Dimensional analysis and the correspondence between classical and quantum uncertainty
DDimensional analysis and the correspondencebetween classical and quantum uncertainty
V Gattus and S Karamitsos School of Physics and Astronomy, University of Manchester,Manchester M13 9PL, United KingdomE-mail: [email protected] , [email protected] May 2020
Abstract.
Heisenberg’s uncertainty principle is often cited as an example of a“purely quantum” relation with no analogue in the classical limit where (cid:126) → (cid:126) is a dimensionful constant, we mayalways work in natural units in which (cid:126) = 1. Dimensional analysis teaches usthat all physical laws can be expressed purely in terms of dimensionless quantities.This indicates that the existence of a dimensionally consistent constraint on ∆ x ∆ p requires the existence of a dimensionful parameter with units of action, and thatany definition of the classical limit must be formulated in terms of dimensionlessquantities (such as quantum numbers). Therefore, bounds on classical uncertainty(formulated in terms of statistical ensembles) can only be written in termsof dimensionful scales of the system under consideration, and can be readilycompared to their quantum counterparts after being non-dimensionalized. Wecompare the uncertainty of certain coupled classical systems and their quantumcounterparts (such as harmonic oscillators and particles in a box), and show thatthey converge in the classical limit. We find that since these systems featureadditional dimensionful scales, the uncertainty bounds are dependent on multipledimensionless parameters, in accordance with dimensional considerations. Keywords : classical limit, dimensional analysis, quantum uncertainty
1. Introduction
Historically, the development of quantum mechanics benefited greatly from theintuition offered by the Lagrangian and Hamiltonian formulations of classicalmechanics. Perhaps the most famous example of this was Schr¨odinger’s derivationof his eponymous equation. Realizing that microscopic particles had a fundamentally“undulatory” (wave-like) nature, Schr¨odinger observed that constant action surfaces ‡ ,being periodic, would play a major role in developing a robust theory of quantummechanics [1]. It is at this point in which dimensional analysis played a crucial rolein Schr¨odinger’s reasoning: indeed, since the action is a dimensionful quantity, it canonly appear as the argument of a periodic function if it is divided by a quantity that ‡ Here, “action” refers to W = (cid:82) Ldt , where L is the Lagrangian. a r X i v : . [ qu a n t - ph ] F e b imensional analysis and quantum-classical correspondence h , which can be converted into a phase if divided by 2 π , leadingto the appearance of the reduced Planck constant (cid:126) ≡ h/ π .Schr¨odinger’s archetypal quantization procedure was later formalized and refinedas the deformation from Poisson brackets to quantum commutators and the promotionof classical expressions to quantum operators for classical systems. It is thereforenatural to ask whether “purely quantum” phenomena can be formulated in a classicalmanner. The question has been approached from multiple angles, and, while itmay be futile to attempt to build a purely classical theory that perfectly replicatesthe observations of quantum mechanics, we at least expect quantum mechanics toreproduce classical predictions corresponding to our everyday experience. This is thecrux of the correspondence principle [2–5].The correspondence principle is often expressed as the idea that quantummechanics must reduce to classical mechanics in some limit, usually referred to asthe classical limit . Stated like this, the correspondence principle is little more thantautological unless we are very careful to expand on how the classical limit is formallydefined. In most undergraduate textbooks, the correspondence principle is formalizedin terms of Ehrenfest’s theorem, which relates the evolution of the expectation valuesof operators to their commutator with the Hamiltonian. However, even Ehrenfest’stheorem alone does not bridge the conceptual gap between quantum and classicalmechanics [6]. This is only possible if we assign a classical meaning to the expectationvalues themselves. In order to do this, consider a quantum system evolving so rapidlythat we can only measure its mean state over many cycles. Hence, the classical limitmay be identified as the limit in which energies (or quantum numbers) are so large thatthe law of large numbers is unquestionably in effect and our macroscopic measurementsare, in effect, averages of a very large number of quantum measurements.While Ehrenfest’s theorem provides a formal way of moving from the quantumrealm of uncertainty to the classical realm of certainty, it is helpful to approach thequestion from the opposite angle and ask how uncertainty may arise in classicalsystems. Of course, for a single system with a fixed, known state, the notion ofan expectation value of a quantity is no different from the quantity itself. However,since quantum mechanics is an inherently probabilistic theory, it makes more senseto formulate classical uncertainty by way of an ensemble of systems, each with adifferent statistical weight. In this framework, the classical probability distributionis the analogue of the quantum mechanical probability density ρ = | ψ | . Therefore,in this way, the statistical average of the ensemble may be identified as the classicalanalogue of the quantum average of the system in question.We turn our attention to the well-known Heisenberg uncertainty principle, apurely quantum mechanical limit on the simultaneous localization of a particle’senergy and momentum ∆ x ∆ p ≥ (cid:126) /
2. A na¨ıve treatment of the classical limitmight lead us to expect that no uncertainty exists in the classical case. However, thiswould be a mistake: the Heisenberg uncertainty principle presupposes the existenceof a fundamental unit of action (i.e. the Planck constant h ), something which doesnot exist in the classical realm. Therefore, a meaningful comparison between thetwo domains can only be achieved by working with non-dimensionalized quantities,such as the quantum numbers, recognized by Schr¨odinger as corresponding to Bohr’sstationary energy levels of the elliptic orbits in the case of the atom. Schr¨odinger evenobliquely refers to the correspondence principle at this stage, noting that even though imensional analysis and quantum-classical correspondence n → ∞ , its quantum natureis restricted to a few angstroms, i.e. for relatively small n [1].The idea that a comparison of the quantum and classical realm must be madethrough the use of pure numbers is by no means exclusive to the correspondenceprinciple: it is a result of the Buckingham- π theorem [7], which states that every lawof physics must be expressible in a non-dimensionalized form. A careful treatmentof the harmonic oscillator, infinite square well, and other single-particle systems withsimple one-dimensional potentials [8–15] reveals that, once position and momentumare properly non-dimensionalized (with the help of x and p , which are referencescales inherent to the system), then it is possible to arrive at a similar limit for theensemble uncertainty, given by ∆¯ x ∆¯ p ≥ c , where c is a pure number.A natural question to ask at this stage is how the relation between quantumand classical uncertainty is modified for interacting one dimensional systems withadditional dimensionful scales (such as coupled systems). Our approach is to defineappropriate dimensionless variables, and seek to obtain the uncertainty relations interms of their dimensionless ratios. In particular, we expect that adding more degreesof freedom to the system will yield more dimensionless ratios in the expression for theuncertainty relations, in accordance with the Buckingham- π theorem.The outline of the paper is as follows: in Section 2, we briefly review dimensionalanalysis and consider the implications of the Buckingham- π theorem for the classicaland quantum uncertainty principle. In Section 3, we outline how the notion of classicaluncertainty may be formalized by providing an overview of the correspondence betweenclassical and quantum uncertainty for one-dimensional harmonic oscillators and theinfinite square well. We then go on to examine how the addition of further dimensionfulscales modifies the uncertainty relations while still respecting the correspondenceprinciple. In Sections 4 and 5, we study coupled harmonic oscillators, whereas inSection 6, we study two particles in a box with a contact potential. Finally, wediscuss our findings in the Conclusions.
2. Dimensional analysis and the classical limit
Physical dimensions are a concept related to but distinct from that of units. Unitsare related to standards (e.g. the length of a rod, the weight of a bearing, etc.) andcorrespond to the way in which the same measurements can be expressed: for instance,1 km and 1000 m both describe the same quantity. Dimensions, however, are inherentqualities of a physical quantity, related to how they are measured within a consistenttheoretical and experimental framework [16].Dimensions of physical quantities can be formalized as vectors living in a “physicaldimension” vector space. The dimension of a quantity Q in a system with three “base”dimensions can be expressed as[ Q ] = M α L β T γ , (1)where M , L , T correspond to the dimensions of mass, length, and time, respectively.Then, [ Q ] is a member of a vector space over R , where vector addition is representedby multiplying [ Q ] and vector multiplication is represented by raising [ Q ] to a power:[ Q a Q b ] = [ Q ] a [ Q ] b . (2)The triplet ( α, β, γ ) is a particular representation of [ Q ], dependent on the choiceof base dimensions. In a system of different fundamental units (and therefore imensional analysis and quantum-classical correspondence T − and therefore wouldquadruple if the definition of a meter were to be halved, whereas a dimensionlessquantity or “pure number” with dimension 1 would not transform. The usualnomenclature of vector spaces can be applied even further: the quantities Q i are saidto have linearly independent dimensions if there is no non-trivial way of constructinga dimensionless quantity out of them, or, equivalently, if (cid:34)(cid:89) i [ Q i ] α i (cid:35) = 1 (3)implies α i = 0 for all i .Dimensional analysis fundamentally rests upon the principle that the laws ofnature should not depend on our choice of units. This simple idea turns out tohave profound implications: it can be used to actually constrain the form of physicalrelations. This constraint is often expressed in terms of the Buckingham- π theorem [7].The simplest statement of this theorem is that a physically meaningful relationbetween quantities q i can always be non-dimensionalized, i.e. written in terms ofdimensionless quantities π j constructed out of q i . It is not difficult to see why thisis the case: since, by definition, only dimensionless quantities are invariant underchanges of units, then any invariant law must be constructed from dimensionlessquantities only. More specifically, if we have a set of m dimensionful quantities with n linearly independent dimensions, and k dimensionless quantities c j , then any non-dimensionalized law can be written as f ( π , . . . , π m − n , c , . . . c k ) = 0 . (4)It is not difficult to see that this relation encapsulates all possible relations thatremain invariant under a change of units. Rayleigh’s method of dimensional analysis(which is what “dimensional analysis” most often refers to in an undergraduatecontext), a procedure in which a system is analyzed by considering all the dimensionfulindependent variables that might influence a dependent variable, is a particularapplication of the Buckingham- π theorem.The Buckingham- π theorem, as expressed through (4), gives us a profound insighton what exactly it means to examine a physical relation in a particular limit. Weobserve that it is meaningless to talk about the limit of a dimensionful quantity: itonly makes sense to consider the limit of the dimensionless quantities π j or c j . Thisis an important distinction, because once we have decided to include a dimensionfulquantity as part of our dimensional analysis, we are considering a different system.Consider, for example, special relativity. Formally (and na¨ıvely) taking the limit c → ∞ in hopes of recovering classical mechanics can lead to unexpected results, suchas infinite rest mass mc for particles. This is because the very concept of “rest mass”presupposes the existence of a finite speed limit, which does not vanish simply bytaking this limit to infinity.For the reasons outlined above, studying the correspondence between classicaland quantum mechanics is more nuanced than it might originally appear. A commonstatement of the correspondence principle is due to Dirac [17], who regarded classicalmechanics as the limiting case of quantum mechanics when the (reduced) Planck imensional analysis and quantum-classical correspondence (cid:126) → x ∆ p > n → ∞ , where n are the quantumnumbers of the system in question) is more precise, and the two statements are notnecessarily equivalent [4].In order to illustrate how the correspondence limit of uncertainty bounds can beviewed through the lens of dimensional analysis, we consider a particle with a one-dimensional trajectory. Immediately, from dimensional analysis, we can see that thereis no way to impose a limit on ∆ x and ∆ p : there are no dimensionless ratios that canbe derived from these quantities. We need to introduce at least one additional scalewith dimensions of action (ML T − ). In quantum mechanics, such a scale alreadyexists: (cid:126) denotes the scale at which quantum effects are relevant. In the classicalensemble, however, any meaningful uncertainty limit will necessarily be expressed interms of some physical scale(s) of the system.We now consider a particle in the quantum realm with characteristic action A (defined as the integral of momentum over length) along with the quantum unit ofaction (cid:126) . We wish to arrive at a relation between ∆ x and ∆ p , and so the dimensionlessquantities are given by π = (cid:126) A , π = (cid:126) ∆ x ∆ p , (5)or, alternatively π = (cid:126) A , π = A ∆ x ∆ p . (6)Any of these two choices is valid; after all, the Buckingham- π theorem does not tellus which choice of dimensionless ratios is more “physically meaningful”. However, itis clear that if the limit (cid:126) /A → π by the quantum number n , since both tell us howprominent quantum effects are. The uncertainty relation is therefore going to satisfy f (cid:18) ∆ x ∆ pA , (cid:126) A (cid:19) = 0 . (7)As a result, the uncertainty equation in the classical limit can be recovered by setting (cid:126) /A → (cid:126) , theuncertainty relation is simply g (cid:18) ∆ x ∆ pA (cid:19) = 0 , (8)and by the correspondence principle, we expectlim (cid:126) /A → f (cid:18) ∆ x ∆ pA , (cid:126) A (cid:19) = g (cid:18) ∆ x ∆ pA (cid:19) . (9)In this limiting case, the minimum uncertainty is necessarily given by g (∆ x ∆ p/A ) = 0,and as such, the uncertainty principle is given by∆ x ∆ pA > c , (10) imensional analysis and quantum-classical correspondence c is some dimensionless parameter, the value of which cannot be determinedvia dimensional analysis. However, we observe that na¨ıvely setting (cid:126) = 0 does notnecessarily recover the classical limit (cid:126) /A →
0; after all, there is no guarantee thatthe function f is continuous.For a simple system of a single particle with energy E and mass m in a box oflength L , we find that the unit of action A = L √ mE . However, in general, we expectthat the uncertainty principle is going to acquire the following form:∆ x ∆ pA > f ( π j ) , (11)where the π j are constructed out of the characteristic action scales A i of the systemand A is some weighted average of the A i . This is as far as dimensional analysiscan take us. We cannot hope of arriving at the precise form of A . Importantly,the above discussion presupposes that the notion of “uncertainty” can exist in theclassical realm. While a truly random variable cannot exist in classical mechanics,chance may be introduced to classical mechanics via ensembles , as discussed in thefollowing section.
3. Classical probability and uncertainty
At first glance, it may seem that there is no room for probability in classical mechanics.Laplace envisioned a being, “Laplace’s demon”, which “at a certain moment wouldknow all forces that set nature in motion”. He argued that for such a being, “nothingwould be uncertain and the future just like the past would be present before itseyes” [19]. However, even in such a perfectly deterministic world, there is room for thenotion of probability. Indeed, the proper classical limit of quantum mechanics (and theonly avenue by which we can describe “classical probabilty” in any meaningful sense) isclassical statistical mechanics rather than classical mechanics [3,6], which necessitatesthe use of the statistical ensemble if we are to extend the notion of uncertainty intothe classical realm.If a single-particle system is viewed as a member of an ensemble of bound-stateparticles, the position of the particle is no longer a trajectory, but rather a distribution.We may view this distribution either in the frequentist sense of a random selection ofone of the members of the ensemble, or in the Bayesian sense of “parametrizing ourignorance” of the particle’s initial conditions. The distribution for the position of aparticle in a classical ensemble is derived by observing that the particle spends a time dt = dx/ | v ( x ) | within the interval dx , where v ( x ) is its speed. Assuming we observethe particle at some unspecified time, the classical probability density for positionmeasurements of a particle uniformly selected from such an ensemble is defined as ρ ( x ) = N (cid:112) m [ E − V ( x )] , (12)where N is a normalization constant, E is the energy of the bound state and V ( x ) thepotential function. This confirms our intuition that the particle is found more often inregions where its speed is low. For periodic systems, N = 2 /τ where τ is the period.From this definition of classical probability density, it follows that the average ofa classical operator A can be defined as an average over one period τ : (cid:104) A (cid:105) CL = (cid:82) τ A ( t ) dt (cid:82) τ dt . (13) imensional analysis and quantum-classical correspondence A = (cid:104) A (cid:105) − (cid:104) A (cid:105) .Having put all preliminaries in place, we may turn our attention to thecorrespondence between the classical and quantum uncertainty in a few select systemsof interest. Single-particle systems have been examined thoroughly in the literature,and from a dimensional analysis perspective, two-body systems may be more robust.However, in order to set the scene and allow for a direct comparison with coupledsystems, we will cite here the one-particle results for the cases of harmonic oscillatorand infinite square well. As discussed in the previous section, it is illuminating toview uncertainty bounds through a non-dimensionalized system of equations, whichmotivates us to define the dimensionless position and momentum variables as follows x i = x i A i , p i = p i m i ω i A i , (14)where A i is the initial amplitude of the system, m i its mass and ω i is the frequencyof oscillation.With this notation, the uncertainty product for position and momentum in theclassical harmonic oscillator case is found to be∆ x ∆ p = 12 . (15)Similarly, for a single particle confined to move under a one-dimensional infinite squarewell potential, the uncertainty product is∆ x ∆ p = 1 √ . (16)This coincides with the quantum mechanical result in the limit of large principalquantum number n . For a full derivation of these uncertainty results, the reader isreferred to [20] and [21].For single-particle systems such as the harmonic oscillator and the infinite squarewell, we note that the uncertainty bound reduces to a pure number in the classicallimit. This is to be expected, since there are just enough dimensionful parameters inorder to define the dimensionless analogues of position and momentum. In the presenceof additional dimensionful parameters, however, we expect the uncertainty bounds todepend on residual dimensionless variables even in the limit of large quantum numbers.Coupled systems feature more dimensionful parameters, and for this reason we turnour attention to them in the next section.In concluding this section, we must stress that recasting quantum mechanics inthe classical phase space (effectively deducing its postulates on statistical grounds)is a much more involved procedure than assigning a classical distribution to aparticle ensemble as outlined above. This procedure usually involves “geometrizing”quantum mechanics by constructing the phase space of a classical system, whichis then endowed with a probability measure. Riccia and Wiener [22], for instance,use stochastic integrals in order to motivate Born’s rule, while Kibble [23] employssymplectic manifolds with a complex structure in order to recover quantum dynamics.Another geometric approach by Heslot [24] reveals (cid:126) to be the curvature of thespace of quantum states, indicating once again that there is a qualitative differencebetween the classical and quantum realms; the former features a state space whichhas a natural unit of distance, while the latter does not. The field of quantuminformation [25, 26] features powerful and sophisticated techniques that make thequantum-classical correspondence much more manifest. These approaches to the imensional analysis and quantum-classical correspondence
4. Two coupled harmonic oscillators with equal masses
We consider a simple coupled system consisting of two harmonic oscillators withspring constant k and equal masses m , coupled via a potential of the form V ( x , x ) = k (cid:48) ( x − x ) , where k (cid:54) = k (cid:48) in general, and x and x denote thedisplacements of the individual oscillators. For similar treatments of this set-up thereader is referred to [27–29]. We model the classical system with two blocks constrained to move in a one-dimensional frictionless surface. Each block is attached to an outer stationary wall bymeans of a spring with force constant k . The inner spring has force constant k (cid:48) . It isassumed that all springs assume their natural length when the system is at rest.This 1D problem can be easily approached utilizing the Lagrangian formalismand is part of the repertoire of any undergraduate dynamics course. The Lagrangianof the system reads L = 12 m ˙ x + 12 m ˙ x − k ( x + x ) − k (cid:48) ( x − x ) . (17)To decouple (17) and obtain the equations of motions, we perform a change of variablesin favour of the normal mode coordinates x c , r defined as linear combinations of theposition variables x , , i.e. x c = x + x √ , x r = x − x √ , (18)which are the centre of mass and relative distance respectively.The characteristic frequencies of the two normal modes of oscillation are givenby ω = k/m and ω = ( k + 2 k (cid:48) ) /m , corresponding to in phase and out of phaseoscillations respectively.The time averages of the normal modes coordinates for position and momentumare then found through (13) over a period τ i = 2 π/ω i with i = c , r : (cid:104) x i (cid:105) = 0 , (cid:104) x i (cid:105) = A i , (19) (cid:104) p i (cid:105) = 0 , (cid:104) p i (cid:105) = 12 m ω i A i . (20)It is now straightforward to change back to the position variables x , in order tofind the uncertainty relation in its conventional formulation as the product of theuncertainties ∆ x and ∆ p .The time averages for the position variable x can be readily obtained via of (19)and (20): (cid:104) x (cid:105) = √ (cid:104) x c + x r (cid:105) = 0 , (21) (cid:104) x (cid:105) = 12 (cid:104) ( x c + x r ) (cid:105) = 14 ( A + A ) . (22) imensional analysis and quantum-classical correspondence x :∆ x = (cid:113) (cid:104) x (cid:105) − (cid:104) x (cid:105) = (cid:112) A + A . (23)Using the definition, the time averages and variance of the momenta are found to be (cid:104) p (cid:105) = 0 , (cid:104) p (cid:105) = 14 m ( A ω + A ω ) , (24)∆ p = m (cid:112) A ω + A ω . (25)Repeating the previous steps for the second block yields identical results. The productof the variances is therefore∆ x ∆ p = m (cid:112) A + A (cid:112) A ω + A ω , (26)and similarly for ∆ x ∆ p . Using the scaled canonical variables defined in (14), (26)can be written as∆ x ∆ p = 14 (cid:115) A A (cid:115) A ω A ω . (27)where the variables have been scaled by A r and mA r ω r respectively. Since the system issymmetric under particle exchange, the expression for ∆ x ∆ p can be easily deducedfrom that of ∆ x ∆ p by swapping indices 1 and 2. We note that this expression,which represents the uncertainty relation in terms of dimensionless position andmomentum for a classical ensemble, comprises of two dimensionless ratios involvingthe amplitudes A c , r and frequencies ω c , r respectively, as expected.We may compare the uncertainty product of this system to that of the 1Duncoupled case of (15) by setting A c = A r and ω c = ω r . As such, we observethat the effect of the coupling is to add another dimensionful degree of freedom to thesystem, which results in the possibility for another dimensionless ratio to appear inthe non-dimensionalized expression for the uncertainty product. The quantum mechanical analogue of the system we have analyzed so far is that of twodistinguishable coupled one-dimensional harmonic oscillators. We treat the oscillatorsas distinguishable to allow for a direct comparison to the uncertainty product of (27)in the previous section. The system is then described by the following Hamiltonian:ˆ H = ˆ p + ˆ p m + 12 k ( x + x ) + 12 k (cid:48) ( x − x ) . (28)Using a variation of the Jacobi coordinates allows to reduce the complexity of theproblem by transforming it to what is essentially a one-dimensional problem in termsof the centre of mass and relative coordinate of (18) [30].After the change of variables, the Hamiltonian becomes separable:ˆ H = ˆ p m + 12 kx + ˆ p m + 12 ( k + 2 k (cid:48) ) x (29)= ˆ H c + ˆ H r . (30)It is therefore possible to determine the evolution of the system by viewing it as twosingle-particle harmonic oscillators with angular frequencies given by ω = k/m and imensional analysis and quantum-classical correspondence (a) (b) (c) Figure 1.
Uncoupled classical (dashed line) and quantum (solid line) probabilitydensities for the harmonic oscillator as functions of the centre of mass coordinate x c for (a) n c = 5, (b) n c = 10 and (c) n c = 20.(a) (b) (c) Figure 2.
Uncoupled classical (dashed line) and quantum (solid line) probabilitydensities for the harmonic oscillator as functions of the relative coordinate x r for(a) n r = 5, (b) n r = 10 and (c) n r = 20. ω = ( k + 2 k (cid:48) ) /m . Since the oscillators are treated as distinguishable, the totalwavefunction of the system is the product of the single-particle wavefunctionsΨ( x c , x r ) = φ n c ( x c ) φ n r ( x r ) , (31)where φ n ( x ) = (cid:18) √ πx n n ! (cid:19) / H n (cid:18) xx (cid:19) exp (cid:18) − x x (cid:19) , (32)with x = (cid:113) (cid:126) mω , and n r and n c are the principal quantum numbers associated withthe wavefunctions φ ( x r ) and φ ( x c ) respectively.Using the definition for quantum mechanical expectation values and making useof the orthogonality properties of the Hermite polynomials, one can calculate theexpectation values for x r and x c and the respective momentum operators for thegeneral wavefunction defined in (31) as (cid:104) x i (cid:105) = 0 , (cid:104) x i (cid:105) = (2 n i + 1) (cid:126) mω i , (33) (cid:104) p i (cid:105) = 0 , (cid:104) p i (cid:105) = (2 n i + 1) m (cid:126) ω i , (34)where i = c , r. We can now transform back to the original spatial variables x and x and obtain the expectation value for positions and momenta using the results (33)and (34) above: (cid:104) x (cid:105) = 0 , (cid:104) x (cid:105) = 12 (cid:20) (2 n r + 1) (cid:126) mω r + (2 n c + 1) (cid:126) mω c (cid:21) , (35) imensional analysis and quantum-classical correspondence (a) (b) (c) Figure 3.
Classical ρ CL ( x , x ) and quantum ρ QM = | ψ n ( x , x ) | probabilitydensities for coupled harmonic oscillators with equal masses vs the spatialvariables x and x for (a) n = 5, (b) n = 10, (c) n = 20. (cid:104) p (cid:105) = 0 , (cid:104) p (cid:105) = 12 (cid:20) m (cid:126) ω r (2 n r + 1)2 + m (cid:126) ω c (2 n c + 1)2 (cid:21) , (36)and similarly for the second oscillator. Hence, the quantum uncertainties in positionand momentum read∆ x = 12 (cid:115) (2 n r + 1) (cid:126) mω r + (2 n c + 1) (cid:126) mω c (37)= 12 (cid:113) A n c + A n r , (38)∆ p = 12 (cid:112) m (cid:126) ω r (2 n r + 1) + m (cid:126) ω c (2 n c + 1) (39)= m (cid:113) ω A n c + ω A n r , (40)respectively, where A n c , r are the classical turning points associated with the energy E n c , r of an harmonic oscillator with mass m and frequency ω c , r : A n c , r = (cid:115) E n c , r mω , r = (cid:115) (2 n c , r + 1) (cid:126) mω c , r n c , r . (41)In terms of the dimensionless variables x i and p i of (14) scaled by means of theamplitude A n c , r , the uncertainty product can be written as∆ x ∆ p = 14 (cid:115) A n c A n r (cid:115) A n c ω A n r ω , (42)or explicitly in terms of the principal quantum numbers n c , r and harmonic oscillatorfrequencies ω c , r as∆ x ∆ p = 14 (cid:115) n c + 1) ω r (2 n r + 1) ω c (cid:115) n c + 1) ω c (2 n r + 1) ω r . (43)We note that the last expression only involves dimensionless ratios of the twofrequencies of oscillation and the characteristic principal quantum numbers. It issuggestive of the correspondence principle that (42) is in complete agreement with imensional analysis and quantum-classical correspondence ρ CL ( x c , r ) be the classical probability density defined by ρ CL ( x ) = 1 π (cid:112) A n − x , (44)and valid in the range x ∈ ( − A n , A n ) where A n is defined in (41) [20, 21]. Theusual representation of 1D quantum and classical probability densities can be foundin the literature (see [21]); as such, we illustrate the coupled case. Figure 1 showsthe quantum probability density, ρ QM ( x c ), along with the classical analogue ρ CL ( x c )in terms of the centre of mass coordinate for various values of the principal quantumnumber n c . Figure 2 shows the probability densities for the same values of the principalquantum number n r as a function of the relative coordinate x r instead.Finally, let the total classical probability density be given by the product of theones associated to the individual modes of oscillation x c , r as ρ CL ( x , x ) = ρ CL ( x c ) ρ CL ( x r ) . (45)Figure 3 offers a comparison between the quantum probability density, ρ QM and theclassical counterpart ρ CL ( x , x ) which are plotted as functions of the original spatialvariables of the problem for value of n equal to 5, 10 and 20. Noticeably, the shape ofthe three dimensional classical probability density is that of a rectangular well withsoft edges. This is the natural and intuitive extension of the 1D distribution shown inFigures 1 and 2, which can be retrieved by slicing the three dimensional distributionsacross lines of x = 0 and x = 0.
5. Two coupled harmonic oscillators with different masses
Let us now consider a slight variation of the problem analyzed in the previous sectionfor which the oscillators have masses m and m with m (cid:54) = m and the potentialfunction is V ( x , x ) = k ( x − x ) . We expect that the results in this casewill have a similar form, even though the dimensionless ratios will differ. For similartreatments of this set-up the reader is referred to [27, 28]. Similarly to Section 4.1, the classical system consists of two blocks connected to threesprings. The motion of the blocks is described by the following Lagrangian L = 12 m ˙ x + 12 m ˙ x − ω ( m x + m x ) − k ( x − x ) , (46)where we assume the angular frequency ω to be the same for both oscillators.The equations of motion obtained from Lagrange’s equations can be decoupled byperforming a variable transformation to the Jacobi coordinates x r = x − x √ , x c = m x + m x M √ , (47)where M = ( m + m ) /
2. Proceeding with this transformation, (46) can be rewrittenas L = 12 M ˙ x + 12 µ ˙ x − M ω x − µω x , (48) imensional analysis and quantum-classical correspondence µ = ( m m ) /M is the reduced mass and ω = ω , ω = ω + 2 k/µ are the twoharmonic oscillator frequencies. The time averages of the relative and centre of masscoordinate are given once again by (19). We may then easily transform back to thespatial coordinates x and x by combining the expressions for x r and x c from (47),i.e. x = 1 √ (cid:16) x c + m M x r (cid:17) , x = 1 √ (cid:16) x c − m M x r (cid:17) . (49)Thus, using the results of (19), the averages for x are found to be (cid:104) x (cid:105) = 0 , (cid:104) x (cid:105) = 14 (cid:20) A + (cid:16) m M (cid:17) A (cid:21) , (50)with variance given by the following expression∆ x = 12 (cid:114) A + (cid:16) m M (cid:17) A . (51)The symmetric properties of the system allow us to easily deduce ∆ x by simplyswapping the indices 1 → x .Repeating the previous steps for the momenta allows to express p , in terms ofthe relative and centre of mass coordinate momenta defined as p r = µ ˙ x r and p c = M ˙ x c respectively. Hence, the time averages and variances of the momenta are (cid:104) p (cid:105) = 0 , (52) (cid:104) p (cid:105) = 14 (cid:20) µ ω A + (cid:16) m M (cid:17) M ω A (cid:21) , (53)∆ p = 12 (cid:114) µ ω A + (cid:16) m M (cid:17) M ω A , (54)for the first oscillator. Once more, the results for the second block are obtained byswapping the subscripts 1 → x i and p i defined in (14), enables us to find thedimensionless uncertainty product, ∆ x ∆ p , as∆ x ∆ p = 14 (cid:115) (cid:16) m M (cid:17) A A (cid:115)(cid:16) m M (cid:17) + µ ω A M ω A , (55)and ∆ x ∆ p = 14 (cid:115) (cid:16) m M (cid:17) A A (cid:115)(cid:16) m M (cid:17) + µ ω A M ω A , (56)for the first and second oscillator respectively.We note that these expressions are symmetric under particle exchange andcomprise of three dimensionless ratios involving the masses, the amplitudes and thefrequencies of the oscillators. The extra term involving the masses in (55) is whatsets it apart from the uncertainty product of the system of coupled oscillators withidentical masses described by (27), where only the other two dimensionless ratiosappeared. This may be surprising, since both systems have the same number offundamental dimensionful variables. The difference in the two expressions can beascribed to the choice of coordinates of (47) employed to decouple the equations ofmotion: since the expression for the centre of mass coordinate is weighted by themasses, this “artificially” introduces the extra term, ( m i /M ) with i = 1 , p i . Such a detail, however, is beyond the capabilities ofdimensional analysis to predict. imensional analysis and quantum-classical correspondence We now turn our attention to the quantum description of a system comprisingof two distinguishable coupled harmonic oscillators with masses m and m . TheHamiltonian of the system readsˆ H = ˆ p m + ˆ p m + 12 m ω x + 12 m ω x + 12 k ( x − x ) . (57)As with the classical counterpart, the Hamiltonian here becomes separable under theJacobi coordinate transformation of (47), i.e.ˆ H = − (cid:126) µ ∂ ∂x + 12 µω r x − (cid:126) M ∂ ∂x + 12 M ω c x (58)= ˆ H r + ˆ H c , (59)where the meaning of the variables is the same as in Section 5.1. The Hamiltonian (58)describes two single particle harmonic oscillators whose solutions are given by (32).As before, the total wavefunction of the system is the product of the single-particlewavefunctions and the expectation values for x r and x c and the respective momentumoperators are simply the one-particle results of (33) and (34). With this information,we may evaluate the expectation values of the original spatial variables of the problemand the respective momenta: (cid:104) x (cid:105) = 0 , (cid:104) x (cid:105) = 14 (cid:20) (2 n c + 1) (cid:126) M ω c + (cid:16) m M (cid:17) (2 n r + 1) (cid:126) µω r (cid:21) , (60) (cid:104) p (cid:105) = 0 , (cid:104) p (cid:105) = 14 (cid:20) (2 n r + 1) µ (cid:126) ω r + (cid:16) m M (cid:17) (2 n c + 1) M (cid:126) ω c (cid:21) , (61)and likewise for the second oscillator variables x and p .In accordance with definition (14), we introduce the variable A n to representthe classical turning point associated with the energy E n of the quantum mechanicalharmonic oscillator. A n is as defined in (41), where now m is either µ or M for A n r or A n c respectively. This allows to find an expression for the dimensionless uncertaintyproduct of position and momentum as∆ x ∆ p = 14 (cid:115) (cid:16) m M (cid:17) A n r A n c (cid:115)(cid:16) m M (cid:17) + µ ω A n r M ω A n c , (62)and ∆ x ∆ p = 14 (cid:115) (cid:16) m M (cid:17) A n r A n c (cid:115)(cid:16) m M (cid:17) + µ ω A n r M ω A n c , (63)for the first and second oscillator respectively, where both canonical variables havebeen scaled by A n c . To highlight the dependence on the principal quantum numbers,(62) and (63) can be written explicitly as∆ x ∆ p = 14 (cid:115) (cid:16) m M (cid:17) (2 n r + 1) M ω c (2 n c + 1) µω r (cid:115) (2 n r + 1) µω r (2 n c + 1) M ω c + (cid:16) m M (cid:17) . (64)Expectedly, the quantum uncertainty relations (62) and (63) match the classical onesgiven by (55) and (56). Furthermore, the considerations made regarding the overalleffect of coupling on the uncertainty product (see end of Section 4.1) and the differenceto the results obtained in Section 4 also apply to the expressions derived above. imensional analysis and quantum-classical correspondence (a) (b) (c) Figure 4.
Classical ρ CL ( x , x ) and quantum ρ QM = | ψ n ( x , x ) | probabilitydensities for coupled harmonic oscillators with different masses vs the spatialvariables x and x for values of n given by (a) n = 5, (b) n = 10, (c) n = 20. The graphical representations of the 1D classical and quantum probabilitydensities are analogous to the ones presented in Figures 1 and 2. Figure 4 offersa comparison between the quantum probability density, ρ QM = | Ψ n ( x , x ) | andthe classical counterpart ρ CL ( x , x ) which are plotted as functions of the originalvariables of the problem for various values of n . Notably, the shape of the classicaldistribution is that of a skewed rectangular well which is not diagonal as the one inFigure 3. It is evident however that the quantum and classical distributions convergein a locally averaged sense for high values of the principal quantum number n in bothcases.
6. Two particles in a box coupled via contact potential
The final set-up we consider is a system of two particles in an infinite potential wellwhich are allowed to interact via a contact potential of the form V ( x ) = (cid:26) λδ ( x − x ) 0 < x , x < L ∞ otherwise , (65)where L is the dimension of the box and the coefficient λ , which has dimensions ofML T − , determines the strength of the interaction. For similar treatments of thisset-up without uncertainty considerations, the reader is referred to [31–35] or to [36]for a treatment with uncertainty relations. Starting with the classical case, the Lagrangian of the system can be written as L = 12 m ˙ x + 12 m ˙ x − V, (66)where V is defined in (65) and, in general, m (cid:54) = m . Once again the equations ofmotion are decoupled via the Jacobi coordinates: x r = x − x , x c = m x + m x M , (67)where M = m + m is the total mass of the system of particles. Proceeding withthis variable transformation allows (66) to be written as L = 12 M ˙ x + 12 µ ˙ x − λδ ( x r ) , (68) imensional analysis and quantum-classical correspondence µ = ( m m ) /M is the reduced mass and the expression for L is valid withinthe spatial domain 0 < x , x < L . The equation of motion for the center of masscoordinate is trivial M ¨ x c = 0, as expected for a classical particle confined in aone-dimensional potential box.Let us now evaluate the classical averages relevant to this set-up. Since 0 Plot of cot ( k n r ) (solid lane) vs − /k n r (dashed line) for a box oflength L = 5 and for µλ/ (cid:126) = 1. k k cot ( k L ) Figure 6. Plot of the self-consistent equation (90) vs k n r for a box of length L = 5 within a range − < k n r < The latter is obtained by integrating the Schr¨odinger equation with Hamiltonianoperator specified by (80) over the small interval (0 − (cid:15), (cid:15) ) [37, 38].From [32, 33], we can see that for a particle confined in a box of length 2 L , i.e. − L < x r < L , with a δ -function potential at position x r = pL , where − < p < 1, thewavefunction can be written as ψ n r ( x r ) = (cid:26) A sin ( k n r ( x r + L )) ( − L ≤ x r ≤ pL ) B sin [ k n r ( x r − L )] ( pL ≤ x r ≤ L ) , (87)where k n r = √ µE r / (cid:126) . Moreover, the continuity condition (84) of the wave functionat x r = pL gives AB = sin [ k n r L ( p − k n r L ( p + 1)] . (88)If p = 0, this condition yields A = − B . Applying now the discontinuity conditionof (86) allows us to obtain a quantization relation for wavenumber k n r , i.e. k n r sin (2 k n r L ) = 2 µλ (cid:126) sin [ k n r L ( p − k n r L ( p + 1)] . (89) imensional analysis and quantum-classical correspondence p = 0 yields the following expression k n r cot ( k n r L ) = − µλ (cid:126) , (90)which is a self-consistent relation for k n r and cannot be solved to give a simpleanalytical answer for the wavenumber. This is shown in Figure 5 where the intersectionpoints between the curves y = cot ( k n r L ) and y = − /k n r represent the solutionsto (90). Clearly, these solutions are not spaced in a regular manner and a numericalapproach becomes necessary. Plotting (90) against k n r however gives us insight intothe way the wavefunction changes over the length of the box. For a box of length L = 5 and within a range − < k n r < 10, the behaviour of the function is shown inFigure 6.We note that for large values of the ordinate, the lines become parallel and equallyspaced, suggesting that in the high quantum number limit, the k n r values satisfying theself-consistent equation will repeat periodically. After normalizing the wavefunctions,the expectation values for the relative spatial coordinate, x r , and momentum areevaluated to be (cid:104) x r (cid:105) = 0 , (cid:104) x (cid:105) = 2 k n r L k n r L − k n r L ) − k n r , (91) (cid:104) p r (cid:105) = 0 , (cid:104) p (cid:105) = (cid:126) k n r = 2 µE n r . (92)The expression for (cid:104) x (cid:105) in (91) seems quite complicated at a first glance but reducesto the familiar L / k n r . This corresponds to the classical regime:lim k n r →∞ (cid:104) x (cid:105) → L . (93)Hence, the expectation values of x and p are found to be (cid:104) x (cid:105) = L , (cid:104) x (cid:105) = L (cid:18) − n c π (cid:19) + (cid:16) m M (cid:17) (cid:104) x (cid:105) , (94) (cid:104) p (cid:105) = 0 , (cid:104) p (cid:105) = 2 µE r + (cid:16) m M (cid:17) M E c , (95)and similarly for the second particle. Using the scaled variables of (77) we can thenevaluate the dimensionless uncertainties in position and momentum as∆ x = (cid:115) − n π + (cid:16) m M (cid:17) (cid:104) x (cid:105) L , (96)∆ p = (cid:115) (cid:16) m M (cid:17) M E c µE r . (97)Taking the limit for large principal quantum number n c and wavenumber k n r , we stepin the classical regime and obtain an equivalent result to (78), i.e.lim n c ,k n r →∞ ∆ x ∆ p = (cid:114) (cid:115) (cid:18) m M (cid:19) (cid:115) (cid:16) m M (cid:17) M E c µE r . (98)This is in complete agreement with the classical result of (78).As it is not possible to obtain a value for the wavenumber k n r in closed form,we will not offer a graphical comparison between quantum and classical probabilitydensities here. The reader is referred to [20] for a comparison of probability densitydistributions for the simple one-particle infinite square well. imensional analysis and quantum-classical correspondence 7. Conclusions Dimensional analysis is a powerful tool for deriving robust results for a host ofsituations, and the correspondence between the quantum and classical domains isno exception. The Buckingham- π theorem of dimensional analysis provides us with afundamental insight regarding the so-called classical limit: its definition cannot involvetaking the limit of a dimensionful quantity such as (cid:126) if it is to be physically meaningful.Even in cases where (cid:126) → n → ∞ , it is notpossible to recover classical mechanics by na¨ıvely setting (cid:126) = 0.There is a stark divide between a classical world where (cid:126) = 0 and a quantumworld viewed by an observer living in the classical limit. The classical probabilitydensity is not a limit of the wavefunction in the classical limit: indeed, the classicalprobability density can be recovered by “smearing out” the quantum probabilitydensity (by an amount that decreases as quantum numbers increase). In a classicalworld, wavefunctions do not exist, but in a quantum world, classical scales are thosein which the smearing required to match the two domains is small (i.e. wavenumbersare very large).In order to illustrate how the correspondence between classical and quantumrealm fares in the context of dimensional analysis, we focused on quantum uncertainty,popularly regarded as a phenomenon with no classical analogue. For simple systems,the role of (cid:126) in the classical limit is assumed by a characteristic action scale, as requiredby the Buckingham- π theorem. For systems with more dimensionful parameters,however, uncertainty limits will in general acquire a more complicated dependenceon dimensionless ratios. To demonstrate this, we examined two distinct systems: acoupled harmonic oscillator and a two-body particle-in-a-box set-up, and showed thatthe uncertainty bounds acquire additional dimensionless terms (and also converge inthe limit of large quantum numbers).Dimensional analysis lends support to the idea that any comparison betweenquantum and classical uncertainties is only possible if made in terms of dimensionlessquantities, since the classical realm lacks a fundamental unit of action (much likeNewtonian mechanics lacks a fundamental unit of speed). Quantum numbers are notsimply the discretized counterparts to the dimensionless ratios of the classical system;they arise precisely because of the existence of the fundamental unit (cid:126) . This highlightsthe fundamental conceptual distinction between quantum and classical mechanics: itis only through the tuning of dimensionless parameters that a bridge between the twodomains can be unambiguously built. This indicates that introducing uncertainty toa system is, in a sense, a one-way street. Uncertainty bounds can only exist alongsidea dimensionful parameter that acts as a “universal certainty limit” (just like relativityrequires a universal speed limit). Once such a parameter is added to a theory, thereis a well-defined limit in which the quantum world appears to be classical, but thereis no limit that will return us to a truly classical world. References [1] Schr¨odinger E 1926 An Undulatory Theory of the Mechanics of Atoms and Molecules Phys. Rev. Phys. Rev. A imensional analysis and quantum-classical correspondence [4] Liboff R L 1984 The correspondence principle revisited Phys. Today Phys. Rev. A Phys. Rev. A Phys. Rev. Am. J. Phys. Eur. J. Phys. Eur. J. Phys. Eur. J. Phys. Eur. J. Phys. Mol. Phys. Eur. J. Phys. Am. J. Phys. 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