Should Undergrads Be Exposed To Modern Physics?

A Loyola University undergrad, ca. 1966, studying for a test on special relativity. The class used a text by Hugh D. Young entitled 'Fundamentals of Mechanics and Heat' (which I reviewed at Amazon) and included at least one chapter (14) on special relativity. 

That is perhaps the most that the majority of bright undergrads would be able to cope with even today.


An ongoing controversy in physics education is just how much modern physics today's undergrads should be exposed to. By definition, 'modern physics'  includes all the physics established since 1900, including: quantum theory, special and general relativity and some high energy particle physics (nature of quarks, neutrinos etc.)

The typical plaintive cry, to judge from occasional letters (say to Physics Today)  amounts to some variation on:

"Why must we be stuck in the tedious, boring physics of the 17th, 18th and 19th centuries? Why always Newton, Faraday and such instead of Bohr, Einstein and Schrodinger?  We want to be taught more of the modern physics less of the antiquated stuff!'

Thus, they are chagrined they're still being taught mostly Newtonian mechanics,  and 'old-style' Electricity and magnetism, thermal physics and only a little bit of special relativity.  What they don't say is that even that "little bit" of special relativity often poses problems.

So the question remains: can they really handle more? Most undergrads even at the top universities are simply not at the level they can grasp quantum mechanics or general relativity. Even Richard Feynman ('Feynman Lectures')  while conducting his famous first year course at Caltech, complained about the difficulty of getting quantum concepts over to many of the students.  Of course, there will always be the select few with plenty of mathematics and physics background for whom the content won't present huge problems. But make no mistake they're in the minority.

I still recall teaching a Calculus physics course for which the final chapter of the text assigned had significant quantum mechanics content, including:  the basic Schrodinger wave equation, quantum square well, atomic energy levels, eigenvalues, probability densities and how to compute expectation values. Attached below  is one problem which I worked out for the class:


But which only 1-2 of the class actually got. Most were stung by: a) not having the requisite mathematical ability (which makes one wonder how they managed to be allowed to move on from the earlier semesters) and b) not being able to conceptualize (despite numerous visual aids).


One of the visual aids which flummoxed them is shown below, including the first 3 wave functions for the H- atom(far left), the corresponding probability densities (middle) and the associated energy levels for an "infinite square well".


Basically, the possible energies of the particle, called energy levels, are quantized. The integer n used to designate them, is called the principal quantum number.

The state with lowest energy, n = 1, is called the ground state. The state with n = 2 is called the first excited state and so on.

There are n/2 de-Broglie waves capable of fitting into the well if the particle is in the nth quantum state.

At once the diagram captures the beauty and simplicity of quantum mechanics, in showing how energy is quantized in "jumps" as it is in the actual hydrogen atom.

In retrospect, the experience at Loyola with Hugh Young's presentation  of special relativity was perhaps at the limits of what a well prepared undergrad could understand.   For example, his derivation of the relativistic equation of energy (W) on pages 394, 395:

ò ^x2 _x1  dp/dt  dx = ò ^t2 _t1  (dp/dt)(dx/dt)  dt =

  ò ^t2 _t1  v(dp/dt)   dt  =  ò ^v2 _v1  v(dp/dv)  dv

W=   ò ^v2 _v1   v  d/dv [m_o v/ Ö (1 - v²/c² ] dv

The preceding equation for W applies since for relativistic momentum:

p  =  m_o v/ Ö (1 - v²/c² )

I argue the above is "at the limit" because of the calculus -based approach with which most undergrads would not be comfortable, even those who've taken AP Calculus.  More to the point, if one veers into quantum theory and general relativity, it doesn't get any easier.

This leaves us asking the question of whether there is an alternative solution or approach, specifically for those less able students who aren't attending Caltech, MIT or Harvard.  I believe there is and came away with one idea from a recent Readers Forum article ('How Black Holes Saved My Astronomy Course')  in Physics Today by Joseph Ribaudo..  Ribaudo - based at Utica College, New York- noted the struggle of his students to master the material  with most underperforming (averages in the "C range")  in his astronomy course.   He realized a large part of the problem was that  the existing astronomy textbooks were all too similar in their writing, content and design elements - and often way too mathematical. So he turned to Neil Degrasse Tyson's 'Death By Black Hole' as his central (core) text aided by "reading and writing assignments derived from popular and historical science publications".

Can a similar strategy be used to incorporate more modern physics into undergrad classical physics courses? I am confident this is feasible. Of course, the extent of integration of the modern physics material will hinge on the time allotted for the course or courses.  In most ('General Physics') courses, the modern physics is fit in at the very end - usually in one or two chapters featuring special relativity and some of the 'original' quantum mechanics, say of Bohr then Schrodinger.  At the maximum level of exposure students will be shown the use of the Schrodinger equation to solve for simple QM systems, say a particle in a box. This assumes, of course, the student has done some work in differential equations. (N.B. It is extremely rare for freshman and sophomore physics majors to take a dedicated modern physics course, far less a QM or Relativity course.)

But let's assume students are roughly at the math and abstraction level of  Ribaudo's astronomy class. What texts might supplement the approach to teach modern physics? I'd include the following:

'The Strange Story of the Quantum' - by Banesh Hofman (Dover, 1964)

'Thirty Years That Shook Physics' - by George Gamow (Dover, 1966)

'Relativity and Common Sense' -  by Herman Bondi (Dover, 1964)

'Sidelights on Relativity' - by Albert Einstein (Dover, 1954)

'The ABC of Relativity' - by Bertrand Russell (Signet, 1959)

 All of the above are eminently readable, and the math is at a minimum. Much of the math that is set out, such as in Bondi's book, is in the form of geometry from which simple derivations of the principles are shown. Einstein's book is terrific in providing an understanding at a non-math level by the master himself.

Ribaudo mentions in his article (p. 11)  setting one mid term exam question to the effect: "Discuss the value of reading 'Death By Black Hole' and whether you'd recommend this book to others".  The same sort of question might be asked of any of the books listed above.  In terms of how much contextual mathematics one's students can handle that has to be left to the instructor. For those who might wish to know some QM but minus all the math, Hoffman's book is probably best. For those who might wish to know some special relativity without the math, Russell's book is likely best. For those who are comfortable with plane geometry and some algebra,  Gamow's book is ideal for QM, and Bondi's ditto for special relativity.

If instructors in special relativity wish to get into more problem solving and math exposure - still largely at the intermediate algebra level (with a tiny bit of differential calculus)-  my own text: 'Modern Physics: Notes, Problems and Solutions' is ideal, namely the first three chapters (pp. 1 -45).

Ribaudo also cites "reading and writing assignments" from popular science magazines (e.g. Discover) and this can also provide fertile material for modern physics. For example, asking students to read the excellent article on quantum entanglement in Discover (July/August, 2016, p. 60) and asking students how they might explain this to a friend. Or better, how they might explain it if they could go back in time and encounter Neils Bohr.

In the end, it is possible to present modern physics to undergrads, but it has to be tailored to their own math and abstraction abilities.  If this is done, and there is the time allotted to do it without having to rush, then both physics instructors and their students will be all the better for it. And we may see much less whining from the undergrads about being denied entry into the "modern stuff".

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Addendum:

The best modern physics text I've found for math proficient students - geared to freshman and sophomores - is 'Space, Time and Quanta: An Introduction to Contemporary Physics' by Robert Mills (1994).  In his Introduction,  Mills writes of the text being a "supplement to a more conventional course" or "designed for the physics side of an interdisciplinary course" - i.e. for other (non-physics) students, but don't believe it. The text is in fact perfect as a standalone, totally complete modern physics text - at least on a par with the analogous content in The Feynman Lectures.  The end of chapter problem sets are especially good, and I simply can't see any "interdisciplinary" student being able to get through them.  Hence, I believe Mills greatly underestimates the book's value as a teaching text for modern physics.

The Vatican's Big Bang Conference: Does George Lemaitre's "Theory" Prove A Supernatural God?

The news this past week out of Rome is the Vatican celebrated the big-bang theory. What? Big Bang? The theory whereby the cosmos incepts from an "explosion" (super dense state) and needs no external divine agents? Yep, one and the same, but the Vatican is not deterred and believes firmly there need be no conflict between its proclamations and the findings of modern cosmology.

So it's not entirely odd that the Vatican Observatory has invited some of the world's leading scientists and cosmologists to talk about black holes, gravitational waves and space-time singularities as it honors a Jesuit cosmologist  (George Lemaitre) who is considered one of the fathers of the idea that the universe began with a gigantic explosion.

The May 9-12 conference honoring Lemaitre was held at the Vatican Observatory, founded by Pope Leo XIII in 1891 to help correct the notion that the Roman Catholic Church was hostile to science. The perception has persisted in some circles since Galileo's heresy trial 400 years ago, but let's face it the perception is not entirely without foundation.  Recall, for reference, how Stephen Hawking noted in his book, 'A Brief History Of Time', how Pope John Paul II warned against trying to investigate the "moment of creation" at one cosmology conference Hawking attended at the Vatican.  Of course, this is the very moment one does want to investigate - or at least as close to it as possible.

Hawking's formulation of the universe's inception spontaneously can be described via the diagram below:

This was explored in detail in the episode God and the Origin of the Universe in the "Planet Green"  'Curiosity' series some five years ago. By the end of the program even a person with no mathematical background could grasp Hawking's simple arguments which followed from gravitational time dilation, e.g. the slowing of clocks as experienced near black holes or other gravitating masses.

Understanding Hawking's arguments even at a basic level, requires a few preliminaries. First, at a basic logic level, we agree and understand that to have causal agents or causality we require time. If there is no time evident or existent, then there can be no nexus or connection from event A to event B. It is time- the chronometric passage of some interval- which allows one to say, e.g.

(Event A) -> (Event B)

If on the other hand, the two events occurred with no time passage, one wouldn't be able to infer causality, say that Event A "caused" B.

Second, we need to grasp at least at a qualitative and rough quantitative level that time slows down in the vicinity of gravitating masses. The way this is approached is usually by using what's called a "metric tensor" such as shown in the bottom half of the lower diagram There are 16 components in this tensor, each of which can be measured by means of clocks...atomic or other. For the form of the metric tensor shown, we have elements conforming to a spherically symmetric distribution with r the radius or distance from the center.

See also my earlier post on general relativity, e.g.

http://brane-space.blogspot.com/2014/12/a-look-at-general-relativity-and_27.html


To operate in a typically curved space-time (x,y,z,t) one will usually use what's called the "interval" such that:

ds² = {1 - 2GM/r}dt² - {1 + 2GM/r}(dx² + dy² + dz ²)

The key component used in gravitational time dilation measurements is the one at the very upper left of the matrix which we denote by g _0o. All experiments on gravitational time dilation can be regarded as direct measurements of the g _0o component of the metric tensor. For all comparisons with actual experiments, it's convenient to express chronometers at two different distances, say:

dτ2/ dτ1 = [g _0o(2)]^1/2  / [g_0o (1)] ^1/2

it's convenient to use fractional deviations, e.g.

Dτ/ τ = (dτ2 - dτ1)/ dt ~ GM {1/r1 - 1/r2}

The clock rate can also be related to the frequency f. If a clock registers a large elapsed time it must be ticking fast, and if a small elapsed time, ticking slow. Thus the frequency difference, D f, must be proportional to D τ, or:

Df/f = (f2 - f1)/f =  D τ/ τ ~ GM{1/r1 - 1/r2}

Cutting to the chase and sparing readers a lot of algebra, one ends up with:

D τ/ τ = g [ D r/ c² ]

As a terrestrial application, say D r = 10 km (the typical height attained by an aircraft) the time dilation is:

Dτ/ τ = 10 ^-12 

Now switch to the Hawking's black hole example, and one finds as one approaches the center of the hole, at which r = 0, there is no time, period. This is also the point of maximal gravitational acceleration, call it g = ¥, or infinite. Thus:

Dτ/τ = ¥ [ 0/c² ]= D f/f = 0

In the above we have assumed the reference clock is right at the black hole center, and the difference is now 0. Then the time elapsed is zero. This is confirmed - well, the tendency of clocks to slow in g-fields is confirmed by hydrogen -maser clocks and others. (See, e.g. 'Gravitation and Spacetime', by Ohanian and Ruffini, p. 183. In effect, one simply extrapolates to the most extreme g-field, present in black holes, to infer no time passage that can be reckoned by any chronometer.

If there is no time passage, there can be NO cause! There can be nothing which causes the universe in the first place, hence...no "supernatural agent" as "creator". Since we already agreed that for causality to operate time must exist, then conversely, if we find time doesn't exist then there can be no cause or causal agent.

Hawking's final argument, illustrated by the top illustration in the lower diagram, is simply that the cosmos (originally with less diameter than a proton) simply "popped" into existence much like we currently observe particles in pair production doing the same via spontaneous quantum fluctuations using the energy-time uncertainty principle. The basic principle underlying these pair production processes is the energy-time uncertainty principle: dE dt > h/2π where h is the Planck constant of action. . Thus, a brief fluctuation occurring over time dt can produce a change in energy dE, which may be sufficient to produce particles. The energy has an estimated magnitude given by the uncertainty principle.

Hawking's other point is that the manifestation of mass in the cosmos always has a complementary action in forming negative energy. He used the analogy of a person forming a large mound or hill by digging into the ground. The larger the hill formed, the bigger the hole from which the hill was excavated. All that negative energy is now scattered across space, as bound systems via gravitational potentials. These potentials represent negative energy reservoirs. For example, gravitational potential energy, say associated with a planet's g-field might be expressed: V(r) = - GMm/r.

If all the positive and negative energy contributions are tallied, the total comes to zero. This zero energy indicates that no agents acted to create the cosmos. We didn't need a "hole digger" to get positive mass from "digging a negative energy hole" because it already inhered in the subatomic cosmic hyperdense state. Moreover, we're still seeing -observing the effects of the initial repulsion of the unstable primeval "atom" as all parts of the cosmos continue to accelerate during the expansion of space.

However, the head of the Vatican Observatory, Brother Guy Consolmagno, says you can believe in both God and the big-bang theory. He bases his argument on Lemaitre's theory of the "primeval atom" and - to give credit where it is due - Lemaitre was the first to explain the recession of distant galaxies (called the "expansion of the universe").   He arrived at this result by solving the equations associated with Einstein's theory of general relativity, see e.g. the content in the link above. According to a Vatican press release:

"He understood that looking backward in time, the universe should have originally been in a state of high energy density, compressed to a point like an original atom from which everything started".

But is this enough basis to believe in both God (as a supernatural entity) and the big bang theory? Not really, and Brother Guy does not bother to go into the further details that it was actually a trio of physicists (Ralph Alpher, Hans Bethe and George Gamow) who get more credit for the big bang theory via their 1948 paper published in Physical Review. That trio actually came closest to predicting the major outcome, the temperature of the cosmic microwave background temperature.  That is, there exists a "relic" radiation comprised of thermal microwaves from the big bang detectable even today. Further, the spectrum must match that of a black body.

They did this by way of a meticulous assessment of primordial nucleosynthesis at that epoch and assuming that the latter was radiation dominated. Gamow himself made an original estimate of the CMB at 10 K (some sources say 50 K) but his collaborating students Ralph Alpher and Robert Herman, later refined that to 5 K in a further paper (Physical Review 74 (12): 1737–1742, 1948).  By 1965, Arno Penzias and Robert Wilson at Bell Laboratories discovered the CMB at 2.7 K. 

There is some dispute here concerning the "ABC" paper as it was called, and that in fact it was a Canadian cosmologist - Andrew McKellar - who predicted the CMB temperature first and most accurately in a 1940 paper. This paper emphasized "excellent coincidences obtained between the wave lengths of three of the unidentified, sharp, interstellar lines and lines arising from the lowest states of the CH, NaH, and CN molecules"   and hence “the maximum ‘effective’ temperature of interstellar space would be 2.7K”. 

My point in respect of the preceding is that while Lemaitre did great original work he did not make any actual predictions, say for what background cosmic temperatures would be detected.  And hence his hypothesis fell short of being an actual big bang THEORY.  Remember, a theory is a hypothesis which has actually made predictions that have been confirmed.  Alpher, Bethe and Gamow met this standard, Lemaitre did not.

Brother Guy Consolmagno's remark to the Vatican and international press that "Christians believe in a supernatural God who is responsible for the existence of the universe" is fine as a statement of faith, but alas, is not scientific. Nor is it based on any objective scientific evidence.  He added that "our science tells us how He did it" but again, that begs the question of why one needs to insert a special supernatural agent-creator at all. Hawking's work - as I showed - indicated this is superfluous.

These distinctions are important and even Lemaitre conceded his conclusions were largely based on faith, not science, i.e. when he reminded Pope Pius XII "the creative act of God was not something that happened 13.8 billion years ago, it is happening continually".   In other words, as a Christian this is what he believes - but it is not a scientific conclusion. At least not in terms of a supernatural deity.

As I noted in my ASTRONOMY magazine article:  ‘The God Factor’ (Astronomy Forum, March, 1990), science selectively excludes problems for which no practical method of inquiry exists. The supernatural, which is neither measurable or verifiable, falls into this category and that includes ‘God’ - if depicted as "causeless" and "supernatural". More to the point, we tend to regard unproven, non-evidentiary agents claimed by virtue of belief alone as evocative of superstition. 

The key fallacy embraced by the Vatican appears to be ignotum per ignotius : postulating an unknown agent or cause to account for a not well understood process or entity, e.g. consciousness or the origin of the cosmos.