# See, how hard I am trying to become an Approved (Full) Professor of Mechanical Engineering in SPPU?—3

I was looking for a certain book on heat transfer which I had (as usual) misplaced somewhere, and while searching for that book at home, I accidentally ran into another book I had—the one on Classical Mechanics by Rana and Joag [^].

After dusting this book a bit, I spent some time in one typical way, viz. by going over some fond memories associated with a suddenly re-found book…. The memories of how enthusiastic I once was when I had bought that book; how I had decided to finish that book right within weeks of buying it several years ago; the number of times I might have picked it up, and soon later on, kept it back aside somewhere, etc.  …

Yes, that’s right. I have not yet managed to finish this book. Why, I have not even managed to begin reading this book the way it should be read—with a paper and pencil at hand to work through the equations and the problems. That was the reason why, I now felt a bit guilty. … It just so happened that it was just the other day (or so) when I was happily mentioning the Poisson brackets on Prof. Scott Aaronson’s blog, at this thread [^]. … To remove (at least some part of) my sense of guilt, I then decided to browse at least through this part (viz., Poisson’s brackets) in this book. … Then, reading a little through this chapter, I decided to browse through the preceding chapters from the Lagrangian mechanics on which it depends, and then, in general, also on the calculus of variations.

It was at this point that I suddenly happened to remember the reason why I had never been able to finish (even the portions relevant to engineering from) this book.

The thing was, the explanation of the $\delta$—the delta of the variational calculus.

The explanation of what the $\delta$ basically means, I had found right back then (many, many years ago), was not satisfactorily given in this book. The book did talk of all those things like the holonomic constraints vs. the nonholonomic constraints, the functionals, integration by parts, etc. etc. etc. But without ever really telling me, in a forth-right and explicit manner, what the hell this $\delta$ was basically supposed to mean! How this $\delta y$ was different from the finite changes ($\Delta y$) and the infinitesimal changes ($\text{d}y$) of the usual calculus, for instance. In terms of its physical meaning, that is. (Hell, this book was supposed to be on physics, wasn’t it?)

Here, I of course fully realize that describing Rana and Joag’s book as “unsatisfactory” is making a rather bold statement, a very courageous one, in fact. This book is extraordinarily well-written. And yet, there I was, many, many years ago, trying to understand the delta, and not getting anywhere, not even with this book in my hand. (OK, a confession. The current copy which I have is not all that old. My old copy is gone by now (i.e., permanently misplaced or so), and so, the current copy is the one which I had bought once again, in 2009. As to my old copy, I think, I had bought it sometime in the mid-1990s.)

It was many years later, guess some time while teaching FEM to the undergraduates in Mumbai, that the concept had finally become clear enough to me. Most especially, while I was going through P. Seshu’s and J. N. Reddy’s books. [Reflected Glory Alert! Professor P. Seshu was my class-mate for a few courses at IIT Madras!] However, even then, even at that time, I remember, I still had this odd feeling that the physical meaning was still not clear to me—not as as clear as it should be. The matter eventually became “fully” clear to me only later on, while musing about the differences between the perspective of Thermodynamics on the one hand and that of Heat Transfer on the other. That was some time last year, while teaching Thermodynamics to the PG students here in Pune.

Thermodynamics deals with systems at equilibria, primarily. Yes, its methods can be extended to handle also the non-equilibrium situations. However, even then, the basis of the approach summarily lies only in the equilibrium states. Heat Transfer, on the other hand, necessarily deals with the non-equilibrium situations. Remove the temperature gradient, and there is no more heat left to speak of. There does remain the thermal energy (as a form of the internal energy), but not heat. (Remember, heat is the thermal energy in transit that appears on a system boundary.) Heat transfer necessarily requires an absence of thermal equilibrium. … Anyway, it was while teaching thermodynamics last year, and only incidentally pondering about its differences from heat transfer, that the idea of the variations (of Cov) had finally become (conceptually) clear to me. (No, CoV does not necessarily deal only with the equilibrium states; it’s just that it was while thinking about the equilibrium vs. the transient that the matter about CoV had suddenly “clicked” to me.)

In this post, let me now note down something on the concept of the variation, i.e., towards understanding the physical meaning of the symbol $\delta$.

Please note, I have made an inline update on 26th December 2016. It makes the presentation of the calculus of variations a bit less dumbed down. The updated portion is clearly marked as such, in the text.

The Problem Description:

The concept of variations is abstract. We would be better off considering a simple, concrete, physical situation first, and only then try to understand the meaning of this abstract concept.

Accordingly, consider a certain idealized system. See its schematic diagram below:

There is a long, rigid cylinder made from some transparent material like glass. The left hand-side end of the cylinder is hermetically sealed with a rigid seal. At the other end of the cylinder, there is a friction-less piston which can be driven by some external means.

Further, there also are a couple of thin, circular, piston-like disks ($D_1$ and $D_2$) placed inside the cylinder, at some $x_1$ and $x_2$ positions along its length. These disks thus divide the cylindrical cavity into three distinct compartments. The disks are assumed to be impermeable, and fitting snugly, they in general permit no movement of gas across their plane. However, they also are assumed to be able to move without any friction.

Initially, all the three compartments are filled with a compressible fluid to the same pressure in each compartment, say 1 atm. Since all the three compartments are at the same pressure, the disks stay stationary.

Then, suppose that the piston on the extreme right end is moved, say from position $P_1$ to $P_2$. The final position $P_2$ may be to the left or to the right of the initial position $P_1$; it doesn’t matter. For the current description, however, let’s suppose that the position $P_2$ is to the left of $P_1$. The effect of the piston movement thus is to increase the pressure inside the system.

The problem is to determine the nature of the resulting displacements that the two disks undergo as measured from their respective initial positions.

There are essentially two entirely different paradigms for conducting an analysis of this problem.

The first paradigm is based on an approach that was put to use so successfully by Newton. Usually, it is called the paradigm of vector analysis.

In this paradigm, we focus on the fact that the forced displacement of the piston with time, $x(t)$, may be described using some function of time that is defined over the interval lying between two instants $t_i$ and $t_f$.

For example, suppose the function is:
$x(t) = x_0 + v t$,
where $v$ is a constant. In other words, the motion of the piston is steady, with a constant velocity, between the initial and final instants. Since the velocity is constant, there is no acceleration over the open interval $(t_i, t_f)$.

However, notice that before the instant $t_i$, the piston velocity was zero. Then, the velocity suddenly became a finite (constant) value. Therefore, if you extend the interval to include the end-instants as well, i.e., if you consider the semi-closed interval $[t_i, t_f)$, then there is an acceleration at the instant $t_i$. Similarly, since the piston comes to a position of rest at $t = t_f$, there also is another acceleration, equal in magnitude and opposite in direction, which appears at the instant $t_f$.

The existence of these two instantaneous accelerations implies that jerks or pressure waves are sent through the system. We may model them as vector quantities, as impulses. [Side Exercise: Work out what happens if we consider only the open interval $(t_i, t_f)$.]

We can now apply Newton’s 3 laws, based on the idea that shock-waves must have begun at the piston at the instant $t = t_i$. They must have got transmitted through the gas kept under pressure, and they must have affected the disk $D_1$ lying closest to the piston, thereby setting this disk into motion. This motion must have passed through the gas in the middle compartment of the system as another pulse in the pressure (generated at the disk $D_1$), thereby setting also the disk $D_2$ in a state of motion a little while later. Finally, the pulse must have got bounced off the seal on the left hand side, and in turn, come back to affect the motion of the disk $D_2$, and then of the disk $D_1$. Continuing their travels to and fro, the pulses, and hence the disks, would thus be put in a back and forth motion.

After a while, these transients would move forth and back, superpose, and some of their constituent frequencies would get cancelled out, leaving only those frequencies operative such that the three compartments are put under some kind of stationary states.

In case the gas is not ideal, there would be damping anyway, and after a sufficiently long while, the disks would move through such small displacements that we could easily ignore the ever-decreasing displacements in a limiting argument.

Thus, assume that, after an elapse of a sufficiently long time, the disks become stationary. Of course, their new positions are not the same as their original positions.

The problem thus can be modeled as basically a transient one. The state of the new equilibrium state is thus primarily seen as an effect or an end-result of a couple of transient processes which occur in the forward and backward directions. The equilibrium is seen as not a primarily existing state, but as a result of two equal and opposite transient causes.

Notice that throughout this process, Newton’s laws can be applied directly. The nature of the analysis is such that the quantities in question—viz. the displacements of the disks—always are real, i.e., they correspond to what actually is supposed to exist in the reality out there.

The (values of) displacements are real in the sense that the mathematical analysis procedure itself involves only those (values of) displacements which can actually occur in reality. The analysis does not concern itself with some other displacements that might have been possible but don’t actually occur. The analysis begins with the forced displacement condition, translates it into pressure waves, which in turn are used in order to derive the predicted displacements in the gas in the system, at each instant. Thus, at any arbitrary instant of time $t > t_i$ (in fact, the analysis here runs for times $t \gg t_f$), the analysis remains concerned only with those displacements that are actually taking place at that instant.

The Method of Calculus of Variations:

The second paradigm follows the energetics program. This program was initiated by Newton himself as well as by Leibnitz. However, it was pursued vigorously not by Newton but rather by Leibnitz, and then by a series of gifted mathematicians-physicists: the Bernoulli brothers, Euler, Lagrange, Hamilton, and others. This paradigm is essentially based on the calculus of variations. The idea here is something like the following.

We do not care for a local description at all. Thus, we do not analyze the situation in terms of the local pressure pulses, their momenta/forces, etc. All that we focus on are just two sets of quantities: the initial positions of the disks, and their final positions.

For instance, focus on the disk $D_1$. It initially is at the position $x_{1_i}$. It is found, after a long elapse of time (i.e., at the next equilibrium state), to have moved to $x_{1_f}$. The question is: how to relate this change in $x_1$ on the one hand, to the displacement that the piston itself undergoes from $P_{x_i}$ to $P_{x_f}$.

To analyze this question, the energetics program (i.e., the calculus of variations) adopts a seemingly strange methodology.

It begins by saying that there is nothing unique to the specific value of the position $x_{1_f}$ as assumed by the disk $D_1$. The disk could have come to a halt at any other (nearby) position, e.g., at some other point $x_{1_1}$, or $x_{1_2}$, or $x_{1_3}$, … etc. In fact, since there are an infinity of points lying in a finite segment of line, there could have been an infinity of positions where the disk could have come to a rest, when the new equilibrium was reached.

Of course, in reality, the disk $D_1$ comes to a halt at none of these other positions; it comes to a halt only at $x_{1_f}$.

Yet, the theory says, we need to be “all-inclusive,” in a way. We need not, just for the aforementioned reason, deny a place in our analysis to these other positions. The analysis must include all such possible positions—even if they be purely hypothetical, imaginary, or unreal. What we do in the analysis, this paradigm says, is to initially include these merely hypothetical, unrealistic positions too on exactly the same footing as that enjoyed by that one position which is realistic, which is given by $x_{1_f}$.

Thus, we take a set of all possible positions for each disk. Then, for each such a position, we calculate the “impact” it would make on the energy of the system taken as a whole.

The energy of the system can be additively decomposed into the energies carried by each of its sub-parts. Thus, focusing on disk $D_1$, for each one of its possible (hypothetical) final position, we should calculate the energies carried by both its adjacent compartments. Since a change in $D_1$‘s position does not affect the compartment 3, we need not include it. However, for the disk $D_1$, we do need to include the energies carried by both the compartments 1 and 2. Similarly, for each of the possible positions occupied by the disk $D_2$, it should include the energies of the compartments 2 and 3, but not of 1.

At this point, to bring simplicity (and thereby better) clarity to this entire procedure, let us further assume that the possible positions of each disk forms a finite set. For instance, each disk can occupy only one of the positions that is some $-5, -4, -3, -2, -1, 0, +1, +2, +3, +4$ or $+5$ distance-units away from its initial position. Thus, a disk is not allowed to come to a rest at, say, $2.3$ units; it must do so either at $2$ or at $3$ units. (We will thus perform the initial analysis in terms of only the integer positions, and only later on extend it to any real-valued positions.) (If you are a mechanical engineering student, suggest a suitable mechanism that can ensure only integer relative displacements.)

The change in energy $E$ of a compartment is given by
$\Delta E = P A \Delta x$,
where $P$ is the pressure, $A$ is the cross-sectional area of the cylinder, and $\Delta x$ is the change in the length of the compartment.

Now, observe that the energy of the middle compartment depends on the relative distance between the two disks lying on its sides. Yet, for the same reason, the energy of the middle compartment does depend on both these positions. Hence, we must take a Cartesian product of the relative displacements undergone by both the disks, and only then calculate the system energy for each such a permutation (i.e. the ordered pair) of their positions. Let us go over the details of the Cartesian product.

The Cartesian product of the two positions may be stated as a row-by-row listing of ordered pairs of the relative positions of $D_1$ and $D_2$, e.g., as follows: the ordered pair $(-5, +2)$ means that the disk $D_1$ is $5$ units to the left of its initial position, and the disk $D_2$ is $+2$ units to the right of its initial position. Since each of the two positions forming an ordered pair can range over any of the above-mentioned $11$ number of different values, there are, in all, $11 \times 11 = 121$ number of such possible ordered pairs in the Cartesian product.

For each one of these $121$ different pairs, we use the above-given formula to determine what the energy of each compartment is like. Then, we add the three energies (of the three compartments) together to get the value of the energy of the system as a whole.

In short, we get a set of $121$ possible values for the energy of the system.

You must have noticed that we have admitted every possible permutation into analysis—all the $121$ number of them.

Of course, out of all these $121$ number of permutations of positions, it should turn out that $120$ number of them have to be discarded because they would be merely hypothetical, i.e. unreal. That, in turn, is because, the relative positions of the disks contained in one and only one ordered pair would actually correspond to the final, equilibrium position. After all, if you conduct this experiment in reality, you would always get a very definite pair of the disk-positions, and it this same pair of relative positions that would be observed every time you conducted the experiment (for the same piston displacement). Real experiments are reproducible, and give rise to the same, unique result. (Even if the system were to be probabilistic, it would have to give rise to an exactly identical probability distribution function.) It can’t be this result today and that result tomorrow, or this result in this lab and that result in some other lab. That simply isn’t science.

Thus, out of all those $121$ different ordered-pairs, one and only one ordered-pair would actually correspond to reality; the rest all would be merely hypothetical.

The question now is, which particular pair corresponds to reality, and which ones are unreal. How to tell the real from the unreal. That is the question.

Here, the variational principle says that the pair of relative positions that actually occurs in reality carries a certain definite, distinguishing attribute.

The system-energy calculated for this pair (of relative displacements) happens to carry the lowest magnitude from among all possible $121$ number of pairs. In other words, any hypothetical or unreal pair has a higher amount of system energy associated with it. (If two pairs give rise to the same lowest value, both would be equally likely to occur. However, that is not what provably happens in the current example, so let us leave this kind of a “degeneracy” aside for the purposes of this post.)

(The update on 26 December 2016 begins here:)

Actually, the description  given in the immediately preceding paragraph was a bit too dumbed down. The variational principle is more subtle than that. Explaining it makes this post even longer, but let me give it a shot anyway, at least today.

To follow the actual idea of the variational principle (in a not dumbed-down manner), the procedure you have to follow is this.

First, make a table of all possible relative-position pairs, and their associated energies. The table has the following columns: a relative-position pair, the associated energy $E$ as calculated above, and one more column which for the time being would be empty. The table may look something like what the following (partial) listing shows:

(0,0) -> say, 115 Joules
(-1,0) -> say, 101 Joules
(-2,0) -> say, 110 Joules

(2,2) -> say, 102 Joules
(2,3) -> say, 100 Joules
(2,4) -> say, 101 Joules
(2,5) -> say, 120 Joules

(5,0) -> say, 135 Joules

(5,5) -> say 117 Joules.

Having created this table (of $121$ rows), you then pick each row one by and one, and for the picked up $n$-th row, you ask a question: What all other row(s) from this table have their relative distance pairs such that these pairs lie closest to the relative distance pair of this given row. Let me illustrate this question with a concrete example. Consider the row which has the relative-distance pair given as (2,3). Then, the relative distance pairs closest to this one would be obtained by adding or subtracting a distance of 1 to each in the pair. Thus, the relative distance pairs closest to this one would be: (3,3), (1,3), (2,4), and (2,2). So, you have to pick up those rows which have these four entries in the relative-distance pairs column. Each of these four pairs represents a variation $\delta$ on the chosen state, viz. the state (2,3).

In symbolic terms, suppose for the $n$-th row being considered, the rows closest to it in terms of the differences in their relative distance pairs, are the $a$-th, $b$-th, $c$-th and $d$-th rows. (Notice that the rows which are closest to a given row in this sense, would not necessarily be found listed just above or below that given row, because the scheme followed while creating the list or the vector that is the table would not necessarily honor the closest-lying criterion (which necessarily involves two numbers)—not at least for all rows in the table.

OK. Then, in the next step, you find the differences in the energies of the $n$-th row from each of these closest rows, viz., the $a$-th, $b$-th, $c$-th and $c$-th rows. That is to say, you find the absolute magnitudes of the energy differences. Let us denote these magnitudes as: $\delta E_{na} = |E_n - E_a|$$\delta E_{nb} = |E_n - E_b|$$\delta E_{nc} = |E_n - E_c|$ and $\delta E_{nd} = |E_n - E_d|$.  Suppose the minimum among these values is $\delta E_{nc}$. So, against the $n$-th row, in the last column of the table, you write the value $\delta E_{nc}$.

Having done this exercise separately for each row in the table, you then ask: Which row has the smallest entry in the last column (the one for $\delta E$), and you pick that up. That is the distinguished (or the physically occurring) state.

In other words, the variational principle asks you to select not the row with the lowest absolute value of energy, but that row which shows the smallest difference of energy from one of its closest neighbours—and these closest neighbours are to be selected according to the differences in each number appearing in the relative-distance pair, and not according to the vertical place of rows in the tabular listing. (It so turns out that in this example, the row thus selected following both criteria—lowest energy as well as lowest variation in energy—are identical, though it would not necessarily always be the case. In short, we can’t always get away with the first, too dumbed down, version.)

Thus, the variational principle is about that change in the relative positions for which the corresponding change in the energy vanishes (or has the minimum possible absolute magnitude, in case the positions form a discretely varying, finite set).

(The update on 26th December 2016 gets over here.)

And, it turns out that this approach, too, is indeed able to perfectly predict the final disk-positions—precisely as they actually are observed in reality.

If you allow a continuum of positions (instead of the discrete set of only the $11$ number of different final positions for one disk, or $121$ number of ordered pairs), then instead of taking a Cartesian product of positions, what you have to do is take into account a tensor product of the position functions. The maths involved is a little more advanced, but the underlying algebraic structure—and the predictive principle which is fundamentally involved in the procedure—remains essentially the same. This principle—the variational principle—says:

Among all possible variations in the system configurations, that system configuration corresponds to reality which has the least variation in energy associated with it.

(This is a very rough statement, but it will do for this post and for a general audience. In particular, we don’t look into the issues of what constitute the kinematically admissible constraints, why the configurations must satisfy the field boundary conditions, the idea of the stationarity vs. of a minimum or a maximum, i.e., the issue of convexity-vs.-concavity, etc. The purpose of this post—and our example here—are both simple enough that we need not get into the whole she-bang of the variational theory as such.)

Notice that in this second paradigm, (i) we did not restrict the analysis to only those quantities that are actually taking place in reality; we also included a host (possibly an infinity) of purely hypothetical combinations of quantities too; (ii) we worked with energy, a scalar quantity, rather than with momentum, a vector quantity; and finally, (iii) in the variational method, we didn’t bother about the local details. We took into account the displacements of the disks, but not any displacement at any other point, say in the gas. We did not look into presence or absence of a pulse at one point in the gas as contrasted from any other point in it. In short, we did not discuss the details local to the system either in space or in time. We did not follow the system evolution, at all—not at least in a detailed, local way. If we were to do that, we would be concerned about what happens in the system at the instants and at spatial points other than the initial and final disk positions. Instead, we looked only at a global property—viz. the energy—whether at the sub-system level of the individual compartments, or at the level of the overall system.

The Two Paradigms Contrasted from Each Other:

If we were to follow Newton’s method, it would be impossible—impossible in principle—to be able to predict the final disk positions unless all their motions over all the intermediate transient dynamics (occurring over each moment of time and at each place of the system) were not be traced. Newton’s (or vectorial) method would require us to follow all the details of the entire evolution of all parts of the system at each point on its evolution path. In the variational approach, the latter is not of any primary concern.

Yet, in following the energetics program, we are able to predict the final disk positions. We are able to do that without worrying about what all happened before the equilibrium gets established. We remain concerned only with certain global quantities (here, system-energy) at each of the hypothetical positions.

The upside of the energetics program, as just noted, is that we don’t have to look into every detail at every stage of the entire transient dynamics.

Its downside is that we are able to talk only of the differences between certain isolated (hypothetical) configurations or states. The formalism is unable to say anything at all about any of the intermediate states—even if these do actually occur in reality. This is a very, very important point to keep in mind.

The Question:

Now, the question with which we began this post. Namely, what does the delta of the variational calculus mean?

Referring to the above discussion, note that the delta of the variational calculus is, here, nothing but a change in the position-pair, and also the corresponding change in the energy.

Thus, in the above example, the difference of the state (2,3) from the other close states such as (3,3), (1,3), (2,4), and (2,2) represents a variation in the system configuration (or state), and for each such a variation in the system configuration (or state), there is a corresponding variation in the energy $\delta E_{ni}$ of the system. That is what the delta refers to, in this example.

Now, with all this discussion and clarification, would it be possible for you to clearly state what the physical meaning of the delta is? To what precisely does the concept refer? How does the variation in energy $\delta E$ differ from both the finite changes ($\Delta E$) as well as the infinitesimal changes ($\text{d}E$) of the usual calculus?

Note, the question is conceptual in nature. And, no, not a single one of the very best books on classical mechanics manages to give a very succinct and accurate answer to it. Not even Rana and Joag (or Goldstein, or Feynman, or…)

I will give my answer in my next post, next year. I will also try to apply it to a couple of more interesting (and somewhat more complicated) physical situations—one from engineering sciences, and another from quantum mechanics!

In the meanwhile, think about it—the delta—the concept itself, its (conceptual) meaning. (If you already know the calculus of variations, note that in my above write-up, I have already supplied the answer, in a way. You just have to think a bit about it, that’s all!)

An Important Note: Do bring this post to the notice of the Officially Approved Full Professors of Mechanical Engineering in SPPU, and the SPPU authorities. I would like to know if the former would be able to state the meaning—at least now that I have already given the necessary context in such great detail.

Ditto, to the Officially Approved Full Professors of Mechanical Engineering at COEP, esp. D. W. Pande, and others like them.

After all, this topic—Lagrangian mechanics—is at the core of Mechanical Engineering, even they would agree. In fact, it comes from a subject that is not taught to the metallurgical engineers, viz., the topic of Theory of Machines. But it is taught to the Mechanical Engineers. That’s why, they should be able to crack it, in no time.

(Let me continue to be honest. I do not expect them to be able to crack it. But I do wish to know if they are able at least to give a try that is good enough!)

Even though I am jobless (and also nearly bank balance-less, and also cashless), what the hell! …

…Season’s greetings and best wishes for a happy new year!

A Song I Like:

[With jobless-ness and all, my mood isn’t likely to stay this upbeat, but anyway, while it lasts, listen to this song… And, yes, this song is like, it’s like, slightly more than 60 years old!]

(Hindi) “yeh raat bhigee bhigee”
Music: Shankar-Jaikishan
Singers: Manna De and Lata Mangeshkar
Lyrics: Shailendra

[E&OE]

/

# The anti-, an anti-anti-, my negativism, and miscellaneous

Prologue:

A better title could very well have been “What I am up against.” However, that title, I thought, would be misleading. … I really am up against many things which I am going to touch on, in this post. But the point is, these are not the only things that I am up against, and so, that title would therefore be too general.

Part I: The Anti-

First, of course, comes the anti.

I stumbled across W. E. Lamb, Jr.’s excellent paper: “Anti-photon” (1995) Appl. Phys. B, vol. 60, p. 77–84. Here is the abstract:

“It should be apparent from the title of this article that the author does not like the use of the word “photon”, which dates from 1926. In his view, there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists. I admit that the word is short and convenient. Its use is also habit forming. Similarly, one might find it convenient to speak of the “aether” or “vacuum” to stand for empty space, even if no such thing existed. There are very good substitute words for “photon”, (e.g., “radiation” or “light”), and for “photonics” (e.g., “optics” or “quantum optics”). Similar objections are possible to use of the word “phonon”, which dates from 1932. Objects like electrons, neutrinos of finite rest mass, or helium atoms can, under suitable conditions, be considered to be particles, since their theories then have viable non-relativistic and non-quantum limits. This paper outlines the main features of the quantum theory of radiation and indicates how they can be used to treat problems in quantum optics.”

BTW, in case you don’t know, W. E. Lamb, Jr., was an American, who received a Nobel in physics, for his work related to the fine structure of hydrogen [^].

So, that’s the first bit of what I am up against.

Also in case you didn’t notice, the initials are important; this isn’t (Sir) Horace Lamb (who, in case you don’t know, was that late 19th–early 20th century British guy who wrote books on hydrodynamics and acoustics that people like me still occasionally refer to [^]. (Lamb and Love continue to remain in circulation (even if a low circulation) among mechanicians even today. (Love, who? … That’s an exercise left for the reader…)))

Oh, BTW, talking of very good books that now have come in the public domain, and (the preparation required for) QM, and all the anti- and un- things, note that Professor Howard Georgi [^]’s excellent book on waves has by now come in the public domain [^].

(Even if only parenthetically, I have to note: I am anti-diversity, too. … This anti thing simply doesn’t leave me alone, though I will try to minimize its usage. Starting right now. … Georgi was born in California. He also maintains a page about women in physics [^].)

… Ummm, I’d better wrap up this part, and so…

… All in all, you can see that I don’t seem to be taking my opposition very seriously, though I admit I should start doing so some day. But the paper is great. (We were talking about the anti-photon paper, remember?) Here is an excerpt in case I haven’t already succeeded in persuading you to go through it, immediately:

“During my eight years in Berkeley, I had just one conversation with Lewis, in 1937, when he called me into his office to give some advice. It was: “When a theorist does not know what to do next, he is useless. An experimental scientist can always go into his laboratory and “polish up the brass”.”

This is the same Lewis who coined the word “photon.” … Now it convinces you to go through the paper, doesn’t it? (The paper is by Lamb; W. E. Lamb.)

[… On a more serious note, this paper has very good notings regarding the history of the idea of the photon.]

Part II: The Anti- Equals the Anti-Anti-

There is no typo here.

Even as I was recoiling off the glow (I won’t use “radiation” or “light”) of [the physics Nobel laureate] Lamb’s reputation, I began wondering precisely how I would counter his anti-photon argument. I even thought of doing a blog post about it. (After all, recently, Roger Schlafly has been hinting at that same idea, too. [May be TBD: insert links])

However, a better sense prevailed, and I did a Google search. I found a good blog post that gives a good rejoinder to the anti-photon arguments. The post is written in simple enough language that any one could understand. … But should I recommend it to you?… The thing is: It comes from a physicist who is reputed to have attempted teaching quantum physics to dogs. Or, at least, teaching people how to teach quantum physics, to dogs.

But of course, in physics, personalities don’t count, and neither do, you know, sort of like, “insults.” [I am also anti-animal rights, BTW [though all in favor of dogs].] And so, let me lead you to the relevant post.

The quantum physics-loving folks would have guessed the man by now (and every one, the fact that the author must be a man, not a woman). So the only remaining part would be which post by Chad Orzel. Here it is [^]. Once you finish reading it (including the comments on the post), then, also go through these couple of others posts by him touching on the same topic [^] [^] (and their blog comments). And, a great post (at wired.com!) by Rhett Allain [^] on the anti-photon side, to which Orzel makes a reference.

Orzel’s basic argument is that anti-bunching equals anti-anti-photon.

That explains the second part of the title.

But, before wrapping up this part, just a word on the PhD guides on the “polishing brass” side, and Indians. The anti-bunching experiments were done by Leonard Mandel [^], who among other things also guided Rupamanjari Ghosh’s PhD thesis. … Rupa…, who? I will save you the trouble of googling; see here: [^ (I am anti-government in education and science, too)] and here [^ (oh well, this post is getting just too long)].

Part III: My Negativism

Roger Schlafly has just recently written an interestingly long post on quantum entanglement. (Very long, by his standards.) In that post [^], he identifies himself as a logical positivist. This isn’t the first time that he has attributed logical positivism to his intellectual positions. Schlafly’s recent post is written, as usual, with good/great clarity

Now consider the premises, this time three, instead of the usual two: (i) Schlafly identifies himself as a logical positivist, (ii) I don’t agree with some part of his positions, and (iii) logic is logic—it cares for completeness.

Ergo, I must be a logical negativist.

That explains the third part of the title.

Some day I plan to write a post on the triplet and singlet states, and quantum entanglement.

Some still later day, I plan to explain how QM is incomplete, by pointing out how it can be made complete. … That is too big a goal to keep, you say?

Well, I do plan to at least explain in simpler terms the phenomenon of quantum entanglement, but only in reference to the text-book treatments. … That should be doable, what say?

… Don’t hold me responsible etc. on this promise; I am careless etc.; and so,  it might very well be in mid-2016 when I might actually deliver on it. … So, for the time being, make do with my logical negativism.

Part IV: Miscellaneous

M1: The preface to Georgi’s book notes the help he received while writing the book inter alia from (the same) Griffiths (as the one who has written very popular undergraduate text-books on electrodynamics and QM). (Griffiths studied at Harvard where Georgi has been a professor, though chances are they were contemporaries.) (No, this Griffiths isn’t the same as the Griffiths of fracture mechanics [^].) (Yes, this Georgi is the same as the one who has advocated the unparticle mechanics [^]. (But why didn’t he use the anti- prefix here?))

M2: The most succinct (and as far as I can make out, correct) treatment of the meaning of “hidden variables” has been not in the recent Internet writings on Bell’s inequalities but in Griffith’s undergraduate text-book on QM.

Why I mention this bit… That’s because, recently, the MIT professor Scott Aaronson had a field day about hidden variables (notably with Travis Norsen) [^], though since then he seems to have moved on to some other things related to theoretical computational complexity, e.g. this graph isomorphism-related thingie [^].

But, no, if you want to know about the so-called hidden variables well (and don’t have my “approach” or at least my “confidence”), then don’t look up the material on the ‘net or blog posts, esp. those by CS folks or complexity theorists. Instead, hit Griffith’s (text-)book.

M3: However, I am unhappy about Griffith’s treatment of the quantum postulates—he (like QChem and most all UG QM books) has only the usual $\Psi$ and doesn’t include the spinor function right while discussing the state definition. Indeed, he continues implicitly treating the two in a somewhat disjoint manner even afterwards (exactly like all UG text-books do). Separable doesn’t mean disjointed.

I am also unhappy about Griffith’s (and every other QM text-book’s) treatment of the basic ideas of identical particles and their states—the treatments are just not conceptually clarifying enough. … May I assist you rewriting this topic, Professor Griffiths? … Oh well… Before I actually make that offer to him, I will try my hand at the task, at this blog…. Sometime in/after mid-2016. (Hopefully earlier.)

But, yes, if you ask me, it’s only the spin and identical particles that still remain truly nebulous topics for the student, today. With single-particle interference experiments and the ubiquity of simulations, one wouldn’t think that people would have too much difficulty with wave-particle duality or interference etc.

Contrast staring at one or two manually drawn static graphs in a book/paper, and imagining how things would change with time, under different governing equations and different boundary conditions, vs. going through simulations on your smartphone, adjusting FPS, changing boundary conditions with the flick of a button… Students (like me) must be having it exponentially easier to learn QM these days, as compared to those hapless 20th century guys.

The points where today’s students are likely to falter would be a bit more advanced ones, like angular momentum. In fact, today’s students don’t know angular momentum well even in the classical mechanics settings. (Ask yourself: how clear and confident are you about, say, Coriolis forces, say, as covered in Shames, or in Timoshenko and Young?).

So, to wrap up, it has to be identical particles and spin that still remain the really difficult topics. Now, it so happens that it is these concepts that underlie popular expositions of entanglement. Little surprise that people never get the confidence that they would be able to deal with entanglement right.

(Focusing on “just” two states of the spin up- and down-, and therefore treating the phenomenon via an abstract two component vector, and then thinking that starting a discussion with this “simple” vector, is a very bad idea, epistemologically speaking. … Yes, I am anti-Susskind’s “theoretical minimum,” too. And yes, Griffiths is right in choosing the traditional way (of the sequence in which to present the QM spin). It’s just that he needs to explain it in (even) better manner, that’s all….)

M4: The day before yesterday was the first time this year that I happened to finally sense that wonderful winter-time air of Pune’s, while returning in the evening from our college. (Monday was a working day for us; no continuous 9-day patch of a vacation.)

It still doesn’t feel like the Diwali air this year in Pune, but it’s getting close: I spotted some nice fog/mist on the nearby nallah (i.e. a small stream) and a nearby canal, a couple of times. …

This has been a year of (heavy) drought. And anyway, these days, there is virtually no difference between the Diwali days and the rest of the year. … Shopping malls are fully Diwali-like at any time of the year for those who have the money, and most women—whether working or otherwise—these days outsource their (Marathi) “chakalee”-making anyway—even during Diwali. So, there isn’t much of a difference between the Diwali days and the other days. Except for the weather. Weather still continues to change in a distinctly perceptible way sometime around Diwali. … So, that’s about all what Diwali means to me, this year.

And, of course, some memories of the magical Diwalis that I have spent in my childhood… Many of these were spent (at least for the (Marathi) “bhau-beej” day and a couple of days more) at my maternal uncle’s place (a very small town, a sub taluka-level place). … As far as I am concerned, those Diwali’s are still real; they would easily remain that way throughout my life.

PS: Having written the post, I just stepped into the kitchen to make me a cup of tea, and that’s when father told me that home-made (Marathi) “chakalee”s had arrived from our family friends just last evening; I didn’t know about it.

Instantaneously, my song-selection collapsed into an anti-previously measured state. (It happens. Real life is more weird than QM.)

Epilogue:

Happy Diwali!

PS (also) to Epilogue:

Excuse me for a couple of weeks now. I will continue studying QM (from text-books), but I will also have to be taking out my notes for an undergraduate course on CFD (computational fluid dynamics, in case you didn’t know) that I should be teaching the next semester—which begins right in mid-December. (In India, we don’t always follow the Christmas–New Year’s–Next Term sequence.) I anyway will also be traveling a bit (just short distances like Mumbai and Nasik or so) over the next couple of weeks. So, I don’t think I will have the time to write a post. (That, in fact, was the reason why I threw in a lot of stuff right in this post.)

… So, there… Take care, and best wishes, once again, for a bright and happy Diwali (and to those of you who start a new year in Diwali, best wishes for a happy and prosperous new year too.)

A Song I Like

(Marathi) “tabakaamadhye ithe tevatee…”  (search on the transcriptionally incorrect “divya divyanchi jyot”)
Singer: Asha Bhosale
Lyrics: Ravindra Bhat

[PS: I kept on adding material after publication of post, and now it has become some 1.5 times the original one. Sorry about that (though I did all the revisions right within 18 hours of publication), but now I am going stop editing any further. Put up with my grammatical mistakes and awkward constructions, as usual. And, if in doubt, ask me! Bye for now.]

[E&OE]

/

# “They don’t even touch a good text-book!”

“They don’t even touch a good text-book!”

This line is a very common refrain that one often hears in faculty rooms or professors’ cabins, in engineering colleges in India.

Speaking in factual terms, there is a lot of truth to it. The assertion itself is overwhelmingly true. The fact that the student has never looked into a good (or “reference” or “foreign authors'”) text is immediately plain and clear to anyone who has ever graded their examination papers, or worked as an examiner on the oral/viva voce examinations.

The undergraduate Indian students these days, esp. those in Pune and Mumbai, and esp. those in the private engineering colleges, always refer to only a locally published text for all their studies.

These texts are published by a few local publishers well known to the students (and their professors). I wouldn’t mind dropping a few names: Nirali, Pragati, TechMax, etc. The books are published at almost throw-away prices (e.g. Rs. 200–300). (There also exists a highly organized market for the second-hand books. No name written, no pencil marks? Some 75% of the cost returned. Etc. There is a bold print, too—provided, the syllabus hasn’t changed in the meanwhile. In that case, there is no resale value whatsoever!)

The authors of these texts themselves are professors in these same private engineering colleges. They know the system in and out. No, I am not even hinting at any deliberate fraud or malpractice here. Quite on the contrary.

The professors who write these local text-books often are enthusiastic teachers themselves. You would have to be very enthusiastic, because the royalties they “command” could be as low as a one-time payment of Rs. 50,000/- or so. The payment is always only a one-time payment (meaning, there are no recurring royalties even if a text book becomes a “hit”), and it never exceeds Rs. 1.5 lakhs lump-sum or so. (My figures are about 5 year old.) Even if each line is copied verbatim from other books, the sheer act of having to write down (and then proof-read) some 200 to 350 pages requires for the author to invest, I have been told, between 2 to 4 months, working overtime, neglecting family and all. The monthly salary of these professors these days can easily approach or exceed Rs. 1 lakh. So, clearly, money is not the prime motivation here. It has to be something else: Enthusiasm, love of teaching, or even just the respect or reputation that an author hopes to derive in the sub-community of these local engineering colleges!

These professors—the authors—also often are well experienced (15–40 years of teaching experience is common), and they know enough to know what kind of examination questions are likely to come up on the university examinations. (They themselves have gone through the same universities.) They write these books targeting only task: writing the marks-scoring answers on those university examinations. Thus, these “text” books are more or less nothing but a student aid (or what earlier used to be called the “guide” books).

It in fact has evolved into a separate genre by itself. Contrary to an impression wide-spread among professors of private engineering colleges in India, there in fact are somewhat similar books also used heavily by the students in the USA. Thus, these local Indian books are nothing but an improvised version of the Schaums’ series in science and engineering (or the Sparks Notes in the humanities, in the US schools).

But there is a further feature here. There is a total customization thrown in here. These local books are now-a-days written (or at least adapted) to exactly match the detailed syllabus of each university separately. So, there are different books, by the same author and for the same subject, but one for Mumbai University, and the other for Pune University, etc. Students never mix up the universities.

The syllabus for each university is followed literally, down to dividing the text into chapters as per the headings of the modules mentioned in the syllabus (usually six per course), and dividing each chapter into sections, with the headings and order of these sections strictly following the order and the letter of the syllabus. The text in each section is followed by a compilation of the past university examination questions (of that same university) pertaining to that particular section alone. Most of these past examination questions are solved in the text—that’s the bulk of the book. When the opening page of a chapter lists the sections in it, the list also carries, in the parentheses, whether this section is “theory” or “numericals”.

Overall, the idea is, even just looking at the “text” book, a student can easily anticipate whether a question is likely to be asked on a given section or not, and if yes, of what kind. The students also work out many logics: “Every semester, they have asked a question on this section. So we have to mug it up well.” Or, playing the “contra”: “Last three semesters, not a single question here? It’s going to come this time round.” Etc. (Yes, I followed this practice in my lectures, too—I did want my students to score well on the final university examinations, after all!)

The customization, for each revision of the syllabus of each university, is done down to that level of detail. So, for the first year course on electrical engineering, you have one text-book of title, say, Electrical Engg. (FE), Pune University, 2012 course, and another text book, now of the title, say, Basic Electrical Technology (FE), Mumbai University, 2011 course. Etc.

That’s what I mean, when I use the phrase the “local” text-books.

I certainly don’t mean the SI Units editions of American texts, or the Indian Standards-adapted editions of reputed texts (such as, say, Shigley’s on design or Thomson and Dahleh’s on vibrations). I don’t mean the inexpensive Indian editions of foreign texts (such as those by Pearson, Wiley, ELBS, etc.) I also don’t mean the text-books written by the well-known Indian authors working right in India (such as those by IIT professors, and published by, say, Universities Press, Narosa, or PHI). I don’t even mean the more general text-books written for Indian universities and/or the AMIE examinations (such as those by S. Chand, Khanna, CBS, etc.). When I say “local” text-books, I specifically mean the books of the kind mentioned above.

Undergraduate students in Pune and Mumbai these days refer only to these local books.

They (really) don’t even bother to touch a good reference text, even if it’s available on the college library shelf.

In contrast, in our times, the problem was, we simply didn’t have the “foreign authors'” texts available to us—not always even in the COEP library. In those days, sometimes, such books happened to be too expensive, even for COEP’s library. And, even back then, Shahani’s text-books anyway were available. But at least, they didn’t cater to only the Pune university (they would list problems from universities as far flung as Madras, Gorakhpur, Agra, Allahabad, etc.) And, in fact, these books were generally looked down upon. Even by the students themselves.

The contrast to today’s situation is too glaring. Naturally, professors sometimes do end up saying the title line with a tone of exasperation.

Yes, I used to sometimes say that line myself, of course with sarcasm, when I taught in the late ’80s in the Pune of those days. (The situation back then was not so acute.) Almost as if by habit, I also repeated the line when I more recently taught a course at COEP (2009, FEM). However, observing students, somehow, my line had somehow begun to lose that cutting edge it once had. First, at COEP, I had the freedom to design this course (on FEM), and they did buy at least Logan and/or Cook. (Even if I was distributing my PDF notes.) And, there was something else to it, too. I somehow got a vague feel that it somehow wouldn’t be fully right to blame students (I mean COEP students in general). However, my COEP stint was only for one semester, only for one course, and only as a visiting faculty. So, the vague feel simply remained what it was—just a vague feel.

Then, recently in 2014, when I began teaching at a private engineering college in Mumbai, I once again heard this line from the other professors. And, I used it myself too. With the usual sarcasm. I did that perhaps for the most part of my first semester there.

However, some way down the line, I once again got that vague feel that, may be, something was “wrong” somewhere, even here, in Mumbai: these kids really were trying to be sincere, and yet, for some reason unknown to me, they still wouldn’t at all refer to good texts.

This is an aside, but I can tell you that it’s very easy to read the faces of the insincere people, esp. when they are young. There are some insincere students too. But, at least going by my own experience, they are in a minority. (It is a headache-some minority. Yet, by numerical magnitude alone, it certainly is in a small minority.) I am not saying this to be politically correct, or to win points from students. What I said is the factual case. In fact, my experience is that when it comes to in-sincerity, parents easily outperform their children. May be because, the specific parents that we mostly end up seeing in college are those whose kids have some problem—low attendance, fee payments, other issues, etc. The parents with whom we get to interact really well, thus, happens to be a self-selected sub-group. They aren’t necessarily representative of all parents… Yet, I am also sure that that’s not the real reason why I think parents can easily be more insincere. I think the real reason is that, at their age, the kids are actually unable to fake too much. It’s far easier for them to be sincere than to be a fake and still get away with it. They just can’t manage it, regardless of their desire. And, looking at it in a better light, I here remember what Ayn Rand had once said in a somewhat similar context, “one doesn’t start out in life by spitting on one’s own face—it’s not in the essential nature of life” or something like that. (Off-hand, I think, it was in the preface to the 25th anniversity edition of The Fountainhead.) So, the kids, by and large, are sincere. … By the time they themselves become parents—well, let’s leave that story right here. (We need them to make all those fee payments, anyway…)

So, coming back to the main thread, I would anyway generally chat with the students, and so, I started asking, esp. some of the more talkative students, the reason why they might not be referring to good texts. After all, in my lectures, I would try to provide very specific references: specific section numbers or even page numbers, in a specific edition of a specific reference text. (And these texts were available in the college library.) Why, I once had even distributed an original research paper. (It was Griffth’s seminal 1920 paper starting the field of fracture mechanics. Griffith’s argument here is rather conceptual, and the paper has surprisingly very little maths. Whatever the maths there is, it is very easily accessible to the SE students, too.)

The result of my initial attempts to understand the reason (why students don’t read good texts) was not so encouraging. The talkative students began dropping by my cabin once in a while, asking which section to use while answering a certain assignment question or so. However, they still only rarely used those better texts, when it came to actually completing their assignments. And, in the unit tests (and in the final end-sem examination), they invariably ended up quoting only the local text books (whether verbatim or not).

The exercise was, thus, futile. And yet, the students’ sincerity—at least the sincerity of their desire, as in contrast to their actions—could not be put in doubt.

So, I took it as a challenge. I set this as a problem for myself: To discover the main reason(s) why my students don’t refer to good text-books. The real underlying reason(s), regardless of whatever they otherwise did to impress me.

It took a while for me to crack the problem. I would anyway generally chat with them, enquiring where they lived, what their parents did, about their friends and brothers and sisters, etc. In addition, I would also observe, now with this new challenge somewhere at the back of my mind, how they behaved (or rushed around) in college: in hallways, labs, canteen, college ground, even at the bus-stop just outside the college, etc.

…Finally, I got it! At least one reason, a main reason, a systemic reason that applied even to those who otherwise were good, talented, curious, or just plain sincere.

As soon as I discovered the reason, I shared it with every one. In fact, I first shared it with my students, before I did with my colleagues or superiors. The answer lies in an Excel spreadsheet, here [^]. (It actually was created in OpenOffice Calc, on Windows 7.)  Go ahead, download it, and play with it a bit. The embedded formulae should be self-explanatory.

The numbers used in the spreadsheet may differ. The specific numbers I have used in the spreadsheet refer to my estimates while working at a college in Mumbai, in particular, in Navi Mumbai. In Mumbai, the time lost commuting is really an issue. If a student lives in Thane or Andheri and attends a college in Navi Mumbai, he easily spends about 3–4 hours in the daily commute (home->bus->railway station/second bus/metro–>another bus or six-seater, all of it taking about 1.5 hours one way, or more). In Pune, the situation is much more heterogeneous. One student could be spending 3 hours commuting both ways (think: from Nigdi to VIT) whereas some other student could be just happily walking to the college campus (think: Paud Phata residents, and MIT). It all depends. In Pune, many students would be using two-wheelers. In any case, for a professor, the only practical guideline for the entire class that he can at all use, would have to be statistical in nature. So, it’s the class average for the daily commute time that matters. For Mumbai in general, it could be 2–3 hours, for Pune students, it could be, say, between 1 to 2 hours (both ways put together).

So Pune is a bit easier on students. In contrast, for many of my Mumbai students, the situation was bad (or even very bad), and they were trying hard (or very hard) to make the best of it. It must have been at least a bit frustrating to them when professors like me, on the top of everything, were demanding making references to good foreign texts, and openly using a sarcastic tone—even if generously laced with humor—if they didn’t. It must have been frustrating to at least 40–60% of them. (The number is my estimate of those who were genuinely interested in referring to good books, even if only for the better-drawn and colorful diagrams, photographs, and also mathematical proofs that came without errors or without arbitrary replacement of $\partial$ by $d$.)

And why do I say that it must have been frustrating? Why didn’t I say it might have been frustrating?

Because, I cannot ever forget that look of that incredibly honest appreciation which slowly appeared on all their faces (including the faces of the “back-benchers”), as I shared my discovery in detail with them.

* * * * *   * * * * *   * * * * *

Do you have the time to read good, lengthy, or conceptually clarifying “reference” texts? Say, Timoshenko (app. mech. and strength of materials); Shames, or Popov (strength of materials); White, or Fox & McDonald, or Som & Biswas (Fluid Mech.); Holman, or Nag, or Sukhatme (Heat Transfer)?

And, if you do, do you spend time reading these texts? If yes, did you complete them (I mean only the portion relevant to the syllabus) in the same semester that you were learning or teaching the subject for the first time? Could you have?

And yes, in my last sentence, I have included “teaching” too. My questions are directed to the professors too. In fact, my questions are directed, first and foremost, only at them.

After all, it is the professors—or at least some of us—who are in the driver’s seat here; the students never are. It is the professors who (i) design the syllabii as well as the examination schemes (including the number of tests to have and their nature), (ii) decide on the number of assignments (and leave no opportunity to level criticism in our capacity as External Examiners, if the length or difficulty of an assignment falls short), (iii) decide on the course text-books (and take due care to list more than 5 prescribed text-books, and more than 10 reference books per course) (iv) decide on the student attendance criteria in detail, up to the individual course level, and report on the defaulting students (and follow through with the meetings with their parents) every two weeks or at least once a month, (v) set the examination papers according to the established pattern—after all, it’s only us who is going to check the papers!, (vi) sometimes, write those local text-books!, and (vii) also keep the expectation that students should somehow show in their final university examination answer books, some evidence of having gone through some good, thick, reference texts, too. Whether we ourselves had managed to do that during our own UG years or not!

And, yes, I also want the IIX professors to ponder over these matters. All their students enjoy a fully residential program; these kids from these private engineering colleges mostly don’t. They at IIXs always get to design all their course syllabii and decide on the examination patterns, and they even get to enjoy the sole responsibility to grade their students. The possibility of adopting a marks normalization scheme, after the examination, always lies at hand, with them, just in case a topic took too long with a certain class or so… Are they then being reasonable in their request demand that the students of these “other” engineering colleges in India be well-read enough, at least by the time the students join them at IIXs for ME/MTech studies?

As to me, no, as I indicated in my earlier posts, while being a professor, I could not always find the time to do that—referring to good text-books. I tried, but basically my situation wasn’t much different from that of my students—we both were short on the available time. So, I didn’t always succeed.

[As to my own UG years, it was mixed: I did hunt for months, and got my hands on, the books like Reed-Hill, White, Holman, etc. However, I would be dishonest if I claimed that it was right during my UG years that I had got whatever I did, from books like these. In my case, the learning continued for years. Yes, I even bought and religiously studied once again even Thomas & Finney’s calculus, when I was in my PhD program at UAB. Despite my attempts during the UG years, I really cannot ascribe a large part, or even a significant part of my current understanding to my UG years. Your case may be different; I was just narrating my own experience.]

… And, as far making references to good books goes, now that I do have time at my hand these days, there is another problem: I don’t know what course in particular I will be teaching the next semester, and where—or for that matter, whether some college will even hire me in the first place, or not.

So, I end up “wasting” my time writing blog posts like this one. Thus, I, too, end up not touching a good reference text!

* * * * *   * * * * *   * * * * *

A Song I Like:

(Hindi) “aane waalaa pal, jaane waalaa hai…”
Lyrics: Gulzaar
Singer: Kishor Kumar
Music: R. D. Burman

[I will go over this post once again, editing it, and may be adding a bit here and there. Done. This post is already too long. So, I will write another post—a brief one—to jot down some tips to make the best possible use of the student’s time—including my suggestions to the engineering colleges as to what they can do to help the students. Also, my take on whether the system as noted above has diluted the quality of education or not—esp. as in contrast to what we had as UG students at COEP more than three decades ago.]

[E&OE]

If you are a working/budding physicist, in all likelihood, you have been taught your quantum physics using books that mangle the historical order of development. Indeed, even McQuarrie’s book on quantum chemistry gives only a sketchy idea about the actual order in which the subject actually got developed. (IMO, the quantum chemistry books are better suited for a self-study of quantum physics. Among them, McQuarrie’s is the easiest to follow, though Levine’s has some topics better covered.)

The conceptual confusion that results out of abandoning the proper hierarchical order is just too huge. For instance, here is a quick question: Every one knows that Bohr’s model came on the scene before the real QM did. But the question is: When did the correspondence principle arrive? During those Bohr-Einstein debates? Or earlier?

Answer: Earlier. Right in 1913, when Bohr put forth his model. The Bohr-Einstein debates, in contrast, came much later, around the 1927 times, i.e., after all the essential principles of QM had already been discovered. And, BTW, it’s the complementarity principle which came during these latter times of the Bohr-Einstein debates.

Interesting? Ready for another question? Ok. Here we go.

Identify which development came first: (i) Dirac’s use of the Poisson brackets in quantum theory, (ii) The application of the matrix mechanics to the calculation of the hydrogen atom spectrum, (iii) The probability interpretation of the wave function?

If you are like 99% of others, you will say: In the order: (iii), (ii) and (i). The correct answer is: (i), (ii) and (iii), precisely in that order!

Don’t let yourself think that such questions are good for those fun quiz competitions or for the generally satisfying trivia. There is a very simple but very profound truth hidden in here: If Pauli could work out the hydrogen atom spectrum before anyone had even an inkling of a probability interpretation, what it obviously means is that there is some way that Pauli used, which is (implicitly or explicitly) more fundamental than some formal system that posits “probability currents” as the first axiom of QM.

More generally, if the historically less-progressed context (i.e. knowledge available by a certain year X) was factually enough (or sufficient) for someone to think of a great new idea, then, among all the conceivable or proposed ordering of topics or contexts that can be taken as foundational to explain that novel idea, the historically least progressed context is the only one that is necessary.

All the rest of the conceivable schemes are either after-thoughts, or organizational devices like mnemonics, or mere deductive tricks, or worse: mere cognitive burden on anyone who takes them seriously as a hierarchically proper scheme.

Having said that, now, pick up any of the introductory textbooks on quantum theory, carefully check out the order in which the topics are progressed in that book, and then ask yourself: How much of an unnecessary, useless cognitive burden is this particular author (i.e. an influential physicist) thrusting on your mind? How much lighter, better, would you feel if the order were something like the following? (The dates in parentheses follow the YYYY/MM format):

• Planck (1900/10): The quantization of energy of the electromagnetic oscillators in the walls of a light-radiating cavity
• Einstein(1905/06): The explanation of the photoelectric effect by quantizing the light radiation itself
• Einstein(1906/12): The first quantum theory of the specific heat of solids
• Bohr(1913/02–09): An explanation of the pattern of the discrete lines in the atomic spectra
• Bohr(1913/02–09): The correspondence principle
• Sommerfeld (1916–1920): Corrections to the Bohr model, introducing additional quantum numbers
• Compton (1923/05): A light scattering experiment, which confirms the quantum nature of light
• de Broglie (1923/09): The hypothesis of the matter waves, with a view to extend the wave-particle duality of light to matter as well
• Pauli (1925/01): The discovery of the exclusion principle, for the electrons in atoms
• Heisenberg (1925/06): The invention of the arrays of observables, to explain the atomic spectra
• Born and Jordon (1925/09): The first physical law stated using non-commuting symbols: $pq - qp = i\hbar I$
• Goudsmit and Uhlenbeck (1925/10): The experimental discovery of the electron spin
• Pauli (1925/11): The first success in applying the matrix mechanics to the line spectrum of hydrogen, including the Stark effect
• Dirac (1925/11): The identification of a Poisson brackets structure in Heisenberg’s analysis.
• Born, Jordon and Heisenberg (1925/11): The “three-man paper” on the mathematics of matrix mechanics submitted for publication (9 days after Dirac’s above paper)
• Schrodinger (1925/12): Formulation of the first ideas of his wave mechanics
• Schrodinger (1926/01): Successful application of his wave equation to the hydrogen atom
• Schrodinger (1926/03): Demonstration of the mathematical equivalence of the matrix mechanics and the wave mechanics
• Born (1926/07): The probability interpretation of the wave function
• Dirac (1926/09): The transformation theory—the wave and matrix mechanics as special cases
• Davisson and Germer (1927/01): Experimental confirmation of diffraction of electrons by a crystal lattice
• Heisenberg (1927/02): Formulation of the uncertainty principle
• Bohr (1927/09): Formulation of the complementarity principle and the Copenhagen interpretation
• Thomson (1927/11): Another experiment which confirms that matter diffracts
• Dirac (1930/05): Publication of the first edition of his book (having its “first chapter missing”)
• Dirac (1939): The third edition of his book introduces the bra-ket notation—the starting point for today’s “Alice and Bob”-obsessed idiots

As you probably know, I have been trying to follow the historical sequence in writing my book. So, in a way, I have been looking a bit carefully into the historical order in which things happened. Still, I had a few surprises in store even for me when I really sat down to compile the above list. Here they are: (i) Einstein’s 1906 paper (which I used to put somewhere in the late teens), and, (ii) Dirac’s 1925 paper.

I am sure that things like the following would come as surprises to many of you: (i) Dirac’s transformation theory being formulated before either the uncertainty principle or the complementarity principle was, (ii) Pauli working out the hydrogen atom line spectrum using the matrix mechanics barely within 2 months of the writing of Heisenberg’s first paper, and in fact before Heisenberg himself could succeed doing so, (iii) Born identifying the matrix nature of the Heisenberg’s non-commutative arrays and Jordon working out the derivations of the mathematics involved in it.

I also think that the help that was both required and received by Heisenberg, might have come as a surprise to many of you, esp. when contrasted with Schrodinger’s single-handed development of all the fundamentals of the wave mechanics—except, of course, the probability interpretation of the wave function, which was supplied by Born.

Most importantly, I think, quite a few must have been shocked to find that Dirac could work out his theory, even predict the existence of anti-matter, without explicitly using the bra-ket notation itself. It has become a fashion to explain this notation right in chapter 1 (though, thankfully, not right in the preface—not yet, anyway).

While writing on the heuristics that he follows while deciding whether a paper on the P-vs-NP issue is worth reading or not, Prof. Scott Aaronson has indicated that any paper not written in LaTeX is suspect.

I have a similar test for books, papers, tutorials etc. on quantum physics, especially the introductory or foundational ones. (Seriously. I have actually followed it over quite a few years in the recent past, and very successfully, too.)

I don’t take any paper/notes/tutorials/book on the foundations of quantum physics for a serious consideration (i.e., I don’t even browse or flip through its abstract) if it has “Alice” and “Bob” written anywhere within it.

Ditto, for any textbook on quantum physics, if it has those two words appearing within the first 90% of the real text matter.

So, hey physicists! Revise your books to follow the kind of an order I have given above!!

Why?

Because, I say so. That’s why.

* * * * *   * * * * *   * * * * *

No “A Song I Like” section, once again. I still go jobless. Keep that in mind.

[This is initial draft, published on September 26, 2012, 8:07 PM, IST. May be I will make some minor corrections/updates later on.]
[E&OE]