Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0445 Acta Polytechnica 62(4):445–450, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague MEASUREMENT OF A QUANTUM PARTICLE POSITION AT TWO DISTANT LOCATIONS: A MODEL Jaroslav Dittrich Czech Academy of Sciences, Nuclear Physics Institute, 250 68 Řež, Czech Republic correspondence: dittrich@ujf.cas.cz Abstract. A simplified one-dimensional model of measurement of the position of a quantum particle by two distant detectors is considered. Detectors are modelled by quantum particles bounded in potential wells with just two bound states, prepared in the excited states. Their de-excitation due to the short range interaction with the measured particle is the signal for the detection. In the approximations of short time or small coupling between the particle and the measuring apparatuses, the simultaneous detection of the particle by both detectors is suppressed. The results extend to other models with two-level detectors. Keywords: Position measurement, EPR paradox, quantum measurement. 1. Introduction The theory of measurement is one of the topics in quantum mechanics from its early days [1]. The development can be found, for example, in reviews [2], [3], [4], [5], [6]. Recently, a comparison of an experimental measurement on a trapped 3-level ion by an interaction with a photon environment with the ideal quantum theoretical measurement was done in [7]. Measurements in quantum field theory with non-relativistic quantum-mechanical detectors coupled to the quantum field were studied in [8]. The famous Einstein-Podolsky-Rosen (EPR) [9] paradox concerns the relation of the measurement results performed on distant parts of a quantum system. Later, the subject evolved to a whole branch of the quantum physics, studies of the quantum entanglements and their applications in quantum informatics (see, e.g., Chapter 16 in [10] for the overview, also extracted in [11], review [12] with an extensive list of references, Chapters 4 and 6 in [13], Chapter 9 in [14], or [15]). For a brief review of the physical background up to 2005 see [16]. Consequences for the black-hole physics are discussed in [17]. Effects in several microscopic and statistical systems were recently studied in [18], [19], [20], [21], [22]. Further possible applications in quantum information are proposed in [23], [24]. The collapse of quantum state due to the measurement was, again, considered in [25], the entanglement in a system chaotic in the classical limit was studied in [26]. Although mostly entanglement of spin states of the two particles is considered, the general idea concerns any simultaneous measurements at distant locations with correlated results. The EPR paradox in the case of a particle position measurement means a simultaneous measurement of the particle presence at two distant places. As the particle can be detected at one place only, a sort of superluminal interaction between the two places seems to be necessary. Another possible interpretation could be that the simultaneous detection at the two places is excluded (or perhaps only strongly suppressed) by the quantum evolution of the complete system, consisting of the measured particle and both detectors (and perhaps even their environment), starting from a given initial state. We try to support such an idea by a simple, perhaps not very realistic, model. We obtain suppression of the simultaneous detection at the two places in an approximation but not exactly. The description of the quantum measurement as the unitary evolution of the complete system was considered already by von Neumann in his classical book [1] (called as process 2 in Section V.1). There exist a number of attempts to model macroscopic measuring apparatuses as quantum objects, let us mention only a sample here. In [1] (Sect. VI.3), [27] (Sect. 139 and Appendix XIV), [28, 29] and essentially also in [30] (Chapter 22), the apparatus is modelled by one very massive body. More close to the quantum description of a macroscopic system are the models of the measuring apparatus as a many-body system, composed of infinite or very large finite number of particles, typically a spin chain - see [31], [32], [33] (Chapter 7). We consider a rather simple model described in the next section. It consists of a particle interacting with two mutually distant detectors. However, we do not attempt to describe the detector as a macroscopic apparatus. We model it by a single quantum particle bounded in a potential well only. 445 https://doi.org/10.14311/AP.2022.62.0445 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en Jaroslav Dittrich Acta Polytechnica 2. The model We consider one light particle and two detectors on a line. Each detector consists of a very massive particle bounded in the square potential well of a width 2a and depth V0 tuned to the existence of just two bound states of energies E1 > E0. The detector is prepared for the measurement in the excited state E1. The detection of the light particle at the detector is modelled by the passing of the detector to the ground state E0 due to the interaction with the light particle. This is the determination of the particle position before the measurement as it gains energy E1 − E0 and is kicked away. The two detectors are located with centres of their potential wells at points −R, R with R ≫ a (see Figure 1). E0 E1 E0 E1 −V0 −V0 −R − a −R + a R − a R + a Figure 1. Potential wells of the two detectors. Remember that they bind two different massive particles (it is not a double-well potential but two different one-well potentials). Let us write the Hamiltonian of the whole three-particle system. Let us denote the potential well V (x) = −V0χ(−a,a)(x) = { −V0 for − a < x < a 0 for x ≤ −a or x ≥ a , (1) where χ(−a,a) is the characteristic function of the interval (−a,a) for definiteness although the detailed shape of the potential V is not really used. Denoting r ∈ R the coordinate of the measured light particle (mass m), x ∈ R the coordinate of the heavy particle in the left detector (mass M) and y ∈ R that in the right detector (mass M), the free Hamiltonian (without an interaction of the measured particle and detectors) reads H0 = − ℏ2 2m ∂2r − ℏ2 2M ∂2x + V (x + R) − ℏ2 2M ∂2y + V (y − R) , (2) understood as a self-adjoint operator in the state space L2(R3) with the domain H2(R3). Let us assume that the measured particle and the particle of the detector interact when their distance is shorter than a with the model interaction Hamiltonian (g is an interaction constant) HI = gχ(−a,a)(r − x) + gχ(−a,a)(r − y) . (3) For simplicity, the interaction potential between the measured particle and the heavy particle in each detector is chosen, again, in the form of a square well (or barrier). Its width 2a is chosen the same as for the potentials binding heavy particles in the detectors, which can be interpreted as the detector size (in the order of magnitude, at the least). The complete Hamiltonian H = H0 + HI . (4) We assume that the detector with Hamiltonian − ℏ2 2M ∂2x + V (x) (5) has just two bound states ψ0, ψ1 with energies E0 < E1. The detectors are prepared for measurement in the excited states ψ1, and the measured particle in a state φ0 so the initial wave function of the complete system is Ψ(0,r,x,y) = φ0(r)ψ1(x + R)ψ1(y − R) . (6) Let us assume that the energy of the light particle is insufficient to release the detector particles from their potential wells and that their states from the continuous spectrum can be neglected in the time evolution. In other words, we assume the wave function of the complete system in the form Ψ(t,r,x,y) = c00(t,r)ψ0(x + R)ψ0(y − R) + c01(t,r)ψ0(x + R)ψ1(y − R) +c10(t,r)ψ1(x + R)ψ0(y − R) + c11(t,r)ψ1(x + R)ψ1(y − R) . (7) 446 vol. 62 no. 4/2022 Measurement of a quantum particle position at two distant locations. . . Inserting into the Schrödinger equation, iℏ∂tΨ = HΨ , (8) and projecting onto ψj (x + R)ψk(y − R) (j,k = 0, 1), we obtain iℏ∂tcjk(t,r) = − ℏ2 2m ∂2r cjk(t,r) + (Ej + Ek)cjk(t,r) +g 1∑ l=0 (fjl(r + R)clk(t,r) + fkl(r − R)cjl(t,r)) , (9) where fjk(z) = ∫ z+a z−a ψj (x)ψk(x) dx (10) come from the matrix elements of interaction term HI (3) and the finite integration range here is a consequence of the finite range of interaction. As eigenfunctions ψj are exponentially decaying for |x| > a, the same is true for fjk at |z| > 2a. In the matrix notation C(t,r) = ( c00(t,r) c01(t,r) c10(t,r) c11(t,r) ) , E = ( E0 0 0 E1 ) , (11) F(r) = ( f00(r + R) f01(r + R) f10(r + R) f11(r + R) ) , (12) G(r) = ( f00(r − R) f10(r − R) f01(r − R) f11(r − R) ) , (13) the equations (9) read iℏ∂tC(t,r) = − ℏ2 2m ∂2r C(t,r) + (E + gF(r))C(t,r) + C(t,r)(E + gG(r)) . (14) The initial condition (6) gives C(0,r) = ( 0 0 0 φ0(r) ) . (15) Matrices F(r) and G(r) are Hermitian and bounded in r ∈ R according to their construction. So it is easily seen that the operator on the right-hand side of (14) is self-adjoint in L2(R,C4) with the domain H2(R,C4) and the solution C(t,r) exists there for every C(0, ·) ∈ H2(R,C4). The probability of finding the left detector in the state ψj and the right detector in the state ψk at the time t is calculated by (7) Pjk(t) = ∫ R |cjk(t,r)|2 dr , (16) i.e., P11 is the probability that no detector reacts (P11(0) = 1 as detectors are initially prepared in the excited state ψ1), P01 the probability of detection by the left detector only, P10 the probability of detection be the right detector only, and P00 the probability of simultaneous detection by both detectors. 3. Short-time evolution In the approximation C(t,r) = C(0,r) + t∂tC(0,r) + O(t2) , (17) C(t,r) =( 0 −igtℏ F01(r)φ0(r) −igtℏ G10(r)φ0(r) i ℏt 2m∂ 2 r φ0(r) + (1 − i t ℏ (2E1 + gF11(r) + gG11(r)))φ0(r) ) , (18) 447 Jaroslav Dittrich Acta Polytechnica assuming at the least that φ0 belongs to the domain of ∂2r . Therefore, the probability P00(t) of the simultaneous detection by the both detectors, indicated by their de-excitations to the ground states, remains zero in the first nontrivial approximation as t → 0, P00(t) = ∫ R |c00(t,r)|2 dr = O(t4) , (19) in comparison with the probabilities of detection by just one detector P01(t), P10(t), which are of the order of O(t2). This is seen from (18), looking by which powers of t the expansions of cjk(t,r) start. This result is an indication that quantum mechanics inherently prefers the detection of a particle at one place only. However, one should not take it more seriously than the model can approximate macroscopic measuring apparatuses. For the larger times, P00(t) surely is not zero, which also corresponds to the physical picture that the particle interacts with one detector and is bounced to the other. 4. Weak coupling approximation In this section, we derive a result similar to the one of the previous section but in the first perturbation approximation according to the coupling constant g between the light particle and the measuring instruments. The procedure follows the standard approach of passing to the interaction picture and then iteration of the integral equivalent of the Schrödinger equation. Let us start from equation (14) and denote as Hm the free Hamiltonian of the light particle, i.e., Hm = − ℏ2 2m ∂2r , (20) where the right hand side is understood as the corresponding self-adjoint operator in L2(R). We transform C(t,r) = ( e− i ℏ Ete− i ℏ HmtC1(t, ·)e− i ℏ Et ) (r) . (21) Here C1(t, ·) denotes the matrix function of r with the value C1(t,r) at the point r (t is just a parameter). The operator e− i ℏ Hmt transforms it to another function C2(t, ·), which is then multiplied by two exponentials depending on E and t only. Finally, the value of the resulting function at the point r, i.e., e− i ℏ EtC2(t,r)e− i ℏ Et, is taken. A similar notation is used in a few following formulas. Inserting (21) into (14), we obtain ∂tC1(t,r) = − ig ℏ (A(t)C1(t, ·))(r) , (22) where the operator A(t) acts as A(t)T = e i ℏ Hmt ( F̃(t)(e− i ℏ HmtT) + (e− i ℏ HmtT)G̃(t) ) (23) on a matrix function T ∈ L2(R,C4) and F̃(t), G̃(t) are multiplications by the matrix functions F̃(t,r) = e i ℏ EtF(r)e− i ℏ Et , G̃(t,r) = e− i ℏ EtG(r)e i ℏ Et . (24) The integral form of (22) reads C1(t,r) = C1(0,r) − ig ℏ ∫ t 0 (A(τ)C1(τ, ·)) (r) dτ . (25) Let us iterate this equation starting with C1(0,r) as the first approximation. Since the operator A(τ) is uniformly bounded in τ, the resulting perturbation series is convergent to the unique solution. Keeping only terms linear in g, C1(t,r) = C1(0,r) − ig ℏ ∫ t 0 (A(τ)C1(0, ·)) (r) dτ + O(g2) . (26) The initial value C1(0,r) = C(0,r) is given in (15) and (26) reads as C1(t,r) = ( 0 gu1(t,r) gu2(t,r) φ0(r) + gu3(t,r) ) + O(g2) , (27) 448 vol. 62 no. 4/2022 Measurement of a quantum particle position at two distant locations. . . where u1(t,r) = − i ℏ ∫ t 0 ( e i ℏ (E0−E1)τ e i ℏ Hmτ f (+) 01 e − iℏ Hmτ φ0 ) (r) dτ , (28) u2(t,r) = − i ℏ ∫ t 0 ( e i ℏ (E0−E1)τ e i ℏ Hmτ f (−) 01 e − iℏ Hmτ φ0 ) (r) dτ , (29) u3(t,r) = − i ℏ ∫ t 0 ( e i ℏ Hmτ (f(+)11 + f (−) 11 )e − iℏ Hmτ φ0 ) (r) dτ , (30) denoting f(±)jk (r) = fjk(r ± R) for j,k = 0, 1. The free propagator e − iℏ Hmτ can be expressed by the wellknown integral formula here (e.g., Eq. (IX.31) in [34]). Passing back to C(t,r), C(t,r) = ( 0 ge− i ℏ (E0+E1)te− i ℏ Hmtu1 ge− i ℏ (E0+E1)te− i ℏ Hmtu2 e −2 iℏ E1te− i ℏ Hmt(φ0 + gu3) ) + O(g2), (31) and the probability of the simultaneous detection by both detectors remains zero in the first nontrivial approximation as g → 0, P00(t) = O(g4) , (32) in comparison with P01(t) and P10(t) which are of the order of O(g2). 5. Concluding remarks The obtained results may be interpreted as a support of the idea that the detection of a particle at one place only is a consequence of the initial conditions for the complete system (particle plus detectors) without any superluminal interaction between two distant places. However, as they are obtained in the approximation of a short time or weak coupling only, it may be questioned whether they do not represent just a kind of initial state stability. A possible objection against this explanation is that C00 is more stable than C01 and C10. Another discussible point is the modelling of a macroscopic detector by a single quantum particle. We tacitly assume that the particles in detectors are heavy and thus have a large action. It was used only in heuristic justification of the assumption that their states remains in the two-dimensional subspace spanned by the vectors ψ0 and ψ1, in other words, the detectors are essentially two-level systems. The calculation for a more realistic models of the detectors and without approximations of small times or weak coupling would be very desirable but seems to be extremely difficult. A very specific model was considered for the definiteness but the main results (19) and (32) hold for any detectors modelled as two-level systems and any interaction separated to the sum of two parts corresponding to the particle interaction with each detector instead of (3). The form of the initial particle state φ0 is also not important here. 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