On the first page of Professor Herbert Pohl's book, "Quantum Mechanics for Science and Engineering," we read that, "Quantum mechanics is a remarkable theory. There seems to be no real doubt that much of physics and all of chemistry should be deducible from its postulates or 'laws'."
A very impressive claim indeed, but then a little further on in the same book and this apparent theory of everything is demoted to being just a model, with the clarification: "Our every attempt to describe or interpret nature involves the use of a model. From it are deduced 'laws' which it is hoped will aid in describing and interpreting natural events. Clearly, as our experience grows, constant revision of these imputed 'laws' will be required. Models, based upon analogies, are often very helpful ... but never complete. In this sense we must always be prepared to abandon it for a better one. ... [In quantum theory,] frequent recourse to approximations of all kinds is required, ranging from mere truncation of series expansions to outright hand-waving. Model after such model will be encountered, but each must be recognized for what it is ... a model."
We can summarize Prof. Pohl's teaching by saying that quantum mechanics has 'laws,' but is a model. "Constant revision of these 'laws' will be required, [because] models [are] never complete. [In addition,] frequent recourse to approximations ... [including] outright hand-waving" are necessary.
Suddenly quantum mechanics is by no means as impressive as we had been led to believe just a few paragraphs earlier. These so-called 'laws', which popular commentaries would have us believe that The Almighty is constrained by, are apparently nothing more than postulates of a model that is not only a set of approximations and assumptions, but can be thrown out any minute for a 'better' model. This ought to come as no surprise, since if Newton's 'laws' of motion, for example, are really laws, then why would we need anything like quantum mechanics and general relativity in the first place?
In this small paper, I shall argue that standard, scientific education misleads the general public, in much the same way that the medical profession does. I.e., it places upon science an aura of mystique and absolute correctness that is totally unjustified. On the contrary, it needs to be realized that our level of understanding of everyday things is exceedingly limited.
Kinetic energy and linear momentum
Consider the following, typical extract from a good high-school physics textbook:
"In collisions, the total momentum of the colliding objects is always conserved. Usually, however, their total kinetic energy is not conserved. Some of it is changed to heat or sound energy, which is not recoverable. Such collisions are said to be inelastic." (Nelkon and Parker, p. 21.)
Here we are instructed, then, that the total linear momentum is always conserved, yet that the total kinetic energy is not conserved in inelastic collisions.
Kinetic energy is a function of only two things; mass and velocity. If we are not discussing nuclear reactions, then we can assume that mass has remained constant during the collision. Hence, something connected with the velocities of the two objects must have been lost and is "not recoverable." However, linear momentum is merely the product of mass and velocity and, since part of the total velocity has been lost, whilst the total mass has remained constant, then total linear momentum has almost certainly changed between what it was before the collision and what it is afterwards. But if it has changed, then linear momentum can not have been conserved in the case of the inelastic collision.
Consider Fig. 1.
Figure 1: Two particles on a collision course with one another.
½ m1u12 + ½ m2u22 = KEB
½ m1v12 + ½ m2v22 = KEA
mu1 + mu2 = LMB
mv1 + mv2 = LMA
where KEB, KEA, LMB and LMA are the kinetic energy before, kinetic energy after, linear momentum before and linear momentum after the collision, respectively.
To rigorously test the values of total linear momentum before and after an inelastic collision, a computer program was written, the heart of which is shown in Fig. 2, and the results from which are shown in Fig. 3. The spheres were varied in mass, such that either one could be significantly greater than the mass of the other. The collision could occur either from left to right or from right to left. The before velocities were iterated through many discreet values and the after velocities varied through twice that number. Only inelastic collisions, where a small percentage of total kinetic energy is lost, were considered.
Figure 2: Section of computer code for testing the assertion that linear momentum is conserved in all inelastic collisions between two objects.
In this code, if statements are used to ensure that a collision will definitely occur, that one particle will not 'pass through' the other particle, that the collision is inelastic and that the loss of energy is at most 3.5%, respectively.
Figure 3: Results obtained by subtracting the total linear momentum after an inelastic collision from that existing before the collision. The solid, black line displays the trend towards decreasing total linear momentum as a function of the number of collisions.
Does Fig. 3 support the contention that linear momentum has been conserved in these 65,053 simulated collisions? No, it does not. On the contrary, the general trend displayed here is clearly that of a loss of linear momentum occurring over time, in complete agreement with what ought to be expected. In fact, only 0.8% of the 65,053 possible inelastic collisions examined conserved linear momentum. These results plainly deny the 'law' of the conservation of linear momentum.
Why, then, do Nelkon and Parker state that "In collisions, the total momentum of the colliding objects is always conserved"? It cannot be because they were blissfully unaware that their statement is an untrue generality, because on the previous page they had written,
"The principle of the conservation of linear momentum states that, if no external forces act on a system of colliding objects, the total momentum of the objects remains constant." (Page 20, emphasis mine.)
I.e., they qualified their statement of the 'law' of the conservation of linear momentum on page 20, but left out any such rider on page 21. Probably, therefore, they were just emphasizing something to the student who they knew would be sitting an 'A'-Level physics examination paper and, in order to answer a mathematical question regarding some collision between solid objects, the student should assume conservation (otherwise the mathematical treatment goes beyond what the student would by that stage have been taught). At this level of understanding of the subject, the scholar is deemed to be better equipped by being taught an approximation as fact, rather than burdened with a complication that only those who are going to study physics at a much higher level will ever need to know about.
However, this approach raises some problems, because to many this level of education becomes terminal. Journalists and writers of 'popular science' will use such texts and place their trust in what they read therein. Even most of those who go on to degree and postgraduate courses will not question the books and papers that they read.
Application to the atmosphere
Figure 4 shows the development of the computer program to facilitate the simulation of specific scenarios. In particular, the statistical nature of collisions within the World's atmosphere was examined in this run.
Figure 4: Small loss of total kinetic energy (0.05%) and large variation in mass simulates collisions within the atmosphere.
A variation of 22 times in particle masses represents the possibility of a collision between a carbon dioxide molecule and a hydrogen atom (and anything inbetween). A wide variation in before velocities was allowed and the total loss in kinetic energy was limited to just 0.05% (atmospheric collisions are approximately elastic). There were 978 collisions that fitted these criteria and we observe the same general trend as was seen in Fig. 3 – namely, that the total linear momentum of the atmosphere, in the absence of external influences, will decrease with time. The information area at the bottom of the graphical user interface shown in Fig. 4, displays the fact that conservation was not observed in even a single instance.
There is a tendency in the media to promote the 'laws of physics', as some sort of mystical framework within which the cosmos came into being and must subsequently operate. The original purpose of any reasonable simplification that went into earlier development of the idea in question then gets lost along the way, and replaced instead by rigid, incontrovertible claims regarding the scope and level of confidence to be attached to the principle. Close examination of these 'laws', however, show them to be transient conveniences, based upon assumptions and approximations that sometimes extend even to "outright hand-waving." Reality very rarely lends itself to being expressed in a neat little equation.
Laws are only an expression of current understanding and we should not be surprised if a previously cherished law is subsequently shown to be errant.
Some laws are stronger than others and it should be noted, for instance, that in a closed system such an increase in entropy as our results allude to is totally to be expected from the Second Law of Thermodynamics, since kinetic energy losses being radiated as heat will mean that total kinetic energy tends to zero over time, which, in turn, means that velocity magnitudes are tending to zero, and hence linear momenta are tending to zero.
In the real world, solar heat and other external influences will affect the velocities of air molecules within the atmosphere, such that those molecules will consistently receive random forces upon them which will ensure that they never actually stop moving.
This work clearly demonstrates a trend of decreasing linear momentum with time in systems of multiple inelastic collisions and thereby denies that conservation of linear momentum is a law in the scientific sense of the word.
This work has benefitted as a result of criticism levelled by Mike Boyd. His comments were highly instrumental in my reassessment of the material presented here.
Nelkon, M. and Parker, P., 1977, "Advanced Level Physics," 4th ed., Heinemann Educational Books, London.Pohl, H.A., 1967, "Quantum Mechanics for Science and Engineering," Prentice-Hall, Inc., New Jersey.