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**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}

**(dp/dt)(dx/dt) dt =**

_{t1}**ò**

^{ t2}

_{t}_{1}^{ }v(dp/dt) dt =

**ò**

^{ v2}

_{v1}^{ }v(dp/dv) dv

**ò**

^{ v2}

_{v1}^{ }v d/dv [m

_{o}v/ Ö (1 - v

^{2}/c

^{2}] dv

The preceding equation for W applies since for relativistic momentum:

**p**= m

_{o}v/ Ö (1 - v

^{2}/c

^{2})

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

*as his central (core) text aided by "*

**'Death By Black Hole'***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 other*s". 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".---------------------

*:*

__Addendum__The best modern physics text I've found for math proficient students - geared to freshman and sophomores - is

**by Robert Mills (1994). In his Introduction, Mills writes of the text being a "**

*'Space, Time and Quanta: An Introduction to Contemporary Physics'**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.

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