## Asymptotic Behavior of Bohmian Trajectories in Scattering Situations and Exit Statistics for Distant Surfaces (2004)

### BibTeX

@MISC{Römer04asymptoticbehavior,

author = {Sarah Römer},

title = {Asymptotic Behavior of Bohmian Trajectories in Scattering Situations and Exit Statistics for Distant Surfaces},

year = {2004}

}

### OpenURL

### Abstract

### Citations

595 |
Speakable and Unspeakable in Quantum Mechanics
- Bell
- 1987
(Show Context)
Citation Context ...ists almost surely and is randomly distributed t→∞ with the density | Ψ out | 2 , where Ψ out is the outgoing asymptote of the scattering part of the wave function. 1 Introduction Bohmian mechanics =-=[6, 3, 8, 9, 10]-=- is a theory of particles in motion that is experimentally equivalent to quantum mechanics whenever the latter makes unambiguous predictions [9]. While Bohmian trajectories are in general highly non-N... |

277 |
A Suggested Interpretation of the Quantum Theory in Terms of “Hidden Variables
- Bohm
(Show Context)
Citation Context ...ists almost surely and is randomly distributed t→∞ with the density | Ψ out | 2 , where Ψ out is the outgoing asymptote of the scattering part of the wave function. 1 Introduction Bohmian mechanics =-=[6, 3, 8, 9, 10]-=- is a theory of particles in motion that is experimentally equivalent to quantum mechanics whenever the latter makes unambiguous predictions [9]. While Bohmian trajectories are in general highly non-N... |

268 | Methods of Modern Mathematical Physics. II: Fourier Analysis, Selfadjointness - Reed, Simon - 1975 |

116 | Quantum equilibrium and the origin of absolute uncertainty
- Dürr, Goldstein, et al.
- 1992
(Show Context)
Citation Context ...ists almost surely and is randomly distributed t→∞ with the density | Ψ out | 2 , where Ψ out is the outgoing asymptote of the scattering part of the wave function. 1 Introduction Bohmian mechanics =-=[6, 3, 8, 9, 10]-=- is a theory of particles in motion that is experimentally equivalent to quantum mechanics whenever the latter makes unambiguous predictions [9]. While Bohmian trajectories are in general highly non-N... |

112 |
Spectral Properties of Schrödinger Operators and Time-Decay of theWaveFunctions.DukeMath
- Jensen, Kato
(Show Context)
Citation Context ...ccurrence of a zero eigenvalue or resonance is an exceptional event: For Hamiltonians H(c) = H0 + cV the set of parameters c ∈ R, for which zero is an eigenvalue or a resonance, is discrete (see e.g. =-=[17]-=- p.589). 6(iii) For PΨ0-almost all Bohmian trajectories the asymptotic velocity is given by v∞, i.e. for all ε > 0 there exists some T > 0 and some C < ∞ such that P Ψ0 ({ q0 ∈ R 3 | ∣ ∣v Ψ (Q(q0, t)... |

73 |
The Wk, p Continuity of Wave Operators for Schrödinger Operators III, Even Dimensional Case m 4
- Yajima
- 1995
(Show Context)
Citation Context ...al projection of H to the finite energy interval [E1, E2] 5 Zero is a resonance of H if there exists a solution f of Hf = 0 such that 〈·〉 −γ f ∈ L 2 (R 3 ) for any γ > 1 2 but not for γ = 0 (see e.g. =-=[21]-=- p.552). The occurrence of a zero eigenvalue or resonance is an exceptional event: For Hamiltonians H(c) = H0 + cV the set of parameters c ∈ R, for which zero is an eigenvalue or a resonance, is discr... |

32 | Schrödinger operators in the twentieth century
- Simon
(Show Context)
Citation Context ...e assumption. Indeed there is a huge amount of literature on the exponential decay of eigenfunctions of Schrödinger operators, although results for the gradient of eigenfunctions are rather rare (see =-=[17, 18]-=- for an overview). We wish to recall here two results on eigenfunctions u ∈ L 2 (R 3 ), i.e. solutions of Hu = Eu with H as above and E < 0 6 . (i) There exist R > 0 and C < ∞ such that sup t∈R for al... |

31 | Quantum equilibrium and the role of operators as observables in quantum theory
- Dürr, Goldstein, et al.
(Show Context)
Citation Context |

30 |
Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory
- Ikebe
- 1960
(Show Context)
Citation Context ... and R0 > 0 such that |V (q)| ≤ C0〈q〉 −n−ε0 for all |q| ≥ R0. Here 〈q〉 := ( 1 + q 1 2) 2 . Clearly the wave operators W± := s − lim t→±∞ eiHt e −iH0t exist 4 and are asymptotically complete (see e.g. =-=[13]-=-). W± are called asymptotically complete if their range fulfills RanW± = Hc(H) = Hac(H), where Hc(H) resp. Hac(H) denotes the spectral subspace of L2 (R3) that belongs to the continuous resp. the abso... |

24 |
Bohmsche Mechanik als Grundlage der Quantenmechanik
- Dürr
- 2001
(Show Context)
Citation Context |

24 |
Bounds on exponential decay of eigenfunctions of Schrödinger operators
- Agmon
- 1984
(Show Context)
Citation Context ...e wish to recall here two results on eigenfunctions u ∈ L 2 (R 3 ), i.e. solutions of Hu = Eu with H as above and E < 0 6 . (i) There exist R > 0 and C < ∞ such that sup t∈R for all |q| ≥ R (see e.g. =-=[1]-=-). |e −iHt u(q)| = sup |e t∈R −iEt u(q)| = |u(q)| ≤ C|q| −1 e −|E|1 2 |q| (ii) If in addition to the above V ∈ K (1) 3 (where we use the notation of [17], p. 467), i.e. if the singularities of V are n... |

22 | On the global existence of Bohmian mechanics
- Berndl, Dürr, et al.
- 1995
(Show Context)
Citation Context ...g (part of the) wave function, i.e. of C. For a discussion of how this comes about see [11]. 4.2 Proof of Theorem 1 and Corollary 1 Proof of Theorem 1. (i) is a direct consequence of Corollary 3.2 in =-=[4]-=- resp. Corollary 4 in [20]. For technical reasons we continue with the proof of (iii). For δ1 > 0, δ2 > 0 and b > a > 0 we define the sets Bδ1ab := { k ∈ R 3 | | Ψ out 0 (k)| > δ1 ∧ a < |k| < b } (2... |

16 |
Scattering Theory by the Enss Method
- Perry
- 1983
(Show Context)
Citation Context ...Ψ this is not a trivial result. However, it is clear heuristically. 0 ‖2 ) of the wave function (bound trajectories). Since the Bohmian equation of motion (2) is On the one hand it is known (see e.g. =-=[15, 14]-=-) that the spatial support of the bound part Ψ pp t of the wave function stays concentrated around the origin (the scattering center) for all times t. of the wave function essentially moves out to inf... |

14 | Münch-Bernandl,K.: The flux across surfaces theorem for short range potentials and wave functions without energy cutoffs
- Teufel, Dürr
- 1999
(Show Context)
Citation Context ...ome R > 0 such that for all T > 0 there exists some CT < ∞ such that |ϕ3(q, t)| ≤ |∇ϕ3(q, t)| ≤ CT |q|(t + |q|) CT |q|(t + |q|) ∀|q| > 0, ∀|q| > R (16a) (16b) for all t ≥ T. The proof can be found in =-=[19]-=-. We give more detailed information in the appendix. Lemma 3. Let H = H0 +V with V ∈ (V )4 and let zero be neither an eigenvalue nor a resonance of H. Let Ψ0 ∈ C. From the (freely evolving) outgoing a... |

13 | Flux-across-surfaces theorem for a Dirac-particle - Dürr, Pickl - 2003 |

13 | Simple proof for global existence of Bohmian trajectories
- Teufel, Tumulka
(Show Context)
Citation Context ...tion, i.e. of C. For a discussion of how this comes about see [11]. 4.2 Proof of Theorem 1 and Corollary 1 Proof of Theorem 1. (i) is a direct consequence of Corollary 3.2 in [4] resp. Corollary 4 in =-=[20]-=-. For technical reasons we continue with the proof of (iii). For δ1 > 0, δ2 > 0 and b > a > 0 we define the sets Bδ1ab := { k ∈ R 3 | | Ψ out 0 (k)| > δ1 ∧ a < |k| < b } (20) and inner subsets there... |

9 | Seven Steps Towards the Classical World
- Allori, Dürr, et al.
- 2002
(Show Context)
Citation Context ...more general, non-scattering situations? When do Bohmian trajectories look like classical ones in general? We consider this question to be the key question of the classical limit in Bohmian mechanics =-=[2]-=-. It is our conviction that the methods developed in this article are naturally fit to give mathematically rigorous results also in the general case and plan to use them to just that end in the future... |

8 | Teta: The flux-across-surfaces theorem for a point interaction Hamiltonian - Panati, A - 2001 |

7 | The Flux-Across-Surfaces Theorem under conditions on the scattering state
- Duerr, Moser, et al.
- 2006
(Show Context)
Citation Context ...1 , |κ∂ η kg(k)| ≤ C〈k〉−1 , |η| = 2 , ∣ ∂ ∂|k| g(k)∣∣ −5 ≤ C〈k〉 , ∣ ∂2 ∂|k| 2 g(k)∣∣ −2 ≤ C〈k〉 , and η is a multi-index. Ψ0 ∈ C ⇒ Ψ out 0 ∈ Ĉ. 9The proof of Lemma 1 is analog to that of Lemma 3 in =-=[11]-=-. For the proof of both Theorem 1 and Theorem 2 we need (pointwise) estimates on ( how fast the 3 − scattering (part of the) wave function tends to the local plane wave ϕ1 = (it) 2 exp i q2 ) ( ) Ψ o... |

7 |
D.: Scattering into Cones I
- Dollard
- 1969
(Show Context)
Citation Context ...asymptote Ψ out t lim t→∞ ‖Ψac t − Ψ out t ‖ = lim ‖e t→∞ −iHt Ψ ac 0 − e −iH0t Ψ out 0 ‖ = = lim ‖Ψ0 − e t→∞ iHt e −iH0t Ψ out 0 ‖ = ‖Ψac The estimation of the second term is also standard (see e.g. =-=[7, 8]-=-). ‖ϕ2(·, t)‖ = ‖(2πt) −3 ∫ 2 q2 i and (e 2t − 1)Ψout 0 L1(R3 ), so R n ·2 i = ‖(e 2t − 1)Ψ out 0 ‖ by dominated convergence. 0 = W −1 + Ψac t and get out − W+Ψ0 ‖ = 0. · −i e t ·y y2 i (e 2t − 1)Ψ ou... |

6 |
Streutheorie aus der Sicht Bohmscher Mechanik
- Daumer
- 1994
(Show Context)
Citation Context ...quivariance (Subsection 2.1) and list some results of potential scattering 2 A paraphrase of his results for Bohmian Mechanics and an appraisal from the viewpoint of Bohmian Mechanics can be found in =-=[10]-=- p. 48. 3~t 1 1+γ ~t Figure 2: Splitting of the Bohmian trajectories made by Ψt = Ψac t +Ψpp t for large times t. Scattering trajectories stay outside some ball with radius growing linear in time (∼ ... |

5 |
Zur Existenz der Dynamik
- Berndl
- 1994
(Show Context)
Citation Context ...T ) at least once and outwards in [T, ∞). Therefore PΨ0 ( Aγ(a, T) ) is bounded from above by the probability that some trajectory crosses SRa,T (t) in any direction in [T, ∞). In Subsection 2.3.2 of =-=[5]-=- Berndl invoked the probabilistic meaning of the quantum probability current density JΨ = ( jΨ , |Ψt| 2) ( ( ) := ∗ Im Ψt ∇Ψt , |Ψt| 2) to prove that the expected number of crossings of a smooth surfa... |

4 | On the flux-across-surfaces theorem for short-range potentials - Nagao |

4 |
A remark on bound states in potential-scattering theory, Il Nuovo Cimento 61 A
- Ruelle
- 1969
(Show Context)
Citation Context ...Ψ this is not a trivial result. However, it is clear heuristically. 0 ‖2 ) of the wave function (bound trajectories). Since the Bohmian equation of motion (2) is On the one hand it is known (see e.g. =-=[15, 14]-=-) that the spatial support of the bound part Ψ pp t of the wave function stays concentrated around the origin (the scattering center) for all times t. of the wave function essentially moves out to inf... |

3 | On the role of the flux in scattering theory - Dürr, Teufel - 2001 |

3 |
The pathwise description of quantum scattering in stochastic mechanics, Stochastic processes in classical and quantum systems
- Carlen
- 1985
(Show Context)
Citation Context ...of establishing what intuitively seems clear, that asymptotically particles move freely on straight lines (for short range potentials), has been addressed before by Shucker [20], Biler [6] and Carlen =-=[8, 9]-=- for stochastic mechanics. Although Shucker proved results for V ≡ 0 only and from those results one cannot infer the existence of an asymptotic velocity, steps in his proof are also useful for our ca... |

2 | A microscopic derivation of the scattering cross section - Dürr, Goldstein, et al. |

2 |
S.: Stochastic mechanics of systems with zero potential
- Shucker
- 1980
(Show Context)
Citation Context ...in the future. The problem of establishing what intuitively seems clear, that asymptotically particles move freely on straight lines (for short range potentials), has been addressed before by Shucker =-=[16]-=- for stochastic mechanics. Although he proved results for V ≡ 0 only, steps in his proof are also useful for our case. Ψ pp t Ψ ac t ~t Figure 1: Splitting of the support of Ψt = Ψ ac t + Ψpp t for la... |

2 |
Potential Scattering in Stochastic Mechanics
- Carlen
- 1985
(Show Context)
Citation Context ...of establishing what intuitively seems clear, that asymptotically particles move freely on straight lines (for short range potentials), has been addressed before by Shucker [20], Biler [6] and Carlen =-=[8, 9]-=- for stochastic mechanics. Although Shucker proved results for V ≡ 0 only and from those results one cannot infer the existence of an asymptotic velocity, steps in his proof are also useful for our ca... |

1 |
Spectral properties of Schödinger operators and time-decay of the wave functions
- Jensen, Kato
- 1979
(Show Context)
Citation Context ...ccurrence of a zero eigenvalue or resonance is an exceptional event: For Hamiltonians H(c) = H0 + cV the set of parameters c ∈ R, for which zero is an eigenvalue or a resonance, is discrete (see e.g. =-=[13]-=- p.589). 6(iii) For PΨ0-almost all Bohmian trajectories the asymptotic velocity is given by v∞, i.e. for all ε > 0 there exists some T > 0 and some C < ∞ such that P Ψ0 ({ q0 ∈ R 3 | ∣ ∣v Ψ (Q(q0, t)... |

1 | Derivation of the scattering cross section in a limit procedure - Moser - 2002 |

1 |
Stochastic interpretation of potential scattering in quantum mechanics
- Biler
- 1984
(Show Context)
Citation Context ...e. The problem of establishing what intuitively seems clear, that asymptotically particles move freely on straight lines (for short range potentials), has been addressed before by Shucker [20], Biler =-=[6]-=- and Carlen [8, 9] for stochastic mechanics. Although Shucker proved results for V ≡ 0 only and from those results one cannot infer the existence of an asymptotic velocity, steps in his proof are also... |