For an early introduction of quantum mechanics concepts in physics curriculum
aa r X i v : . [ phy s i c s . e d - ph ] J a n For an early introduction of quantum mechanicsconcepts in physics curriculum
Giovanni Organtini
Sapienza Universit`a di Roma & INFN-Sez. di RomaPiazzale Aldo Moro 5 - 00185 ROMA (Italy)E-mail: [email protected]
July 2018
Abstract.
In this paper we suggest to anticipate the introduction of concepts suchas quantum state and the operators connected to their transformations well in advanceof what is usually done.
Keywords : quantum mechanics, quantum state, operators
Submitted to:
Phys. Ed.
1. Introduction
With the advent of special relativity [1], physics knew one of the deepest revolutionsin its history. The revolution we are talking about does not consist in the fact thatconcepts like space and time, till then considered absolute, cannot be taken as such.In our opinion, the most important revolution happening with the formulation ofspecial relativity consists in the fact that reality cannot be considered independent from measurements . In this respect, the revolutionary breadth of special relativity is far moreimportant than that of quantum mechanics, although this is little recognised.Special relativity is based on the experimental observation according to which thespeed of light appears to be the same in every reference frame. This observation, to allintents and purposes, is incomprehensible, in the sense that it is completely extraneousto common sense. In order to accept the conclusions of special relativity, one mustaccept the fact that Galilean transformations are wrong and that, irrespective of thefact that it appears meaningful or not, what we measure is what we should consider as real . A frequent question posed by students when we teach them about length contractionand time dilation is: are those real effects or do they just appear as such? In other words,students are still considering the possibility that time and space are in fact absolute,however, when we measure them, they appear to be relative, possibly because of thetechnologies used. or an early introduction of quantum mechanics concepts
2. The concept of the state
A very common statement in classical physics is that the state of a pointlike particle isgiven when its position ~x and its velocity ~v are known.On the other hand, we could not find a single textbook in which is explained whatis intended for state .Our proposal is to define the state of a physical system starting from the definitionfor the word “state” given in the vocabulary [3], according to which a state is a modeor condition of being . In other words it describes, as precisely as possible, which are theconditions in which a given system can be observed at a given time.In order to provide such information for a pointlike particle, it is enough to providejust few numbers: one for its mass, three for its position and three for its velocity. Thereis no other meaningful quantities to express the mode or condition of being of such aparticle, in the sense that any other quantity that can be measured or predicted for sucha particle, either is meaningless or can be derived from the knowledge of the latter.For example, the particle momentum ~p = m~v can be computed directly from thestate variables. If it is taken as a state variable, either m or ~v can be removed from it, or an early introduction of quantum mechanics concepts state frequently appears in physics textbooksis in the study of the laws of ideal gases, where the ideal gas law is often called the gasequation of state : pV = nRT . (1)In this case the state is implicitly given when three of the four quantities (pressure p ,volume V , quantity of gas n and temperature T ) appearing in the equation are known.Again, each of the state variables can be predicted once the initial state is knownand external conditions applies. For example, if forces act on the container reducing, e.g.,its volume, the pressure and the temperature of the gas change accordingly dependingon external conditions.It is worth noting that, as long as we speak about a gas and we do not interpretit as a set of pointlike particles using the relatively recent kinetic theory, the state ofthe system is not given in terms of positions and velocities, but, in contrast, in terms ofpressure, volume and temperature.For a gas, terms like position and velocities are meaningless, since there is no wayto measure the position of a gas, nor its velocity. A gas is contained in a volume: it doesnot hold a position. Since a gas, when at the equilibrium, can just sit still inside thevolume, it is impossible to define a procedure to measure its velocity. Even if the gasexpands irreversibly as in the Joule experiment, there are no procedures to provide ameasurement of its speed . In fact there is not even a definition of it. As a consequence,the state of a gas cannot be given in terms of position and velocity, but only in terms ofquantities that can be measured on it independently: pressure, volume and temperature,for example.Similar considerations apply in other fields, too. Just to make one example in anunusual field, while introducing the electric current, one can provide the state of a wirewhose ends are connected to the plates of a charged capacitor providing the electric or an early introduction of quantum mechanics concepts I flowing in the wire at t = 0, its resistance R and the capacitance C of thecapacitor. Once these quantities are known, the state of the wire can be told at anytime. The capacitance C can be taken as a parameter governing the way in which thestate changes, just as a force determines how the state of a pointlike particle changesor the contact with en energy reservoir at fixed temperature determines how the stateof a gas transforms. Hence, the state of a wire can be given in terms of I and R . Inother terms, the Ohm’s Law is a sort of equation of state for a conductor.A final remark about the state: the state of being of, e.g., a wire, may include thecolour of its sheathing. Even for a gas, the color can be part of its description. Thereason for which we usually do not include it in the state is that it is not relevant for thetransformations we are interested in. In order to describe a chemical reaction, however,in which a gas in a container changes its color, the latter can be interesting and mustbe included in it.
3. Introducing the operators
Moreover, one can also introduce as early as possible the concept of an operator as aformal way to describe the effect of something external to the system under investigationwhich causes a change of its state. An external force ~F , for example, transforms a statethat can be represented as | ~x, ~v i at time t = 0 into a state (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~x + ~vt + ~F m t , ~v + ~Fm t + = O | ~x, ~v i (2)at time t . The transformation from the state at time t = 0 and the one at time t = 0is represented by the application of an operator O that provides a transformation ofthe variables in the state according to some rule that can be inferred from observations,expressed in terms of physics laws.Putting a gas, whose state is given by | p, V, T i , contained in a rigid container incontact with a temperature reservoir T r causes its state to change into a new one (cid:12)(cid:12)(cid:12)(cid:12) p ′ = nRT r V , V, T r (cid:29) = O | p, V, T i . (3)In any case the physics law can be regarded as the application of an operator thatassociates a state (i.e. an element of a set) to another state (i.e. another element of thesame set). Often, classical physics operators can be represented by a scalar or vectorialfunction acting on the space of the state variables. However, that may not be alwaysthe case.This is the case as long as we pretend that physical quantities are real numbers thatcan attain any value, and as long as we believe that physical quantities are continuousand deterministic functions. or an early introduction of quantum mechanics concepts
4. The concept of a force
The intuitive concept of a force is something that either pull or push. In general sucha model is appropriate for pointlike particles because our eyes, in fact, “measure”distances. Forces on pointlike particles (or bodies that can be modelled as pointlikeparticles) just move their point of application when they don’t cancel each other.In other words, these kind of forces alter the state of pointlike particle. However,in Newtonian dynamics, forces are responsible for producing accelerations, i.e. formodifying the state of motion of bodies. Then, the observation of a displacementof a pointlike particle cannot be ascribed to forces, unless a change is its velocity isobserved. The intuitive concept of a force is what causes the observed problems inconceptual understanding of Newtonian dynamics [4]. The old Aristotelian concept,according to which a moving particle is always subject to a force, still persists in manystudents.The application of a force on a gas is only apparently similar to what happens topointlike particles. While the force can still be regarded as something that push orpull, it cannot be represented by a vector. At least, it cannot be always representedby just one vector. Forces like those applied to a pointlike particle can be applied tothe gas container. The container, in turn, exerts a force on the gas. However, whilethe force on the container can be applied on just one of its point, the forces exerted onthe gas by the container must be applied to something that has a volume and is alwaysdirected perpendicularly to the container’s walls. Moreover, the effect of forces on thecontainer is distributed over its whole surface. In this case there is no acceleration, nora displacement of the gas (there is, of course, a displacement of the particles composingthe gas that, however, are not the gas as a whole). That is why one is forced to introducethe concept of pressure, defined as a scalar quantity that can be measured combiningthe measurement of a force and of a surface.A similar problem appears when we talk about electric currents in wires. Untilwe do not have a model of electricity in terms of particles subject to the electromotiveforce, one cannot clearly identify any force acting on the wire. However, it is clearthat some kind of force must develop when connecting the wire to a capacitor. In fact,the temperature of the wire, i.e. its internal energy, increases and the first principle ofthermodynamics states that the increment can only be due to the work done by thecapacitor. From the definition of work it follows that the capacitor must apply somesort of force to the wire, that, however, does not move at all nor changes its shape orsize. From the above observations it follows that forces are those entities that tend tocause a change in the state of a system. Forces are not necessarily vectors. They arevectors only when they result in pulling or pushing a point. In other cases they mustbe described in some other way.The operators mentioned above, in fact, are a possible way to express the effectof a force. One can clearly see that, in the above examples, vectors only appear when or an early introduction of quantum mechanics concepts Z and A , i.e., | Z, A i . At a later time, thestate of the system has changed into a new state | Z ± , A, e, ν i = O |
Z, A i . (4)A change in the state requires the intervention of a force (or, better, an interaction)represented by the action of the operator O on the initial state.An even better introduction to this problem consists in making a difference between interactions and forces . Interactions are produced by a source (the mass for gravity, theelectric charge for electricity, the electric current for magnetism, the color for the stronginteraction and the hypercharge for the weak interaction) and, depending on the way inwhich the state is described, may results in the appearing of a force. This way, forcesdefined as “fictitious”, like those appearing in non–inertial reference frames, appear tobe as real as non–fictitious ones. In fact, this description makes it possible to overcomeanother difficulty in understanding Newtonian dynamics, coming from our experiencethat, as a matter of fact, fictitious forces are among the few forces that we really feel ‡ .The operator O must be able to transform the initial state from | Z, A i to | Z ± , A i ,but it also must create a pair of new particles (the electron e and the correspondingneutrino ν ). Forces, then, can do much more than just pushing or pulling. It isworth noting that, despite this possibility was implicitly contained already in the earlyquantum mechanics, until the work of Enrico Fermi [5] the whole scientific communitybelieved that the electrons emitted in beta decay pre-existed in the nuclei. That wascausing many problems to the interpretation of the phenomenon. The adoption of thispoint of view, in contrast, was proven to be correct in the sense that almost all of thepredictions made using the Fermi’s theory turned out to be correct. The possibility to create particles not existing in the initial state was considered shocking at the beginning,but as a matter of fact there is nothing in classical physics preventing it, provided thatforces (interactions) are correctly introduced. ‡ Being our senses immersed in the gravitational field, gravity is not usually perceived as a force, infact. or an early introduction of quantum mechanics concepts
5. From classical physics to quantum mechanics
The introduction of the weak interactions is just the first step into a gradual formulationof quantum mechanics. Presenting quantum mechanics as an abrupt revolution withrespect to classical physics may be fascinating for many reasons. However, it leavesthe conviction in students that there must be something wrong in it, being so muchdifferent to what was learnt so far. This was, in fact, the same mood in the scientificcommunity at the very beginning of the new mechanics. With time, we realised that,in fact, quantum mechanics is indeed much more solid than classical physics. Thereis no field in physics in which the predictions of the theory are verified at precisionlevels comparable with those attained in quantum mechanics. A smoother transitionbetween classical physics and quantum mechanics, then, is desirable because it may helpin reducing the difficulties that students have in accepting its results.In fact, we believe that it is even possible to do the opposite, i.e., startingfrom quantum mechanics to land into classical physics. However, this is a long termprogramme that we will pursue in the future.Once the picture described in the previous sections has been consolidated, one canintroduce quantum mechanics in many ways. For example, starting from the uncertaintyprinciple, one can see that there is not a solid procedure to measure the position andthe velocity of an electron bound in an atom. Hence, its state cannot be described inthese terms. As in the case of the gases and of the wires, we are forced to find otherphysical quantities that can be used to describe the state of such electrons, i.e. theirmode of being.One can see, for example, that the binding energy of the electron can be obtained bymeans of the photoelectric effect and it can be experimentally shown that the electronsin all the atoms of each chemical species share the same set of energies. At the sametime, one can show that the angular momentum of the electrons are quantised andthat the value of such a quantity is independent on the energy. As a consequence, notdifferently from classical physics, the state of electrons in atoms must be described bythe only meaningful quantities describing their mode of being: | E, J, J z i where E istheir energy and J and J z their total angular momentum and their third component,respectively.The interaction with a field causes the appearance of a force that in turn induces,as usual, a change in the state. Photoelectric effect, Compton scattering and otherphenomena can all be described in terms of an interaction causing a change in the stateof the electron.Moreover, assuming this picture, one can easily explain phenomena considered veryunnatural, like electron diffraction by two slits. The latter is simply explained by thefact that what emerges from the screen with the slits is not a flux of pointlike particles(we cannot tell the position of each single electron), but something whose state includea given momentum. Such an object is characterised by the fact that at t = 0 (thetime of the crossing), the distribution of the number of electrons is known to be a pair or an early introduction of quantum mechanics concepts
6. Testing the hypothesis
The hypothesis under the above considerations is that quantum mechanics appears tobe much more acceptable to students, if they have been prepared to deal with it whilelearning classical physics. We performed a first test of this hypothesis teaching physicsto a class of students in biotechnology, applying the methods outlined above.The physics course is given during the first semester of the first year, so the studentsattending the class have just exited the high school. Most of them (84 %) alreadyreceived a first introduction to physics, often including elementary quantum mechanics.It is worth noting that the population can be regarded to be almost homogeneous tothat of high school students, since, besides the age, those students share with them avery soft interest for physics.At the end of our course we administered a questionnaire that was answered by108 students, among which 17 were not exposed to physics in high school. To eachquestion we proposed five possible answers: from “not at all” to “definitely”, with “no”,“neutral” and “yes” as possible intermediate values. Table ?? shows the answers tothe question “do you believe you understood physics more deeply with the approachadopted in this course”? 35 % of the respondents answered positively, 45 % are neutral,while 20 % did not agree.To the question “Do you think that the way the course is designed is more difficultthan a traditional course?” those who answered positively were 17 %, while 40 % areneutral and 43 % did not agree.Interestingly enough, those who declared that physics were not among thedisciplines they liked while at school, changed from 42 % to 18 %. Such a result dependson many variables, however, since the whole course was designed around the principlesoutlined in the previous sections, we believe that a great part of this achievement is tobe ascribed to their application, who made classical physics more and more appealingas the students understand the connection between classical and modern physics.We intend to continue investigating on this point, to understand what, in particular,make students change their mind about physics among the various actions taken duringthe course. or an early introduction of quantum mechanics concepts
7. Conclusion
We propose to introduce quantum physics terminology and principles rather early inthe study of classical physics. In particular, a rather more formal definition of the stateof a system and the introduction of operators as something that produces a change inthe state, to be identified with the presence of some interaction, might be made at thevery beginning of a physics course and greatly help in introducing quantum mechanics.A very preliminary investigation has been done about student’s perception of thesubject. First results are encouraging, hence we look forward to search for partnersfor a physics education research aiming at identifying how the proposed approach caneffectively increase the interest and the comprehension of physics in high school andcollege students.
Bibliography [1] Albert Einstein (1905) ”Zur Elektrodynamik bewegter K¨orper”, Annalen der Physik 17: 891.[2] Letter to Max Born (4 December 1926); The Born-Einstein Letters; translated by Irene Born(Walker and Company, New York, 1971).[3] ”state” Merriam-Webster.com. 2018. (November 2018)[4] D. Hestenes, M. Wells and G. Swackhamer, “Force Concept Inventory”, The Physics Teacher30