Understanding the relation between classical and quantum mechanics: prospects for undergraduate teaching

Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.


Introduction
Both classical and quantum mechanics successfully describe the world within their specific range of validity.While the first one is a reliable description of phenomena at a macroscopic scale, the second one concerns the microscopic level and is characterized by many counterintuitive features.As a matter of fact, the relation between classical and quantum mechanics is often expressed in terms of Planck's correspondence principle [1], according to which classical physics is recovered when a dimensionless parameter containing Planck's constant h goes to zero.This formulation was originally set up in order to show how to get the classical Rayleigh-Jeans energy density for black body radiation from the quantum Planck formula.In fact, there exists another formulation of this relation, due to Bohr, in which the limit of large quantum numbers is taken [2].As a consequence of these limits, classical mechanics can be considered as an approximation of a more refined and general theory, which is thought to be fundamental: quantum mechanics.In principle this should point to a reduction relation between the two theories [3,4].However, the situation is much more involved, as claimed by some authors (see, for instance, Ref. [5] and references therein).While much interesting work on this issue has been done in the didactic literature (see e.g.[6][7][8][9][10][11][12][13] and references therein), in both courses and textbooks it is almost always simply claimed that classical mechanics can be recovered by performing the ℎ → 0 limit.This may instill and reinforce in most students the misconception that there is no problem whatsoever with that limit.In fact, this topic is part of the wide debate on the issue of possible relations between scientific theories within the philosophy of science.The semiclassical limit problem is also linked with important conceptual issues such as the emergence of the macroscopic from the quantum and the measurement problem [14], but here we shall focus only on the technical problem of the classical limit.
In this paper we show, by a critical analysis of some concrete examples, why establishing the relation between classical and quantum mechanics is a challenging task.Indeed, the limit operation is no more than a heuristic tool which in several cases gives valid results, while in other cases it does not work.Accordingly, undergraduate teaching should consider these features and moreover suggest a plurilinear view of the history of theoretical physics [15].In Section 2, we discuss the issue of undergraduate teaching of quantum mechanics in general and concerning the relation of quantum to classical mechanics in particular.The discussion of several critical issues underlying the relation between classical and quantum mechanics is given in Section 3, while in Section 4 implications and perspectives for undergraduate teaching are drawn.Finally, our conclusions end the paper.

Students' difficulties with quantum mechanics
Quantum mechanics is rightly considered as a difficult and counterintuitive subject.Recent literature in Physics Education (see e.g.[16][17][18][19][20][21][22] and references therein for as sample of the vast and growing dedicated literature) deals with the immensely challenging of teaching the basics of this subject in high school, where this need has arisen quite recently.However, the difficulties in making sense of this strange theory have long been present also at higher levels, such as undergraduate education [22][23][24][25].
Here, despite students are now proficient with the mathematical techniques needed for fully formulating the theory and doing all sorts of computations (at least physics majors), the conceptual and interpretational problems continue to make life hard to students and teachers.In fact, major changes in understanding the world and the physical reality are required [20,21] when dealing with concepts such as probability, uncertainty and state superposition and strange and intriguing issues such as non-locality and entanglement.As a matter of fact, students naturally tend to apply classical reasoning in understanding quantum phenomena such as for instance tunneling or wave-particle duality, or in interpreting expectation values for observables such as energy or angular momentum, so that the main source of misconceptions comes from mixing classical and quantum concepts [22].According to some scholars [24], while thinking about quantum mechanics a common difficulty, and a challenge, for learners is to shift from mental models based on sensory experience towards models essentially based on abstract properties.Furthermore, as pointed out by several studies [23,25] students' difficulties are quite universal, e.g.independent of teaching style, background and institution.For many years, the attention of instructors for conceptual matters has not been very high.Older textbooks such as [26], which typically approach the subject following the standard set by Dirac's masterpiece [27], tend to privilege computational aspects and the classic applications to atoms, molecules, solids, etc. over more conceptual aspects.More recently, the great progress of the last decades in quantum information, quantum computation, quantum communication, and the like [28], has made necessary a renewed greater attention towards conceptual and fundamental issues.As a consequence, more modern textbooks (cf.e.g.[29,30]) develop the subject in a different fashion, more focused towards the above-cited themes, typically starting from two-level systems, whose simplicity allows to introduce complex topics such as entanglement, teleportation, quantum algorithms etc. early on.
When shifting to more subtle and conceptual matters, with deep philosophical implication as well, even more difficulties in students' understanding are expected.Among such issues, it is certainly possible to include that of the correspondence principle and the transition from the quantum to classical world [1][2][3][4][5].This is a problem that was central in the early days of quantum mechanics, namely when and how the theory recovers the description of the world given by good old classical mechanics, is still not often treated in detail in textbooks, limiting to a few paragraphs on the correspondence principle (often in sections dealing with the old quantum theory, which many modern books tend to avoid altogether), on the Ehrenfest theorem (which usually deserves no more than a short section), on the WKB approximation, and maybe a handwaving discussion of how the classical path becomes dominant in the path integral when the action is large compared with ℏ.In addition to the lack of proper treatment in textbooks, this topic, to the best of our knowledge, has still not been treated in the physics education research literature.
In fact, that of the classical limit and of the semiclassical picture of quantum mechanics is still a very important issue, which is not only of philosophical interest.Starting from the sixties, in fact, there has been a new surge of interest in this subject (called the "postmodern era" [6,31]), also in view of its relevance to quantum chaos and the emergence of classical chaos from quantum mechanics [6,31].Also, its careful treatment can help in identifying those situations where a classical visualization of the phenomena can be helpful [31].While indeed one of the most common misconceptions consists in considering the wave function as an object living in three dimensions also when more than one particle is involved [23], it is also true that, as Heller and Tomsovic [31] put it, "unavoidably, we think classically about systems of more than a couple of particles".The same authors then argue that an enhanced understanding of semiclassical quantum mechanics can help to develop physical insight into quantum mechanics, and acts as a sort of "scaffolding".Hence, the classical limit should be properly treated, and its complexity and subtlety should be appreciated by all students who undertake quantum mechanics.According to [23], a very effective strategy for addressing misconceptions is to assign a problem to be solved that makes the misconception concrete.Therefore, to address the misconception that the classical limit is a straightforward matter, in the following section we provide a couple of examples that can form the basis of such problems.

Quantum mechanics versus classical mechanics
Historically, the first place where the limit ℏ/S → 0 is encountered is Planck's classic book Theory of Heat Radiation [1].The quantity S is chosen case by case in such a way to make the limiting parameter dimensionless.Typically, it is some characteristic action of the corresponding classical motion.In fact, the limit is to be understood as formal, its actual meaning being that ℏ/S → 0 when ℏ << S, since ℏ is a universal constant, whose value is (as far as we know) fixed, and it cannot be tuned.In the theory of heat radiation, the relevant inequality reads ℎ   ⁄ ≪ 1, and characterizes the regime where the classical Rayleigh-Jeans formula well approximates the Planck formula for the spectral energy density of black body radiation at frequency ν: ( This limiting relation is known as Planck's correspondence principle. Unfortunately, as first recognized by Berry [32], the classical limit is singular for the majority of physical systems, which means that the nature of the solution near the limit is different from its nature at the limit [5].Really smooth transitions between theories take place only in a few cases, while a singularity is the usual situation [33], unveiling a bunch of novel phenomena in the region of transition.A paradigmatic example of such a phenomenon [32,34] is the relationship between the wave theory of light and geometrical optics.Here the latter is obtained as the short wavelength limit of the former, so that the limit λ/a → 0 is considered, a being the typical linear dimension of the obstacles with which light interacts.Let us take as a simple example a plane wave traveling along the x-axis with speed v, i.e. (, ) =  � 2  ( − )�.Here the singularity of the limit λ/a → 0 clearly emerges, as  is a trigonometric function; as such, it is non-analytic at λ = 0, but exhibits fast oscillations and takes all the values between -1 and +1 for every interval of x and t.The same feature shows up when considering the simple phenomenon of superposition of two plane waves with speeds v and -v respectively, whose final formula is always built of trigonometric functions with an argument proportional to 1/λ.By calculating its time average one gets 2 2 � 2  �, whose argument still includes 1/λ: it describes an interference pattern fixed in space.Only by taking a further spatial average, one eventually gets a finite result.As a further peculiar feature, the limit to geometrical optics is characterized by the presence of caustics, which are envelopes of families of rays marking the boundary between regions with different numbers of rays [32,34].
Let us now illustrate the meaning of the above statements by considering some concrete examples within the realm of quantum mechanics.
Step barrier.Let us take a particle with energy E in the presence of a step potential barrier () =  0 1+ −  ⁄ , featuring a smoothness threshold L, and consider the case E > V(x) [35].It is well known that the classical reflection coefficient is always zero, while quantum mechanical predictions are quite counterintuitive.Indeed, there is a small probability for the particle to be reflected by the barrier, resulting in a non-vanishing reflection coefficient.Let's now consider the semiclassical regime, which implies the limit ℏ  2 ⁄  ⟶ 0, where  2 is the momentum of the particle in the region x >> 0. By taking this limit, the reflection coefficient becomes: As a key feature, this expression cannot be expressed as a power series in ℏ, so we are in the presence of a singular limit.The situation gets even worse for a sharp barrier (L = 0), whose reflection coefficient is independent of ℏ and can never give rise to the expected classical limit.Systems with classical chaos.Classical chaos emerges in the long-time limit, so in order to find the classical limit of a chaotic system one has to take both the t → ∞ and ℏ/S → 0 limits.However, these limits do not commute and this is a critical issue, giving rise to a complex behavior.In general, for an isolated system the quantum counterpart is non chaotic, i.e. quantum mechanics suppresses chaos.But decoherence acts by making it to reappear: an example is the chaotic tumbling of Hyperion, one of the satellites of Saturn [36][37].
As a further relevant issue, interesting emergent phenomena arise in such systems, whose fingerprint is the energy-level statistics of semiclassical spectra [38].In fact, it is well known that isolated bounded systems have discrete energy levels and that the corresponding semiclassical spectra are built of highly excited states.Thus, they exhibit universality features, which allow one to distinguish between integrable and chaotic systems.In fact, the level spacings (s) distribution follows the Poisson distribution in the first case, whose result is a clustering of levels: while being well approximated by the Wigner distribution in the second one, where a repulsion of levels takes place: This spectral universality is a peculiar emergent phenomenon, which is clearly non classical because it is a property of discrete energy levels.However, it is semiclassical, because only in the limit ℏ/S → 0 it is possible to get many energy levels within a classically small interval.This feature is another relevant consequence of the singularity of the semiclassical limit.Systems with a discrete energy spectrum.Let's take, for the sake of simplicity, a paradigmatic example, the harmonic oscillator (notice however that a particle in a box or the hydrogen atom show the same behavior), whose energy eigenvalues are: Here, by simply taking the limit ℏ/S → 0, the energy will be zero for all n.So, a meaningful result can be obtained by taking also the limit n → ∞ (Bohr's formulation of the correspondence principle) and requiring that the product is equal to a fixed value which is the classical action J = nh.Only in this way the classical energy is recovered [39]: where   is the energy of the classical oscillator with angular frequency ω and amplitude A. Indeed, it has been pointed out by some authors that the limit of large quantum numbers is not a sufficient condition to recover the classical behavior, even for quantum phenomena characterized by a discrete spectrum.For instance, Messiah, in his book Quantum Mechanics [26], makes the following enlightening consideration: "In order that the approximation be justified, it is necessary that this spacing could be considered negligible; that is the case if large quantum numbers are involved…The condition is certainly not sufficient; thus, some purely quantum-mechanical effects such as the uncertainty relations are not related to the discreteness of certain spectra".Here Messiah hints to quantum phenomena which do not exhibit a dependence on n, so that the limit n → ∞ is not effective.
In general, it has been shown [40] that Planck's correspondence principle does apply to all periodic systems, while this is not the case for Bohr's correspondence principle, so that the limits n → ∞ and ℏ/S → 0 are not universally equivalent.
Ehrenfest's Theorem.The statement of the theorem is that, under given conditions, the average values of position and momentum of a quantum system follow a classical trajectory.For instance, for a one-dimensional particle in a scalar potential V(x) one gets: if the Ehrenfest substitution, namely the replacement of 〈F(q)〉 with F(〈q〉), is valid.When this approximation holds on, the resulting equation ( 7) tells us that the centroid of the quantum wave packet will follow a classical trajectory.But what is the domain of validity of Ehrenfest substitution?In fact, it is quite restricted: one can replace the average of a function of position with the function of the average position only for systems described by a Hamiltonian which is a second order polynomial or less.Hence, there are systems where this substitution doesn't work, such as a particle scattering off a potential step [26].In this respect, Ballentine [41] shows that Ehrenfest theorem is neither a necessary nor a sufficient condition for the classical limit, the main problem lying in the assumption that the classical limit of a wave function is a single trajectory.Instead, it should be an ensemble of classical orbits.Finally, Rohrlich [42] points out that the Ehrenfest substitution holds on only for negligible fluctuations of the canonical coordinates.As such, the Ehrenfest theorem only provides a reduction of point particle mechanics.

Discussion and perspective for teaching
What is the classical limit of quantum mechanics?When trying to provide an answer to this question, a widespread belief is that the right answer should be found in the correspondence principle.But even at this level there is some ambiguity: one may refer either to Planck's [1] or to Bohr's [2] correspondence principle, so that a dualistic vision emerges.Indeed, many quantum mechanics textbooks show Planck's formulation within the discussion of blackbody radiation, in order to get the classical Rayleigh-Jeans energy density from Planck's one by taking the limit ℎ   ⁄ ≪ 1, which is an instance of the general limit ℏ/S → 0. On the other hand, Bohr's formulation deals with the limit of large quantum numbers and is usually considered with reference to the hydrogen atom.Thus, another question naturally arises: is there a relation between these two formulations of the correspondence principle?The answer is nontrivial.In fact, on the one hand they have been shown to be not universally equivalent [40].On the other hand, there exist simple quantum mechanical systems such as the harmonic oscillator, for which a meaningful classical limit can be obtained only carrying out both limits, i.e. putting together the two formulations, and constraining the product of the quantum number and the Planck constant to be equal to the appropriate classical action [26, 39][43].This synthesis may be effective in ending the above dualistic picture.
Another critical issue is related to the nature of the limit ℏ/S → 0, which is singular for a number of physical systems.This feature can be easily recognized in the wave representation of quantum mechanics (dealing with probability waves).Considering, as for the case of wave optics, the superposition of two equal beams moving in opposite directions (now particle beams have to be considered), it is immediately evident that also here trigonometric functions come into play within the intensity formula, which undergoes fast oscillations between 0 and 4 in arbitrarily small intervals of x as ℏ/S → 0. Furthermore, when dealing with the quantum counterpart of classically chaotic systems, also the long-time limit must be considered, which brings into play new puzzling emergent phenomena.
Finally, textbooks often present the Ehrenfest theorem as a further statement of the correspondence between quantum and classical mechanics.It allows to obtain classical dynamical equations by averaging the corresponding quantum mechanical observables.Unfortunately, the so-called Ehrenfest substitution has a restricted validity, because it applies only to small fluctuations [42].
Summing up, the relation between quantum and classical mechanics is highly non-trivial and involves deep epistemological considerations.In fact, the main lesson is that in such a case a successful reduction is not possible, but it is more appropriate to recognize a plurality of theories' foundations [15,44].This should imply a deep change of perspective in quantum mechanics teaching, which is expected to shift from a strict reductionism-based approach, which is currently standard, to a new one also open to a theoretical pluralism.In particular, the role of semiclassical mechanics in providing further physical insight into the structure of quantum dynamics itself and uncovering novel emergent phenomena should be properly emphasized.Some care should also be taken, of course, in underlining how the insight provided by the classical limit and by semiclassical mechanics is limited in scope, and that there are several situations, especially when more than one particle is involved, where it fails.In other words, there is the risk of enforcing the misconception that wave functions always live in three-dimensional space which we referred to above [23].These are of course the situations which are most puzzling and counterintuitive, and should be properly and carefully treated.

Conclusions
In this contribution, we have discussed an important but not often emphasized topic in quantum mechanics, neither in routine teaching nor in the physics education literature, which is its formal relationship with classical mechanics.As underlined in the philosophical and in the mathematical physics literature, and as we have seen by analysing explicit examples, this relation is by no means simple or straightforward as is often stated in physics textbooks.In fact, a similar statement can be made for another correspondence which can be stated in formally analogous terms, that between geometric and wave optics, and a careful discussion of the limit can be fruitfully included also in optics courses.
Our proposal is thus that the teaching of quantum mechanics at the undergraduate level should take this aspect into account, by explicit discussion of examples, including but not limited to the ones we described.The development of a teaching-learning sequence based on the above considerations is ongoing, and it will be discussed elsewhere, together with its implementation and results.