aa r X i v : . [ qu a n t - ph ] F e b Hydrogen atom as a quantum-classical hybrid system
Fei Zhan , and Biao Wu International Center for Quantum Materials, Peking University, 100871, Beijing, China Centre for Engineered Quantum Systems, School of Mathematics and Physics, TheUniversity of Queensland, St Lucia QLD 4072, AustraliaE-mail: [email protected]
Abstract.
Hydrogen atom is studied as a quantum-classical hybrid system, where the protonis treated as a classical object while the electron is regarded as a quantum object. We use awell known mean-field approach to describe this hybrid hydrogen atom; the resulting dynamicsfor the electron and the proton is compared to their full quantum dynamics. The electrondynamics in the hybrid description is found to be only marginally different from its full quantumcounterpart. The situation is very different for the proton: in the hybrid description, the protonbehaves like a free particle; in the fully quantum description, the wave packet center of the protonorbits around the center of mass. Furthermore, we find that the failure to describe the protondynamics properly can be regarded as a manifestation of the fact that there is no conservationof momentum in the mean-field hybrid approach. We expect that such a failure is a commonfeature for all existing approaches for quantum-classical hybrid systems of Born-Oppenheimertype.
1. Introduction
Though quantum mechanics and its ensuing developments have been verified by all theexperiments to be the correct description of the physical universe, classical physics is still a useful,convenient, and sometimes irreplaceable tool to describe many systems. For example, it is veryconvenient and also accurate to describe the motion of the Earth around the Sun with Newton’sequations of motion instead of the Schr¨odinger equation. There are even various situations,where one finds it convenient and necessary to separate one system into two subsystems: onesubsystem is described quantum mechanically and the other is described classically. This givesrise to quantum-classical hybrid systems.Quantum-classical hybrid systems come in three main categories: ( i ) In unifying gravitywith the other fundamental interactions, one simply gives up quantizing gravity and proposesa unification theory where gravity is treated as a classical field and the other three forces asquantum fields [1, 2, 3]. ( ii ) In the standard Copenhagen interpretation of quantum mechanics,one always deals with a hybrid system, where a quantum system interacts with a classicalmeasuring apparatus. In such a case, the classical apparatus can cause the collapse of thewave function of the quantum system, which is still beyond mathematical description [4]. ( iii )The hybrid system in this category is typified by systems in the field of solid state physics andchemistry [5, 6, 8]. In the Born-Oppenheimer approximation [9], the nuclei are treated classicallybecause their motion is much slower due to their large masses while the electrons are treatedas quantum objects. This kind of systems are now also arising in nano-science [10, 11], wherea classical detector of tens of nanometers in size interacts with a quantum object. For theseystems, the interaction from the classical subsystem does not cause dramatic changes in thequantum subsystem such as the collapse of the wave function. For systems in this category, wecall them Born-Oppenheimer systems.We are only interested in the Born-Oppenheimer hybrid systems. There have been manydifferent approaches proposed to describe these hybrid systems[5, 6, 7, 8, 12, 13, 14, 15, 16, 17,18, 19, 20, 21, 22]; some of these approaches are shown to be equivalent to each other [23]. Thereare discussions on what conditions need to be satisfied by a self-consistent theory for a hybridsystem[24, 25, 26, 27, 28]. In this work, we use hydrogen atom as a concrete example to explorethe possible inconsistency or inadequacy of a theory for hybrid systems. As is well known, thefull quantum mechanical description of hydrogen atom, the simplest Born-Oppenheimer systemin nature, can be found analytically. This full quantum description can be used to benchmarkthe hybrid description.We use the mean-field approach that was proposed in Ref. [11] to describe hydrogen atom.Its theoretical structure was fully analyzed in Ref. [27]. With this approach, we find that theelectron dynamics is very similar to the corresponding full quantum description. However, wefind that the proton in the hybrid approach behaves like a free particle, very different from itsfull quantum description where the wave packet center of the proton orbits around the center ofmass. This failure to describe the proton motion properly is rooted in the fact that there is noconservation of momentum in the hybrid approach. Our further analysis shows that this failureor inadequacy in the hybrid approach is intrinsic: it is caused by the loss of entanglement of theelectron and proton dynamics in the mean-field hybrid approach. As this loss of entanglementexists in all known hybrid approaches, we expect that all the hybrid approaches fail to describeproperly the dynamics of the classical subsystem.
2. Full quantum solution for hydrogen atom
Before we study the hydrogen atom as a hybrid system, we briefly review its general full quantumsolution and then apply it to a special case, where the electron is described by a wave packetthat moves and evolves around a circle [29].The Schr¨odinger equation for a hydrogen atom can be written as i ¯ h ∂∂t ψ ( r e , r p , t ) = (cid:20) − ¯ h m e ∇ r e − ¯ h m p ∇ r p + V ( | r e − r p | ) (cid:21) ψ ( r e , r p , t ) , (1)where m p ( r p ) is the mass (coordinate) of the proton, m e ( r e ) is the mass (coordinate) of theelectron, and V ( | r e − r p | ) is the Coulomb potential that depends only on the distance betweenthe proton and the electron. As done in every textbook, we introduce the relative coordinate r and the coordinate of the center of mass R , r = r e − r p , R = m e r e + m p r p M , (2)where M = m e + m p is the total mass of the hydrogen atom. In these new coordinates, theSchr¨odinger equation (1) becomes i ¯ h ∂∂t ψ ( R , r , t ) = (cid:20) − ¯ h M ∇ R − ¯ h µ ∇ r + V ( | r | ) (cid:21) ψ ( R , r , t ) , (3)where µ = m e m p / ( m e + m p ) is the reduced mass. So, the wave function of a hydrogen atomcan be written as ψ ( R , r , t ) = ψ r ( r , t ) ψ c ( R , t ) , (4)here ψ r ( r , t ) and ψ c ( R , t ) satisfy, respectively, i ¯ h ∂∂t ψ r ( r , t ) = − ¯ h µ ∇ r ψ r ( r , t ) + V ( | r | ) ψ r ( r , t ) , (5) i ¯ h ∂∂t ψ c ( R , t ) = − ¯ h M ∇ R ψ c ( R , t ) . (6)This shows that the motion of a hydrogen atom can be separated into two independent parts: ψ r ( r , t ) describes the motion of a particle in a Coulomb potential V ( | r | ) while ψ c ( R , t ) describesthe motion of a free particle. However, this by no means implies that the motions of the electronand the proton can be described by two independent wave functions, which will become muchclearer in the later analysis.We go back to the coordinate system of r e and r p . The density distribution of the electronis | ψ e ( r e , t ) | = Z | ψ r ( r , t ) | | ψ c ( R , t ) | d r p = Z | ψ r ( r e − r p , t ) | | ψ c ( m e r e + m p r p M , t ) | d r p = Z | ψ r ( r e − r p , t ) | | ψ c ( r e − m p M ( r e − r p ) , t ) | d ( r p − r e )= Z | ψ r ( x , t ) | | ψ c ( r e − m p M x , t ) | d x , (7)where we have plugged in Eq. (2). Similarly, the proton density is given by | ψ e ( r p , t ) | = Z | ψ r ( x , t ) | | ψ c ( r p − m e M x , t ) | d x . (8)The above results are very illuminating. As the center of mass motion is free particle-like, weassume that ψ c ( R , t ) is a Gaussian wave packet with width σ . So, the density of electron is justthe density of the relative motion | ψ r ( r , t ) | coarse-grained with a Gaussian function of width M σ/m p ≈ σ . In contrast, the density of proton is the density of the relative motion | ψ r ( r , t ) | coarse-grained with a Gaussian function of a much larger width M σ/m e ≈ σ . In otherwords, the electron density and the proton density are very similar to each other but the protondensity looks about 1837 times fuzzier.In the reference frame where the hydrogen atom is motionless, the center of the wave packetof electron is h r e i = m p M Z | ψ r ( r , t ) | r d r = m p M h r i , (9)where h r i is the center of the wave packet for the relative motion. The center of the wave packetof proton is h r p i = − m e M h r i . (10)We now consider a special case. In this case, the initial state of the center of mass of thehydrogen atom is described by a Gaussian function ψ c ( R , t = 0) = 1(2 πσ ) / exp (cid:20) − | R | σ (cid:21) (11)ith σ being the width; the relative motion at the beginning is depicted by the following wavefunction [29], ψ r ( r , t = 0) = 1(2 πσ n ) / ∞ X n =1 exp (cid:20) − ( n − ¯ n ) σ n (cid:21) u n ( n − n − ( r ) , (12)where ¯ n and σ ¯ n are the mean and the width of the Gaussian distribution, respectively, and u nlm is the standard energy-eigenstate for the hydrogen atom[30] u nlm = s(cid:18) na B (cid:19) ( n − l − n [( n + 1)!] e − r/na B (cid:18) rna B (cid:19) l L l +1 n − l − (cid:18) rna B (cid:19) Y ml ( θ, φ ) (13)with L being associated Laguerre polynomial, Y the spherical harmonics, and a B the Bohrradius.The ensuing dynamics of the circular wave packet in Eq.(12) under the influence of theCoulomb potential V ( | r | ) has been studied in detail in Ref.[29]. For the sake of self-containment,we summarize their results here. The wave packet is localized on a circle with radius ∼ ¯ n a B .This wave packet remains localized and moves on the circle for several T Kepler , the period of thecorresponding classical motion on the same circle. At time T spread ∼ T Kepler , the spreadingof the wave packet becomes so severe that it distributes rather uniformly on the circle. Thisis characterized by h r i = 0. The wave packet can recover its localized form and revive at T rev = (¯ n/ T Kepler , and repeats its previous dynamics afterwards. With this in mind, it isstraightforward to picture the quantum dynamic motion for both the electron and the protonin this special case.For the electron, its wave packet is the wave packet in Eq.(12) coarse-grained with theGaussian wave packet for the center of mass motion (see Eq.(7)). As long as the width σ is not too large, its wave packet dynamics should be very similar: it is localized and orbits on acircle of radius ∼ ¯ n a B before T spread . During this period, the wave packet center h r e i oscillatesperiodically with its amplitude decreasing. After T spread and before T rev , the wave packet spreadsover the circle, which is characterized by h r e i = 0. This dynamics repeats itself after T rev .For the proton, its wave packet center has a similar motion as the electron’s according toEqs.(9,10). The difference is that the proton moves in the opposite direction and h r p i varieswith time on a circle of much smaller radius. However, the dynamics of the proton wave packetis very different. The wave packet of the proton is the result of coarse-graining with a Gaussianfunction of large width M σ/m e ≈ σ (see Eq.(8)). This large-size coarse-graining makesthe wave packet much less localized. At the same time, the proton moves on a much smallercircular orbit (about 1837 times smaller). With these two factors combined, it is clear that thewave packet of the proton is always spread and smeared out over the entire circular orbit. Nodistinct peak and other structure can be seen. The three different time scales of the wave packetdynamics, T Kepler , T spread , and T rev , which are used to characterize the distinct features of thewave packet at different times, become rather meaningless for the proton.
3. Hybrid dynamics in hydrogen atom
We now treat the hydrogen atom as a hybrid system, where the proton is regarded as a classicalobject while the electron is treated quantum mechanically. With the approach in Ref.[11], thehybrid Hamiltonian for the hydrogen atom is H = h ϕ e ( r e , t ) | − ¯ h m e ∇ r e + V ( | r e − r p | ) | ϕ e ( r e , t ) i + p p m p . (14)e expand the wave function | ϕ e i in a set of complete orthonormal basis, | ϕ e ( r e , t ) i = X j ϕ j ( t ) | j i . (15)With the classical canonical Hamiltonian structure introduced by Heslot[31], we have thefollowing Poisson brackets, { ϕ j , ϕ ∗ k } = iδ jk / ¯ h, { r pj , p pk } = δ jk , (16) { ϕ j , ϕ k } = { r pj , r pk } = { p pj , p pk } = 0 , (17)where r pj ( p pj ) is the j th component of coordinate (momentum) vector of the proton. With thesePoisson brackets, we can obtain the hybrid equations of motion, i ¯ h ddt | ϕ e ( r e , t ) i = (cid:20) − ¯ h m e ∇ r e + V ( r e − r p ) (cid:21) | ϕ e ( r e , t ) i , (18)˙ r p = ∂H∂ p p = p p m p , (19)˙ p p = − ∂H∂ r p = −∇ r p h ϕ e ( r e , t ) | − ¯ h m e ∇ r e + V ( r e − r p ) | ϕ e ( r e , t ) i . (20)Note that p p and r p are independent dynamical variables in the above equations of motionwhile r e is just some external parameter. The dynamical variables for the electrons are ϕ j ’s inEq.(15).Similar to the full quantum treatment, we focus on the case where the initial state for theelectron is given by the circular wave packet in Eq.(12). In terms of the electron coordinate r e ,the circular wave packet has the following form, ϕ e ( r e , t = 0) = 1(2 πσ n ) / ∞ X n =1 exp (cid:2) − ( n − ¯ n ) σ n (cid:3) u n ( n − n − ( r e − r p ) , (21)where r p is the initial position of the proton. Since the proton is much more massive thanthe electron, its motion is rather slow and will not cause quantum transition between differentquantum states of the electron. In other words, the weight before each eigenstate u n ( n − n − willnot be changed by the proton motion. This is just the famous quantum adiabatic theorem[32] orthe essence of the Born-Oppenheimer approximation. As a result, the dynamics of the electronwave packet is just the same as the electron dynamics in the full quantum description. The onlydifference is that the wave function of the electron in this hybrid approach is not coarse-grained.In contrast, the proton is very different: the classical motion of a proton in the hybridtreatment does not agree with the motion of the wave packet center h r p i for any meaningfulperiod of time. Let us examine the right hand side of Eq.(20). E e = h ϕ e | − ¯ h m e ∇ r e + V ( r e − r p ) | ϕ e i is the electron energy. It is clear that the eigen-energy E nlm for each eigenstate u nlm isindependent of the proton position r p . As mentioned above, the proton motion is so slow that itwill not cause quantum transition between different electronic states u nlm . This means that E e is independent of r p . As a result, the right hand side of Eq.(20) is zero and the momentum of theproton does not change with time. So, the proton in the hybrid dynamics is like a free-particle,motionless or moves along a straight line. In contrast, the wave packet center h r p i makes circularmotion around the center of mass before time T spread . This shows that the hybrid approach failsto describe the proton dynamics properly.his failure to describe the proton dynamics can be illustrated from a different angle. Ouranalysis shows that this failure essentially has the root in the fact that there is no conservationof momentum in the hybrid dynamics of hydrogen atom. We define the total momentum forthis system as P = p p + h ϕ e | ˆ p e | ϕ e i . (22)In the adiabatic limit, we know that p p is a constant while h ϕ e | ˆ p e | ϕ e i changes with timesignificantly at least for the first several T Kepler . This means that the total momentum P changes with time. The easiest way to appreciate this result to set the initial momentum p p of the proton be zero. Then according to Eqs.(18,19,20), the proton remains motionless whilethe electron wave packet moves on the circle and its change of momentum d h ϕ e | ˆ p e | ϕ e i /dt has afinite value for the first several T Kepler .
4. Self-consistency of hybrid approaches
The above discussion leads to immediate questions, such as, “How general is the conclusion thatthe hybrid approach is inadequate in describing the dynamics of the classical subsystem?”, “Isit possible that hybrid approaches are intrinsically flawed?” There is already some attempt toanswer these questions [28]. Again we do not do general analysis here; we instead use hydrogenatom as an example and hope that our analysis with hydrogen atom may shed some light onthe general questions.First, the analysis on hydrogen atom can be generalized to multi-nucleus systems. Theelectronic states clearly do not depend on the position of the center of mass of all the nuclei.When the nuclear center of mass moves slowly as in the usual case, it will not cause quantumtransition between different electronic states. As a result, the center of mass of all the nuclei feelsno force and there is no conservation of the total momentum. It seems that this shortcoming canbe remedied by moving the derivative in Eq.(20) inside the bra-ket, that is, replacing Eq.(20)with ˙ p p = − ∂H∂ r p = −h ϕ e ( r e , t ) |∇ r p (cid:20) − ¯ h m e ∇ r e + V ( r e − r p ) (cid:21) | ϕ e ( r e , t ) i . (23)This is nothing but the well-known Ehrenfest equation[33, 34]. It is easy to show that the totalmomentum is conserved if this Ehrenfest equation is used instead of Eq.(20). As the Ehrenfestequation is derived from Eq.(20) with an argument that appears right but is fundamentally flawedupon close examination[34], we now have a very intriguing situation: the total momentum isnot conserved for the “correct” Eq.(20) while the total momentum is conserved for the “flawed”Eq.(23). This may be regarded as an indication that there exists intrinsic inconsistency in ahybrid theory.Secondly, there is loss of quantum entanglement in any hybrid theory. In the full classicaltreatment, a pair of independent dynamical variables { r e ( t ) , r p ( t ) } , one for the electron andthe other for the proton, are enough to specify completely the dynamics of the system. In thehybrid approach, there also exists such a dynamical pair { r p ( t ) , ϕ e ( r e , t ) } , for the proton andthe electron, respectively, which can completely describe the whole dynamics.In contrast, in the full quantum mechanical approach, there exists no such a dynamical pair,one for the electron and the other for the proton, with which the full quantum dynamics iscompletely determined. It is true according to Eq.(4) that the full quantum dynamics can bespecified by two independent wave functions. However, one wave function is for the relativemotion and the other is for the center of mass, instead of for the electron and the proton,respectively. The motion of the electron and the proton is always entangled together. Thisentanglement between the electron and the proton is manifested at two levels. For the firstlevel, it is the entanglement understood in the usual sense that one can not have the totalwave function as the product of the electron wave function and the proton wave function, ( t ) = ψ e ( r e , t ) ψ p ( r p , t ). Maybe temporarily at a given moment, one can have this kind of directproduct but not for any meaningful period of time. For the second level, which is much weaker,the total wave function can not be the functional of two independent dynamical functions, onefor the electron f e ( r e , t ) and the other for the proton f p ( r p , t ). In other words, we can neverwrite the total wave function as ψ ( t ) = ψ [ f e ( r e , t ) , f p ( r p , t )]. To see this clearly, we introducethe reduced wave functions for the electron and the proton. Similar to the densities, they arealso coarse-grained from the wave function for the relative motion, e ψ e ( r e , t ) = Z ψ r ( r , t ) ψ c ( R , t ) d r p = Z ψ r ( x , t ) ψ c ( r e − m p M x , t ) d x , (24) e ψ p ( r p , t ) = Z ψ r ( r , t ) ψ c ( R , t ) d r e = Z ψ r ( x , t ) ψ c ( r p − m e M x , t ) d x . (25)It is clear that these two functions contain all the dynamical information that we can have forthe electron and the proton. However, as they are obtained from the coarse-graining, these tworeduced wave functions alone can not specify the total wave function ψ ( t ). The entanglementbetween the electron and the proton at the second level is immediately destroyed in the hybridapproach, where the dynamics of the whole system can be determined completely by the protondynamics r p and the electron dynamics ϕ e ( r e , t ). Although our analysis is done for hydrogenatom, it can be generalized. In any hybrid approach, there is no entanglement between thequantum subsystem and the classical subsystem at the second level. This loss of entanglementis general and intrinsic.
5. Conclusion
In summary, we have studied the dynamics in a hydrogen atom with two different approaches.One is the full quantum theory that can be found in the standard textbook and the other is thehybrid mean-field approach, where the proton is regarded as a classical object and the electron isa quantum object. We have found that there is only marginal difference for the dynamics of theelectron between these two approaches. However, the dynamics for the proton is very differentbetween the two: the proton in the hybrid approach behaves like a free particle, not moving ormoving along a straight line; the wave packet center for the proton in the full quantum approachcan make circular motion for a limited time. These differences are summarized in Table 1.full quantum approach hybrid approachelectron dynamics coarse-grained ψ r ( r , t ) ψ r ( r , t )proton dynamics h r p i moves around the center of mass behaves as a free particleconservation of the total momentum no conservation of the total momentum Table 1.
Comparison between the full quantum dynamics and the hybrid dynamics of hydrogenatomThe failure of proper description of the proton dynamics is the manifestation that there is noconservation of total momentum in the hybrid approach. We acknowledge that this failure mayjust be to the interest of theorists as there is no experimental way to probe the proton dynamics.For systems with more than one nuclei, the non-physical artifacts caused by the hybrid treatmentmay also be not essential to experiments. The reason is that the hybrid approach may only fail tocapture the proper motion for the center of mass of the nuclei. For the relative motions betweenthe nuclei, the hybrid approach may be adequate. All the vibrational and rotational frequenciesfor the relative nuclear motion are computed by treating the nuclei as classical objects. If thisapproximation can not capture physics that can be measured experimentally, it would be foundand known for a long time. cknowledgments
We thank Yinhan Zhang and Shaoqi Zhu for useful discussions. B.W. is supported by theNational Basic Research Program of MOST (NBRP) of China (2012CB921300, 2013CB921900),the National Natural Science Foundation (NSF) of China (11274024), the Research Fund for theDoctoral Program of Higher Education (RFDP) of China (20110001110091).
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