"Calculus is so last century"? Hardly!

Two months ago a WSJ op-ed ('Calculus Is So Last Century') by Tianhul Michael Li caught the eyes of many mathematicians as well as physicists (and I can assure you, many space and solar physicists). In it Li basically argued that calculus is primarily of the last century as it "is the handmaiden of physics - invented by Newton to explain planetary and projectile motion."

Adding:

"While its place at the core of math education may have made sense for Cold War adversaries engaged in a missile and space race, 'Minute Man' and 'Apollo' no longer occupy the same prominent role in national security and prosperity they once did."

Which makes one wonder what alternative universe and planet this guy is living on? Surely not the one that I'm on!

A feature story in Saturday's NY Times, for example, expatiated on the latest developments including "hypersonic targeted warheads" not only under development by the U.S. but China and Russia as well. The U.S. has a non-nuclear version but the take is that the Russians and Chinese are developing nuclear warheads- independent of missiles- that can be hurled into space then come down on an enemy with little or no warning time. This is in contrast to Cold War ICBMs that general gave about 30 minutes advance notice, unless launched by submarines)

Most alarming is the drive by a number of nations, including the U.S., China and Russia, to develop low yield nukes which - as the article put it- could make their use much more likely than the "mutually assured destruction"  (MAD) model of the past allowed. Indeed, the Russian already have a military posture firmly in place that permits the use of tactical nukes (with 10- 25 kt warheads) in the event that NATO forces overwhelm Russian positions in Eastern Europe. (And cruise missiles are also being outfitted with low yield nukes.)

The current U.S. proposal to "modernize" its nuclear warheads is also seen as possibly destabilizing the existing system and rousing the Chinese and Russians to action in their own warhead development.

Far from national security having switched away from H-bomb Cold War worries, we are evidently entering a new era wherein nuclear "swords of Damocles" literally hang over our collective heads, in the form of targeted hypersonic warheads, or even nuclear-armed satellites,

None of these systems will use linear algebra (the math Li mainly advocates teaching) to reach their targets, but plain old brute differential calculus, of a form I described in earlier blog posts, e.g. from May 6, 2013 (the rocket problem).

Surely then, Calculus isn't passé if it is still central to the ultimate form of nullifying any national security or 'prosperity': nuclear war,

Meanwhile, in fields as diverse as astrophysics, astronomy, quantum mechanics and plasma physics - not to mention climate science -  differential calculus is as critical as ever. While it is true algorithms and numerical simulations, models have been developed in all these fields, it is also true that differential calculus underscores that development and continued tweaking of those models depends on the actual math, not just crunching numbers into numerical or statistical data sets.

Li is correct when he asks at the outset 'Can you remember the last time you did calculus?'  But that question could also be asked about linear algebra or finite mathematics, or a course in multivariate statistics.  The point is that any form of advanced learned in high school or college will fall by the wayside if it is not used in some way later on, irrespective of how it was taught.

Li argues that he isn't saying calculus shouldn't be taught, as it is "great mental training" which is most gracious of him. But if nuclear warheads are going to be set off in detached form and land on my head, I'd be curious as to the math underlying the dynamics. I don't want to just read about it in the NY Times.

Where I do agree with him more, is when he criticizes the "single drive toward calculus in high  school and college" which "displaces other topics more important for today's economy and society".  These include "statistics, linear algebra and algorithmic thinking" (such as revealed in finite math.)

I concur because I've always been skeptical of the high school AP Calculus curricula and whether students are really of sufficient mental maturity that the courses serve any useful purpose (apart from the usual academic 'feather' in the cap to expand their choice of college). And I never saw the reason for pre-med students to be taking calculus. In each case then, perhaps some finite math course or statistics would better serve these populations as opposed to differential and integral calculus.

Calculus then, ought to be here to stay, certainly for most college math and science majors and perhaps even some others (e.g. Philosophy) interested in exploring the role of quantum mechanics in modern expositions, say involving quantum nonlocality and entanglement.

Solutions to Quantum Mechanics Problems (2)

1) We see that from the diagram below that  if L = 3 we get one value for ℓ. and three for j :

So ℓ 1 = 1 and ℓ 2 = 2 therefore:  ℓ 1 + ℓ 2 = 1 + 2 = 3.

Meanwhile, S can be defined by only one value of s, or s = ½

The possible j-values are:

j = ℓ + s = ℓ 1 + s = 1 + ½ = 3/2

j = ℓ2 +s = 2 + ½ = 5/2

j = ℓ - s = ℓ 1 - s =  1 - ½ = ½

j = ℓ 2 - s =  2 - ½ = 3/2

So in total: j = ½, 3/2 and 5/2

Note that, conforming to j-selection rules, all the j's differ by an integral amount, though they are half-integral (e.g. 3/2, 5/2) themselves.

To obtain any J (total angular momentum) we need an L-S coupling vector that yields J = 5/2. Two possible L-S couplings are available: [L + S] and [L + S – 1] and it is the last that yields the appropriate result: [3 + ½ - 1] = 5/2

This means we need values such that  ℓ 1 = 1,  ℓ 2 = 2 and s = ½ to make this work.

2) The vector solutions (which the reader can do) following the same tack as the previous probme, will show:

j = ℓ + s = 3 + ½ = 7/2

and:

m _J  = -7/2, -5/2, -3/2, -½, ½, 3/2, 5/2 and 7/2

meanwhile:

j = ℓ - s = 3 -  ½ = 5/2

So:

m _J = -5/2, -3/2, -½, ½, 3/2, 5/2

Note that these last m _J quantum numbers would also be the ones applicable to the original problem for which: L = 3 , S = ½, and J = 5/2.

3)  a) The numerical value of the total angular momentum is given by:

L = [ℓ (ℓ + 1)]^1/2 (ħ)

Where ℓ = 3, then:

L = [3 (3 + 1)]^1/2 (ħ)  =    [3 (4)]^1/2 (ħ)   =   3Ö2  ( ħ)

b) The z-component of the orbital angular momentum is given by:

L(z) = m _ℓ ħ

For this election,  m _ℓ  = 3, so that:

L(z) = 3 ħ

(4) We have ℓ1 =3 and ℓ2 = 2, then:

Therefore, the possible values of L will be found  from letting ℓ1 =3 and adding each next descending value of m _ℓ  from 2, to 1, to 0, to -1, to -2:

(3) + 1 =   4

(3) + 0 = 3

(3) + (-1) = 2

(3) +  (-2)  = 1

So the total angular momentum L can have the values:

5, 4, 3, 2 and 1.

The f electron has ℓ =3  so that the total angular momentum quantum number possibilities are:

j = ℓ + ½,   ℓ - ½

Then: j = 7/2,  5/2

(5) For 4s 3d we have:

ℓ 1 = 2, s1 = 1/2, ℓ 1 = 0 and s1 = 1/2. Then for the maximum value:

ℓ 1 + ℓ 2 = 2 + 0 = 2

and: s= s1 + s2 = 1/2 + 1/2 = 1

The lowest  energy level is then:

4s 3d (³D1)

Since 2s' + 1 = 3, leading to minimum:

j' = [s' - ℓ '] = 1.

Using the assorted combinations, for ℓ '= 0 and ℓ ' = 2, to get the respective j' values (in combination with s' = 0) and then further for s' = 1, we arrive at the energy configuration diagram shown below:

Quantum Mechanics: As Mystifying Now as 100 Years Ago

A new survey of physicists working in the field of quantum mechanics (including a poll reported in a recent preprint on the physics arXiv server¹),) discloses that the so-called experts remain as mystified as ever. The semi-serious poll of 33 key thinkers on the fundamentals of quantum theory shows that opinions on some of the most profound questions are fairly evenly split over several quite different answers.


For example, votes were roughly evenly split between those who believe that, in some cases, “physical objects have their properties well defined prior to and independent of measurement” and those who believe that they never do. Despite the famous idea that observation of quantum systems plays a key role in determining their behavior, 21% felt that “the observer should play no fundamental role whatsoever”.

Regrettably, QM is not a descriptive field, i.e. amenable to straightforward English interpretation. It is primarily a mathematical theory. If one ventures outside the bounds of the mathematical descriptions to offer English interpretations, one risks nonsense. I believe it was Feynman, in a Preface to his 'Lectures in Physics', Vol. III, who remarked that once one tries to use purely English descriptions to button hole QM he or she will "disappear down a rabbit hole, never to appear again."

For readers who want a comprehensive and understandable book that provides the basis for the Copenhagen Interpretation, I recommend Heinz Pagels `The Cosmic Code' (Bantam, 1982) - which will dispel a lot of incorrect perceptions and assumptions that have accumulated over the past 25 years. In his book, Pagels endorses the best policy in quantum mechanics as simply being a `fair witness'. That means absolutely avoiding embellishment and exaggeration of the results, including projection of personal `fantasies'. If one insists on reading more into quantum measurement results than their statistical significance allows, self delusion ensues.

Back to the poll devised by Anton Zeilinger of the University of Vienna, together with Maximilian Schlosshauer, now at the University of Portland, Oregon, and Johannes Kofler at the Max-Planck-Institute of Quantum Optics in Garching, Germany. It was then disseminated at a meeting of the Templeton Foundation where attendees were given 16 multiple-choice questions on key foundational issues in quantum theory.

Disagreements over the theory’s interpretation have existed ever since it was first developed, but Zeilinger and his colleagues believe that their poll might be the first to interrogate the full range of views held by experts. A previous poll at a 1997 quantum mechanics workshop in Baltimore asked attendees the single question of which interpretation of quantum theory they favored most.

Probably the most famous dispute about what quantum theory means was that between Albert Einstein and his peers, especially the Danish physicist Niels Bohr, on the question of whether the world was fundamentally probabilistic rather than deterministic, as quantum theory seemed to imply. One of the few issues in the new poll on which there was something like a consensus was that Einstein was wrong. Quantum theory IS probabilistic!

This is because most quantum physicists still adhere to the comprehensive, original interpretation of quantum theory developed in the 1920s: the so-called Copenhagen interpretation. This proposed that the physical world is unknowable and in some sense indeterminate, and the only meaningful reality is what we can access experimentally. The mathematical underpinning was perhaps first explicated by Max Born, who showed the quantum wave function (PSI) was a statistical artifact, not a real physical wave.

Consider: you want to find the probability that some particle will be found in a region of length a. The probability is given as an integral in terms of the wave function (PSI):  INT (-oo to + oo) PSI* (PSI) dx

where PSI =

A sin (2π x/ a) exp (- iEt/h/2π)

and PSI* = A sin (2π x/ a) exp (iEt/ h/2π)

with h/ 2π = 1.054 x 10 ^-34   J-s (modified Planck constant h)


Obviously, without doing the full integration, the complex functions for PSI and its complex conjugate (PSI*) cannot be identified with any real world entity. Hence (Max) Born's conclusion to treat the wave function as primarily statistical in nature is amply justified.

As the earlier Baltimore meeting, the Austrian poll found the Copenhagen interpretation to be favored over all others, including the Stochastic interpretation of David Bohm and Brian Hiley (which treats PSI as a physically real wave) and the ‘Many World’ interpretation of Hugh Everett. Even so, only 42% of the voters endorsed the Copenhagen. However, the same 42% also admitted that they had switched interpretation at least once. And whereas a few decades ago the options were very few, says Schlosshauer, “today there are more ‘sub-views’.”

The most striking implication of the poll is that, while quantum theory is one of the most successful and quantitatively accurate theories in science, interpreting it is as plagued with as many controversies as it was at the outset.  According to Schlosshauer “Nothing has really changed, even though we have seen some pretty radical new developments happening in quantum physics, from quantum information theory to experiments that demonstrate quantum phenomena for ever-larger objects. Some thought such developments would push people one way or the other in their interpretations, but I don't think there’s much evidence of that happening.

However, he says there was pretty good agreement on some questions, adding.

“More than two-thirds believed that there is no fundamental limit to quantum theory — that it should be possible for objects, no matter how big, to be prepared in quantum superpositions like Schrödinger’s cat. So the era where quantum theory was associated only with the atomic realm appears finally over.”

Wow! So we can now look at the superposition effect experienced in rockets and automobiles?

Other notable views: 42% thought that it would take 10–25 years to develop a useful quantum computer, whereas 30% placed the estimate at 25–50 years. Meanwhile in the much debated role of measurement in quantum theory — how and why measurements affect outcomes —the votes split with 24% regarding it as a severe difficulty and 27% as a “pseudoproblem”.

Zeilinger and colleagues do not claim that their poll is rigorous or necessarily representative of all quantum researchers. John Preskill, a specialist in quantum information theory at the California Institute of Technology in Pasadena, suspects that “a broader poll of physicists might have given rather different results”.

Perhaps the most amazing line of agreement from all participants is that quantum theory does its job so well yet stubbornly resists answering our deeper questions. Maybe this "contains a lesson in itself,” according to Schlosshauer. In fact, the most revealing answer may well have been that 48% believe that there will still be conferences on the foundations of quantum theory in 50 years time.

I would not dispute that at all!