interchange i ↔ j [1]. It is gen-erally the case that the two body operator f (2) ij is sym-metric under i ↔ j , so that we can rewrite F (2) as F (2) = X i According to the symmetrization postulate [1], we haveto antisymmetrize the reference state to describe N iden-tical fermions. This is achieved by the standard antisym-metrization operator P f : P f | φ P i | φ P i · · · | φ P N i N ≡ X π ( − σ ( π ) | φ P i π (1) · · · | φ P N i π ( N ) = det | ψ P i | ψ P i · · · | ψ P N i | ψ P i | ψ P i · · · | ψ P N i ... ... . . . ... | ψ P i N | ψ P i N · · · | ψ P N i N , (3)where the sum is over all the N ! permutations of theparticle indices { , , · · · , N } and σ ( π ) = +1 ( − 1) if π is an even (odd) permutation and we have writtenthe result in the form of a determinant. Thus, thesymmetrization postulate implies the Pauli exclusionprinciple, saying that all the P i ’s must be distinct if P f | φ P i | φ P i · · · | φ P N i N describes a physical state. Inturn, the SPS indices in the reference state actually sat-isfy 1 ≤ P < P < · · · < P N ≤ M , so that the normal-ized antisymmetrized state can be written as | ψ i ( M,N ) f ≡ r N ! det | ψ P i | ψ P i · · · | ψ P N i | ψ P i | ψ P i · · · | ψ P N i ... ... . . . ... | ψ P i N | ψ P i N · · · | ψ P N i N . (4)Note that | ψ i ( M,N ) f is not only antisymmetric under theinterchange i ↔ j of two particle indices (correspond-ing to interchanging two rows), but also antisymmetricunder the interchange P i ↔ P j of two state indices (cor-responding to interchanging two columns). C. Fock states: filling the vacuum Since the N fermions are indistinguishable, we haveto avoid saying “which fermion”, but to say “ how many fermions are there in a SPS ”. Suppose the SPS | φ α i is occupied by N α (which can be either 0 or 1 and sat-isfy P Mα =1 N α = N ) fermions. Such a state for the N -fermion system can be uniquely identified by the indices { P , · · · , P N } of all the occupied SPSs (i.e., those with N α = 1), and can be written as | ψ i ( M,N )Fock ≡ | P , · · · , P N i . (5)Note that the particles indices have disappeared in theFock state | ψ i ( M,N )Fock and only the SPS indices are in-volved. This suggests us to introduce the following M SPS creation operators : a † α , α = 1 , , · · · , M in terms of which | ψ i ( M,N )Fock can be generated by succes-sively operating a † P , a † P , · · · , a † P N onto a vacuum state | ~ i ≡ | , , · · · , i without any fermion: | ψ i ( M,N )Fock = a † P a † P · · · a † P N | ~ i . (6)Note that we only used the concept of “filling the vac-uum” [1] and have not yet assigned any commutation re-lations among the a † α ’s . D. Equivalence between | ψ i ( M,N ) f and | ψ i ( M,N )Fock :relation between a † α and a † β A key observation for the development of the formal-ism is the equivalence between the fully antisymmetricstate | ψ i ( M,N ) f (in the “first-quantization language”) andthe Fock state | ψ i ( M,N )Fock (in the “second-quantization lan-guage”) [1]: | ψ i ( M,N ) f = | ψ i ( M,N )Fock . (7)Since | ψ i ( M,N ) f is antisymmetric under the interchange P i ↔ P j , so must be | ψ i ( M,N )Fock : | ψ i ( M,N )Fock = a † P · · · a † P i · · · a † P j · · · a † P N | ~ i = − a † P · · · a † P j · · · a † P i · · · a † P N | ~ i . This can be achieved by imposing anticommutation rela-tions among the creation operators: { a † α , a † β } = 0 , ∀ α, β = 1 , , · · · , M, (8)and hence { a α , a β } = 0 , ∀ α, β = 1 , , · · · , M, (9)where a α ≡ ( a † α ) † . A direct consequence of Eq. (8) is( a † α ) = 0, which is just an alternative description of thePauli exclusion principle.Having introduced the anticommutation relationsamong the a † α ’s, the equivalence between the antisym-metric states and the corresponding Fock state is nolonger restricted to the reference state with 1 ≤ P
2) onto | ψ i ( M,N ) f must be equivalentto operating ˆ F ( a ) onto | ψ i ( M,N )Fock : F ( a ) | ψ i ( M,N ) f = ˆ F ( a ) | ψ i ( M,N )Fock , a = 1 , . (11)To derive the explicit form of ˆ F ( a ) in Eq. (11), let uscalculate the left-hand-side of this equation. We firstfocus on the one-body operator F (1) .Note that F (1) ( | ψ i ( M,N ) f ) is symmetric (antisymmet-ric) under the interchange of the particle labels, i ↔ j ,the state F (1) | ψ i ( M,N ) f is also antisymmetric under i ↔ j .Actually, it is easy to show that F (1) | ψ i ( M,N ) f = r N ! N X i =1 P f [ | φ P i · · · ( f (1) i | φ P i i i ) · · · | φ P N i N ] , (12) which is nothing but an antisymmetrization of the newstate | φ P i · · · ( f (1) i | φ P i i i ) · · · | φ P N i N , in which | φ P i i i isreplaced by the new SPS f (1) i | φ P i i i .As an illustration, we verify Eq. (12) for a simple ex-ample with M = 3 and N = 2. In this case | ψ i (3 , f = q ( | φ P i | φ P i − | φ P i | φ P i ), so F (1) | ψ i (3 , f = ( f (1)1 + f (1)2 ) r 12 ( | φ P i | φ P i − | φ P i | φ P i )= r 12 [( f (1)1 | φ P i ) | φ P i − ( f (1)1 | φ P i ) | φ P i + | φ P i ( f (1)2 | φ P i ) − | φ P i ( f (1)2 | φ P i )] . Note that P f ( f (1)1 | φ P i ) | φ P i = ( f (1)1 | φ P i ) | φ P i − | φ P i ( f (1)2 | φ P i ) , P f | φ P i ( f (1)2 | φ P i )= | φ P i ( f (1)2 | φ P i ) − ( f (1)1 | φ P i ) | φ P i , we thus obtain F (1) | ψ i (3 , f = r 12 [ P f ( f (1)1 | φ P i ) | φ P i + P f | φ P i ( f (1)2 | φ P i )] , consistent with Eq. (12).From Eq. (12) we see that the one-body operator F (1) can only change the state of a single particle. Thus,the matrix elements of ˆ F (1) do not vanish only fortransitions from the Fock state | P , · · · , P i , · · · , P N i to | P , · · · , P ′ i , · · · , P N i , say. As pointed out by Landau andLifshitz [6], “The calculation of these matrix elements isin principle very simple; it is easier to do it oneself thanto follow an account of it”. Nevertheless, we believe thisstatement might be obscure for a beginner.To proceed, we expand the SPS f (1) i | φ P i i i in Eq. (12)in terms of the complete set of the SPSs for particle i : F (1) | ψ i ( M,N ) f = r N ! N X i =1 P f | φ P i · · · M X Q i =1 i h φ Q i | f (1) i | φ P i i i | φ Q i i i · · · | φ P N i N . (13)It is now important observe that the matrix element i h φ Q i | f (1) i | φ P i i i actually depends only on the state indices Q i and P i , but not on the particle label i . To see this,we insert the completeness relation R d x | ~x i ih ~x i | = 1 i for particle i and get i h φ Q i | f (1) i | φ P i i i = Z Z d x i d x ′ i i h φ Q i | ~x i ih ~x i | f (1) i | ~x ′ i ih ~x ′ i | φ P i i i = Z Z d x i d x ′ i φ ∗ Q i ( ~x i ) h ~x i | f (1) i | ~x ′ i i φ P i ( ~x ′ i )= Z Z d xd x ′ φ ∗ Q i ( ~x ) h ~x | f (1) | ~x ′ i φ P i ( ~x ′ ) , (14)where in the last line we have dropped out the par-ticle index i from ~x i , ~x ′ i , and f (1) i . It is clear that i h φ Q i | f (1) i | φ P i i i is the same for all the particles and we can write i h φ Q i | f (1) i | φ P i i i = h φ Q i | f (1) | φ P i i . (15)By inserting Eq. (15) into Eq. (13) and note that thematrix element h φ Q i | f (1) | φ P i i , which has noting to dowith the particle indices, is not affected by the antisym-metrization operator P f , we obtain F (1) | ψ i ( M,N ) f = N X i =1 M X Q i =1 h φ Q i | f (1) | φ P i i r N ! P f [ | φ P i · · · ( | φ Q i i i ) · · · | φ P N i N ]= N X i =1 M X Q i =1 h φ Q i | f (1) | φ P i i a † P · · · a † P i − a † Q i a † P i +1 · · · a † P N | ~ i , (16)where we have used Eq. (10). According to Eq. (11),we expect the last line of Eq. (16) to be equal toˆ F (1) | ψ i ( M,N )Fock = ˆ F (1) a † P a † P · · · a † P N | ~ i . It is therefore de-sirable to replace the a † Q i in Eq. (16) by a † P i through ap-propriate manipulations. To this end, we have to intro-duce the following properties for the fermion annihilationoperators: a α | ~ i = 0 , ∀ α, (17) { a α , a † β } = δ αβ , ∀ α, β. (18)Using the above properties, we can show that a † P · · · a † P i − a † Q i a † P i +1 · · · a † P N | ~ i = a † Q i a P i a † P · · · a † P i − a † P i a † P i +1 · · · a † P N | ~ i . (19)It is not difficult to prove Eq. (19) by using Eqs. (8),(17), and (18) (see Appendix A). Applying Eq. (19) inEq. (16) gives F (1) | ψ i ( M,N ) f = N X i =1 M X Q i =1 h φ Q i | f (1) | φ P i i a † Q i a P i | ψ i ( M,N )Fock = N X i =1 M X α =1 h φ α | f (1) | φ P i i a † α a P i | ψ i ( M,N )Fock , (20)where we have changed the summation index Q i into α as it runs over all the SPSs. Note now that the sum over i in the last line of Eq. (20) can also be extended to allthe SPSs since a P ′ i | ψ i ( M,N )Fock = 0 if P ′ i / ∈ { P , · · · , P N } .So, F (1) | ψ i ( M,N ) f = M X β =1 M X α =1 h φ α | f (1) | φ β i a † α a β | ψ i ( M,N )Fock . (21)By comparing the above equation with Eq. (11), we fi-nally obtain the explicit form of the one-body operatorˆ F (1) : ˆ F (1) = M X α,β =1 h φ α | f (1) | φ β i a † α a β . (22)This completes our derivation of the one-body operatorwith the help of the newly introduced relations given byEqs. (17) and (18). IV. DERIVATION OF THE TWO-BODYOPERATOR: A STRAIGHTFORWARDEXTENSION The derivation of the two-body operator ˆ F (2) closelyfollows that of the one-body operator. Similar to theaction of F (1) onto | ψ i ( M,N ) f , the state F (2) | ψ i ( M,N ) f isantisymmetric under the interchange of the particle in-dices and has the form F (2) | ψ i ( M,N ) f = r N ! X i In this note, we have provide a intuitive and lessad hoc derivation of the second-quantization formal- ism from the its first-quantization version. The deriva-tion avoids invoking any priori knowledge of the alge-braic solution of the simple harmonic oscillator and re-lies only on the concept of filling the vacuum in theFock space, as well as the equivalence between thefully symmetrized/antisymmetrized many-particle stateand the corresponding Fock space. Taking the case ofFermi statistics as an example, the anticommutation re-lations among the fermionic creation operators naturallyarise from the antisymmetrization property of the many-fermion state in first quantization. The commutationrelations among the creation and annihilation operatorsare imposed by demanding that operating the one-bodyoperator F (1) onto the antisymmetric state | ψ i ( M,N ) f re-covers the action of ˆ F (1) onto the Fock state | ψ i ( M,N )Fock .The derivation of the two-body operator turns out to be astraightforward extension of the foregoing constructions.We believe the method presented in this note wouldoffer a clear and transparent interpretation of the secondquantization formalism that would be friendly to gradu-ate students or beginners who first study this topic. Appendix A: Proof of Eq. (19) We start with the left-hand-side of Eq. (19): a † P · · · a † P i − a † Q i a † P i +1 · · · a † P N | ~ i = ( − i − a † Q i a † P · · · a † P i − a † P i +1 · · · a † P N | ~ i = ( − i − a † Q i a † P · · · a † P i − (1 − a † P i a P i ) a † P i +1 · · · a † P N | ~ i = ( − i − a † Q i a † P · · · a † P i − a P i a † P i a † P i +1 · · · a † P N | ~ i = ( − i − ( − i − a † Q i a P i a † P · · · a † P i − a † P i a † P i +1 · · · a † P N | ~ i = a † Q i a P i a † P · · · a † P i − a † P i a † P i +1 · · · a † P N | ~ i . (A1) Appendix B: Proof of Eq. (26) We start with the left-hand-side of Eq. (26): a † P · · · a † Q i · · · a † Q l · · · a † P N | ~ i = ( − i − a † Q i a † P · · · a † P i − a † P i +1 · · · a † Q l · · · a † P N | ~ i = ( − i − ( − l − a † Q i a † Q l a † P · · · a † P i − a † P i +1 · · · a † P l − a † P l +1 · · · a † P N | ~ i = ( − i − ( − l − a † Q i a † Q l a † P · · · a † P i − ( a P i a † P i + a † P i a P i ) a † P i +1 · · · a † P l − ( a P l a † P l + a † P l a P l ) a † P l +1 · · · a † P N | ~ i = ( − i − ( − l − a † Q i a † Q l a † P · · · a † P i − ( a P i a † P i ) a † P i +1 · · · a † P l − ( a P l a † P l ) a † P l +1 · · · a † P N | ~ i = ( − i − ( − l − ( − i − a † Q i a † Q l a P i a † P · · · a † P i − ( a † P i ) a † P i +1 · · · a † P l − ( a P l a † P l ) a † P l +1 · · · a † P N | ~ i = ( − i − ( − l − ( − i − ( − l − a † Q i a † Q l a P i a P l a † P · · · a † P i − ( a † P i ) a † P i +1 · · · a † P l − ( a † P l ) a † P l +1 · · · a † P N | ~ i = − a † Q i a † Q l a P i a P l a † P · · · a † P i − ( a † P i ) a † P i +1 · · · a † P l − ( a † P l ) a † P l +1 · · · a † P N | ~ i = a † Q i a † Q l a P l a P i a † P · · · a † P i − ( a † P i ) a † P i +1 · · · a † P l − ( a † P l ) a † P l +1 · · · a † P N | ~ i . 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