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July 28, 2019
###### Abstract

The notion of “closed systems” in Quantum Mechanics is discussed. For this purpose, we study two models of a quantum-mechanical system spatially far separated from the “rest of the universe” . Under reasonable assumptions on the interaction between and , we show that the system behaves as a closed system if the initial state of belongs to a large class of states, including ones exhibiting entanglement between and . We use our results to illustrate the non-deterministic nature of quantum mechanics. Studying a specific example, we show that assigning an initial state and a unitary time evolution to a quantum system is generally not sufficient to predict the results of a measurement with certainty.

Isolated systems]On the probabilistic nature of quantum mechanics and the notion of closed systems

J. Faupin]Jérémy Faupin J. Fröhlich]Jürg Fröhlich B. Schubnel]Baptiste Schubnel

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A key reason why, in science, we are able to successfully describe natural processes quantitatively is that if some process of interest is far isolated from the rest of the world it can be described as if nothing else were present in the universe; i.e., it can be viewed as a process happening in a “closed system”. This means, for example, that a condensed-matter experimentalist studying a magnetic material does not have to worry about astrophysical processes inside the sun, in order to understand the magnetic properties of the material in his earthly laboratory. Nor, for that matter, does he have to worry about what his colleague in the laboratory next door is doing, provided he is not experimenting with strong magnetic fields. It is the purpose of our paper to show that the notion of “closed systems”, in the sense just sketched, makes sense in quantum mechanics – in spite of the phenomena of entanglement and of the “non-locality” of Bell correlations.

Rather than engaging in a general, abstract discussion, we propose to study some concrete models of quantum systems, , composed of two spatially far separated subsystems and coupled to a common environment (that can be empty). We discuss various sufficient conditions implying that, for a large class of initial states of including ones exhibiting entanglement between and , the time-evolution of expectation values of observables, , of the subsystem behaves as if the subsystem were absent. The interesting ones among our sufficient conditions turn out to be uniform in the number of degrees of freedom of the subsystem . Our results can be interpreted as saying that there is “no signaling” between and , provided that these subsystems are spatially far separated from one another, independently of whether the initial state of is entangled, or not, and independently of the number of degrees of freedom of . In other words, our conditions guarantee that can be considered to be a “closed system” (or “isolated system”). (Absence of signaling has previously been discussed, e.g., in [27, 8, 25].)

We will discuss two models. In a first model, we choose to describe a quantum particle moving away from a system that may have very many degrees of freedom; the environment is absent. It will be assumed that, in a sense to be made precise below, interactions between and become weaker and weaker, as the distance between the two subsystems increases. One purpose of our discussion of this model is to show that quantum mechanics does not admit a realistic interpretation – in the sense that knowing the unitary time evolution of a system and its initial state does not enable one to predict what happens in the future – and that it is intrinsically probabilistic. In a second, more elaborate model, the subsystems and are allowed to exchange quanta of a quantum field (such as photons or phonons), i.e., and can“communicate” by emitting and absorbing field quanta; accordingly, the environment is chosen to consist of a quantum field, e.g., the electromagnetic field or a field of lattice vibrations. The goal of our discussion is to isolate conditions that enable us to derive an “effective dynamics” of the subsystem that does not explicitly involve the environment and is independent of .

To keep our analysis down to earth, we will only study systems and (with finitely many, albeit arbitrarily many degrees of freedom) that can be described in the usual Hilbert-space framework of non-relativistic quantum mechanics, with the time evolution given by a unitary one-parameter group. The “observables” are taken to be bounded selfadjoint operators on a Hilbert space. (For simplicity, the environment will be assumed to have temperature zero, with pure states corresponding to unit rays in Fock space.)

Concretely, the Hilbert space of pure state vectors of the system is given by

 H=HP⊗HQ⊗HE, (1.1)

where , and are separable Hilbert spaces. General states of are given by density matrices, i.e, positive trace-class operators, , of trace acting on . General observables of the entire system are self-adjoint operators in , where is the algebra of all bounded operators on the Hilbert space . Observables refering to the subsystem are selfadjoint operators of the form

 OP=O⊗\mathds1HQ∨E,O=O∗∈B(HP). (1.2)

A state of the entire system determines a state of the subsystem (a reduced density matrix) by

 TrHP(ρPA):=Tr(ρ(A⊗\mathds1HQ∨E)), (1.3)

for an arbitrary operator .

Time evolution of is given by a unitary one-parameter group on .

We are now ready to clarify what we mean by “closed systems”: Informally, can be viewed as a closed subsystem of if there exists a one-parameter unitary group on such that

 Tr(ρU(t)∗(A⊗\mathds1HQ∨E)U(t))≈TrHP(ρPUP(t)∗AUP(t)), (1.4)

for a suitably chosen subset of density matrices and all times in some interval contained in . Mathematically precise notions of “closed subsystems” will be proposed in the context of the two models analyzed in this paper, and we will subsequently present sufficient conditions for to be a closed subsystem of .

The plan of our article is as follows. In subsections \@setrefSec2-1 and \@setrefSec2-2 we introduce the models analyzed in this paper. The first model describes a quantum particle, , with spin interacting with a large quantum system and moving away from . (The subsystem may consist of another quantum particle entangled with and a “detector”. The two particles are prepared in an initial state chosen such that they move away from each other, with moving away from the detector.) This example will be useful in a discussion of some aspects of the foundations of quantum mechanics, in particular of the intrinsically probabilistic nature of quantum mechanics. The second model describes a neutral atom with a non-vanishing electric dipole moment that interacts with a large quantum system . Both and are coupled to the quantized electromagnetic field, . In this model, corresponds to the composition . The point is to identify an effective dynamics for that does not make explicit reference to the electromagnetic field . Our results on these models are stated and interpreted in subsections \@setrefSec2-1r and \@setrefSec2-2r, respectively. In subsection \@setrefSec2-1e, we sketch some concrete experimental situations described, at least approximately, by our models.

Proofs of our main results are presented in section \@setrefproofs. Many of the techniques used in our proofs are inspired by ones used in previous works on scattering theory; see, e.g., [26, 7, 15, 16, 11, 12]. Some technical lemmas are proven in two appendices.

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J. Fr. thanks P. Pickl and Chr. Schilling for numerous stimulating discussions on models closely related to the first model discussed in our paper. J. Fa. and J. Fr. are grateful to I.M. Sigal for many useful discussions on problems related to the second model and, in particular, on scattering theory. J. Fa.’s research is supported by ANR grant ANR-12-JS01-0008-01.

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We consider a quantum particle, , of mass and spin ; (throughout this paper, we employ units where ). The particle interacts with a large quantum system, , which we keep as general as possible. The pure states of the composed system, , correspond to unit rays in the Hilbert space , where and is a separable Hilbert space. The dynamics of is specified by a selfadjoint Hamiltonian

 H=HP⊗\mathds1HQ+\mathds1HP⊗HQ+HP,Q (2.1)

defined on a dense domain . In (2.1),

 HP:=−Δ2⊗\mathds1C2.

The operator and , defined on their respective domains, are self-adjoint.

Remark. It is not important to exclude the presence of external fields or potentials acting on the particle. All that matters is that the propagation of the particle approaches the one of a free particle as time tends to . To keep our analysis simple we assume that if the interaction between and is turned off then propagates freely.

To identify as a closed subsystem of , one assumes that

1. the strength of the interaction between and (described by the operator ) decays to zero rapidly as the “distance” between and tends to ; and

2. the initial state of the system is chosen such that the particle propagates away from , the distance between and growing ever larger. (We will actually choose the initial state such that, with very high probability, the particle is scattered into a cone far separated from the subsystem .)

A graphical illustration of Assumptions (1) and (2) is given below.

Next, we reformulate Assumptions (1) and (2) in mathematically precise terms. It is convenient to identify with . We denote by three orthonormal vectors in .

• (Location of and properties of the interaction Hamiltonian) There is an open subset (possibly unbounded), the “spatial location” of the subsystem , separated from the cone

 C2θ0:={→k∈R3∣→k⋅→ex≥|→k|cos(2θ0)} (2.2)

with , by a distance , and a covering

 Ω=⋃n∈IΩn,I⊆N

of by open cubes of uniformly bounded diameter such that

1. the interaction Hamiltonian can be written as a strongly convergent sum of operators,

 HP,Q=∑n∈IHP,Qn, (2.3)

on the dense domain .

The operator encodes the interaction between the particle and the subsystem of located in the cube . The distance between and the cone is denoted by and is supposed to tends to , as tends to ;

2. there is a constant and a sequence of operators on with the properties that

 ∥(HP,QnΨ)(→x)∥C2⊗HQ≤∥(\mathds1HP⊗Nn)Ψ(→x)∥C2⊗HQ[dist(Ωn,→x)]α,→x∈Ωc, (2.4)

for all and for all , and

 ∑n∈Id1−α2n≤Cd−β, for some β>0,C<∞. (2.5)

Furthermore, for all .

• (Choice of initial state) The initial state , , is a smooth function of of rapid decay with values in . Its Fourier transform,

 ˆΨ0(→k):=1(2π)3/2∫R3e−i→k⋅→xΨ0(→x) d3x, (2.6)

has support in the conical region defined by

 Cθ0;v:={→k∈R3∣→k⋅→ex≥|→k|cos(θ0),|→k|>v} (2.7)

for some .

• (Bound on the number of particles in ) For ,

 ∥(\mathds1HP⊗Nn)e−itHQΨ0∥Ls(R3;C2⊗HQ)
###### Remarks.

Assumption (A2) guarantees that the distance between and grows in time with very high probability. The hypotheses (A1) and (A3) are mathematical reformulations of Assumption (1). The operator can be thought of as counting the number of “particles” of the system contained in the subset , for all . If the system is composed of identical particles, is the bosonic/fermionic Fock space over , () and the operator is the second quantization of the multiplication operator by the characteristic function .

The decomposition of into cubes is used to get bounds that are uniform in the number of degrees of freedom of the system .

We observe that the decay in (2.4) is faster than the one of the Coulomb potential. To justify (2.4) one would have to invoke screening.

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The reduced density matrix, , of the particle corresponding to the state of the entire system is defined by

 ⟨φ1,ρPφ2⟩L2(R3;C2):=∑j∈J⟨φ1⊗ej,Ψ0⟩⟨Ψ0,φ2⊗ej⟩,

where are arbitrary vectors in and is an orthonormal basis in .

###### Lemma 2.1.

We require assumptions (A1), (A2) and (A3). Then, for all , there exists a length such that, for any ,

 ∣∣⟨e−itHΨ0,(OP⊗\mathds1HQ)e−itHΨ0⟩−TrHP(ρPeitHPOPe−itHP)∣∣≤η∥OP∥, (2.9)

for all and for all .

Lemma \@setref1.1 justifies considering as a closed subsystem: any observable of the subsystem evolves as if were absent, up to an error term that can be made arbitrarily small by increasing the separation between and . A similar result was already discussed in [26], but with a finite range interaction between and . The proof of Lemma \@setref1.1 is given in Appendix \@setrefD.

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We choose to be composed of a particle of spin (electron) and of a spin filter . The particles and are scattered into opposite cones, and the spin filter selects the particle according to its spin component along the axis corresponding to a unit vector . A Stern-Gerlach-type experiment is added to the setup to measure a component of the spin of the particle .

The Hilbert space of the system is , where . We set

 |↑⟩:=(10),|↓⟩:=(01).

We assume that the initial state is an entangled state of the form

 Ψ0=1√2nn∑j=1(|ϕj,P,↓;ψj,P′,↑⟩−|ϕj,P,↑;ψj,P′,↓⟩)⊗|χj⟩, (2.10)

where , for all . We assume that Assumptions (A1), (A2) and (A3) of Section \@setrefSec2-1 are fulfilled by the Hamiltonian and by the initial state of the composed system , with . For simplicity, we neglect interactions between and . However, it is not hard to generalize Lemma \@setref1.1 to a situation where and interact via a repulsive two-body potential, assuming that the momentum space support of the wave function is contained in a cone opposit to , for all .

We denote by the spin operator of the particle , where , , are the Pauli matrices. The next corollary is a direct consequence of Lemma \@setref1.1.

###### Corollary 2.2 (No-signaling).

Let . We require assumptions (A1), (A2) and (A3) of Paragraph \@setrefSec2-1. Then there is a distance such that, for any ,

 |⟨e−itHΨ0,→SPe−itHΨ0⟩|<η,∀t≥0. (2.11)

For sufficiently large values of , Corollary \@setrefcoco shows that the mean value of the spin operator of the particle very nearly vanishes for all times , regardless of the initial state vectors of the filter . In particular, the expectation value of the spin operator of is independent of the kind of measurement on performed by the spin filter . Here we assume that a particle with passes the filter, while a particle with is absorbed by , with probability very close to . A realistic interpretation of quantum mechanics, in the sense that the time evolution of pure states in the Schrödinger picture would completely predict what will happen, necessary fails. It would lead to the prediction that

 ⟨e−itHΨ0,(→SP⋅→n)e−itHΨ0⟩≈−12 (2.12)

for sufficiently large times if the particle has passed the filter . This contradicts Eq. (2.11). It shows that choosing a unitary time evolution and specifying an initial state does not predict the results of measurement, but only probabilities for the outcomes of such measurement. Our conclusion remains valid if the particles and are indistinguishable particles; see [18, 26].

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We consider a neutral atom that interacts with a quantum system and the quantized electromagnetic field. The atom either moves freely or moves in a slowly varying external potential. We assume that, initially, it is localized (with a probability close to one) far away from the system , and we allow the system to create and annihilate photons. Our aim is to prove a result of the form of (1.4). Our estimates for this model are however not uniform in the number of degrees of freedom of the subsystem . This problem could be solved by decomposing into small subsystems. This complication is avoided to keep our exposition as simple as possible. We do not specify the nature of , but we emphasize that it could represent another atom or a molecule. The internal degrees of freedom of the atom are described by a two-level system. The total Hilbert space of the system is the tensor product space

 H:=HP⊗HQ⊗HE,

where

 HP:=L2(R3)⊗C2 and HE:=F+(L2(R––3))

are the Hilbert spaces associated to the atom and to the electromagnetic field, respectively. Here is the (symmetric) Fock space over . We use the notation

 R––3:=R3×{1,2}={k––:=(→k,λ)∈R3×{1,2}},dk––=∑λ=1,2d3k,

where is the photon momentum and denotes the polarization of the photon. Any element can be represented as a sequence of totally symmetric -photons functions. The scalar product on is defined by

 ⟨Φ,Ψ⟩=∑n≥0∫R––3n¯¯¯¯Φ(n)(k––1,...,k––n)Ψ(n)(k––1,...,k––n)dk––1...dk––n

for all .

The Hamiltonian of the total system is written as

 H:= HP⊗\mathds1HQ⊗\mathds1HE+\mathds1HP⊗HQ⊗\mathds1HE+\mathds1HP⊗\mathds1HQ⊗HE +HP,E+HP,Q+HQ,E,

where

 HP:=−Δ2+(ω0000)

is the free atomic Hamiltonian, with the energy of the excited internal state of the atom, the Hamiltonian for the system , and

 HE:=dΓ(|→k|)≡∫R––3|→k|a∗(k––)a(k––)dk––

is the second quantized Hamiltonian of the free electromagnetic field. The operator-valued distributions and are the photon annihilation and creation operators. We suppose that is a semi-bounded self-adjoint operator on . In what follows, we write for , and likewise for and , unless confusion may arise.

The interaction Hamiltonians, , and describe the interactions between the atom, the system , and the quantized field. The atom-field interaction is of the form , where is the dipole moment of the atom, is the vector of Pauli matrices, and is the quantized electric field, i.e.,

 HP,E:=iλ0∫R––3χ(→k)|→k|12→ε(k––)⋅→σ(ei→k⋅→xa(k––)−e−i→k⋅→xa∗(k––))dk––,

where is an ultraviolet-cutoff function such that on and on , and are polarization vectors of the electromagnetic field in the Coulomb gauge. With the usual notations, can be rewritten in the form

 HP,E=Φ(hx)≡a∗(hx)+a(hx), (2.13)

with

 hx(k––):=−iλ0χ(→k)|→k|12→ε(k––)⋅→σe−i→k⋅→x. (2.14)

By standard estimates (see Lemma \@setreflm:B1), is -bounded with relative bound . We suppose that and are symmetric operators relatively bounded with respect to and , respectively, and that is a self-adjoint operator with domain . Further technical assumptions on and needed to state our main theorem will be described below.

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We assume that:

1. The support of the initial atomic wave function (at time ) is contained inside a ball of radius .

2. There is a large distance such that the interaction Hamiltonian between the ball of radius , , centered at the same point as the ball containing the support of the initial wave function of the atom and the region containing is bounded in norm by , for some finite constant and some exponent .

3. With very high probability, there aren’t any photons emitted by the subsystem towards, nor absorbed by from the ball of radius centered at the same point as the ball, , containing the support of the initial wave function of the atom.

To simplify the analysis, we suppose that the initial atomic wave function is contained inside a ball (of radius ) centered at the origin, and that is located outside the ball of radius centered at , for some fixed . Assumptions (2) and (3) are then replaced by the hypotheses that is bounded by , and that does not emit nor absorb photons inside the ball of radius centered at the origin. These assumptions imply that the system does not penetrate into the ball of radius centered at the origin. This hypothesis can be weakened for concrete choices of the subsystem .

We recall the definition of the scattering identification operator (see [23], [7] or [16] for more details) and a few other standard tools from scattering theory to rewrite Assumptions (1) through (3) in mathematically precise terms. Let denote the set of all vectors such that for all but finitely many ’s. The map is defined as the extension by linearity of the map

 I:a∗(g1)⋯a∗(gm)Ω⊗a∗(h1)⋯a∗(hn)Ω↦a∗(h1)⋯a∗(hn)a∗(g1)⋯a∗(gm)Ω, (2.15)

for all . The closure of on is denoted by the same symbol and is called the scattering identification operator. Observe that is unbounded. Let

 H0:=HP⊗HE,H∞:=HQ⊗HE.

The Hilbert space corresponds to the atom together with photons located near the origin, whereas corresponds to the system together with photons located far from the origin. We extend the operator to the space by setting

 I:H0⊗H∞→H.

We use to “amalgamate” with .

We recall that the Hamiltonian on associated with the atom and the quantized radiation field,

 HP∨E:=HP+HE+HP,E,

is translation-invariant, in the sense that commutes with each component of the total momentum operator

 →PP∨E:=→PP+→PE=−i→∇x+∫–R3→ka∗(k––)a(k––)dk––.

This implies (see e.g. [2] or [10] for more details) that there exists a unitary map

 U:HP⊗HE→∫⊕R3C2⊗HEd3p,

such that

 UHP∨EU−1=∫⊕R3H(→p)d3p.

For any fixed total momentum , the Hamiltonian is a self-adjoint, semi-bounded operator on . Its expression is given in Appendix \@setrefB. It turns out that, for and for small coupling , has a ground state with associated eigenvalue , and that this ground state, , is real analytic in , for ; see [10] and Theorem \@setrefthm:anal for a more precise statement. Given , we assume in the rest of this text that , where is the critical coupling constant such that has a ground state for all with . We introduce a dressing transformation , defined, for all and for a.e. , by the expression

 J(u)(→x):=1(2π)32∫R3^u(→p)ei→x⋅(→p−→PE)χ¯Bν/2(→p)ψ(→p)d3p, (2.16)

where is such that on , and , outside . The state describes a dressed single-atom state. We recall that, for any operator on , the second quantization of , , is the operator defined on by its restriction to the -photons Hilbert space, which is given by

 Γ(a)|L2(R––3)⊗ns:=⊗na,n=0,1,2,... (2.17)

and . We denote by

 N:=∫R––3a∗(k––)a(k––)dk––

the photon number operator on Fock Space. We are ready to state our main assumptions.

• (Initial state of atom) Let be such that . The initial orbital wave function of the atom is supposed to be of the form

 u(→x)=R−3/2v(R−1→x),

for some . In particular,

 supp(u)⊂{→x∈R3,|→x|≤R},

and is independent of .

• (Initial state of photons far from the atom) The state satisfies

 (\mathds1HQ⊗Γ(\mathds1|→y|≥3d))φ=φ,

for some , where denotes the “photon position variable”, and

 φ∈D(HQ∨E)∩D(\mathds1HQ⊗eδN),

for some .

• (Interaction ) The interaction Hamiltonian between the atom and the subsystem , , is a symmetric operator on , relatively bounded with respect to , and satisfying

 ∥HP,Q\mathds1|→x|≤d∥≤Cd−β (2.18)

for some constants and .

• (Interaction ) The interaction Hamiltonian between the subsystem and the radiation field, , is a symmetric operator on such that is self-adjoint on

 D(HQ∨E)=D(HQ+HE).

Moreover, in the sense of quadratic forms, satisfies

 (2.19)

for all and for all Fourier multiplication operators on such that , where stands for or .

• (Number of photons emitted by ) The initial state satisfies for all times , and

 ∥∥dΓ(\mathds1|→y|≥ct)e−itHQ∨Eφ∥∥ ≤C⟨t⟩, (2.20)

for some , where is a positive constant depending on and .

###### Remarks.

(B1), (B3) and (B4) are direct mathematical reformulations of the hypotheses (1), (2) and (3) above. In (B2), we assume that, initially, photons in contact with are “localized” outside the ball of radius centered at the origin.

The constant in (2.18) depends a priori on the number of degrees of freedom of the subsystem . This problem could be circumvented by decomposing into subsystems located ever further away from , as we did for the first model.

Assumption (2.19) is very strong and can be relaxed in concrete examples for the subsystem . For instance, if is linear in annihilation and creation operators, Eq. (2.19) is not relevant and the estimate of the norm of the commutator of with other operators on Fock space can be carried out directly. The calculations are the same as for the operator .

Assumption (B5) implies that the number of photons created by does not grow faster than linearly in time. Indeed, using Hardy’s inequality and the fact that , we have that

 ∥∥dΓ(\mathds1|→y|≤ct)e−itHQ∨Eφ∥∥≤ct∥∥dΓ(|→y|−1)e−itHQ∨Eφ∥∥≤ct∥HEe−itHQ∨Eφ∥∥≤Ct.

(2.20) says that the number of photons emitted by and traveling faster than light grows at most linearly in time. Eq. (2.20) could be weakened by a polynomial growth. This would lead to worse estimates in Theorem \@setrefthm:isolated below. Assumption (B5) is not fully satisfactory, since the upper bound may depend on the number of degrees of freedom of . The main reason why we impose (2.20) is that photons are massless. The operator is not -bounded, and some of our estimates cannot be proven if we do not control the time evolution of the total number of photons.

For massive particles, the dispersion law is replaced by , where is the mass of the particles of the field. Since is -bounded, and since, under our assumptions, , we have that

 ∥∥Ne−itHQ∨Eφ∥∥≤C(∥HQ∨Eφ∥+∥φ∥).

Hypothesis (B5) is therefore obviously satisfied for massive particles. To simplify our presentation, we only state and prove our main result for photons.

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Our aim is to show that, under Assumptions (B1)-(B5), behaves as a closed system over a finite interval of times. For , we write if there is a constant independent of , and , such that .

###### Theorem 2.3.

Consider an initial state of the form

 ψ=1∥∑li=1I(J(ui)⊗φi)∥l∑j=1I(J(uj)⊗φj),

where and satisfy Assumptions (B1), (B2) and (B5) with , for , and for all . We introduce the density matrix

 ρP:=1∑li=1∥J(ui)∥2l∑j=1|J(uj)⟩⟨J(uj)|∈B(H0).

Suppose, moreover, that Asumptions (B3) and (B4) are satisfied. Then

 ⟨e−itHψ, (OP⊗\mathds1HQ⊗\mathds1HE)e−itHψ⟩=TrH0(ρPeitHP∨E(OP⊗\mathds1HE)e−itHP∨E)

for all , all , and all .

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We now assume that the atom is placed in a slowly varying external potential, , with . We set

 HεP:=HP+Vε(→x),HεP∨E:=HP∨E+Vε(→x),Hε:=H+Vε(→x).

We define the effective Hamiltonian on as

 HεP,eff:=E(−i→∇x)+Vε(→x),

where is the ground state energy of the fiber Hamiltonian