The scare quotes in the title are to distinguish "Modern Physics" classes like the one I'm teaching this term from modern physics as a general subject, which, of course, all right-thinking people should study in depth. The question comes from a comment by Coriolis on last week's post about what "Modern Physics" is as a class:

Having passed through those classes (I'm now a grad student), I have to say I didn't see much worth in the Modern physics class (and your description of it is pretty much how I remember it, except without the relativity). It's basically in that middle ground trying to give you a "taste" of QM, without actually going into the math and subject in a way that makes sense like the first real QM class does. Maybe it somehow helped subconsciously when I took the first QM class, but I don't think so.

The existence of these classes-- a whirlwind tour of (special) relativity and quantum mechanics, and applications thereof-- is a little problematic, for exactly the reasons Coriolis notes: there isn't enough time spent on each topic to really learn it in depth. And, in fact, there are other ways of approaching the same material. I think that sophomore-level "modern physics" courses do serve a couple of purposes, though.

The whole reason for "modern physics" classes at the sophomore level is as a bridge between the intro-level courses in Newtonian mechanics and E&M and the upper-level physics classes. While there are some weird phenomena that crop up in those classes, for the most part, they're pretty approachable: the physics involved deals with ordinary objects and everyday (or, at least, not ridiculously odd) situations, and the math involved is very straightforward calculus.

When you get to quantum mechanics, though, you need to make two big jumps. One is mathematical-- the Schrödinger Equation is a second-order differential equation, the idea of eigenstates and so on brings in a lot of linear algebra, quantum wavefunctions are necessarily complex numbers, etc.-- and the other is conceptual. The way things behave in the quantum regime is so utterly unlike the way that things behave in everyday life that it takes a little while to get your head around the whole idea.

"Modern Physics" classes in the sophomore level try to bridge this gap. Many of the students taking them are only just getting to the point in the usual math curriculum where they can begin to understand the techniques involved, so these classes typically try to ease into the relevant topics. I spend a whole class on complex exponentials (because the math department doesn't), for example, and do a computer exercise in Mathematica to introduce the idea of Fourier series before diving into the uncertainty principle.

At the same time, the classes try to hit most of the essential concepts: the idea that particles need to be described as wave-like in some respects, the idea of energy quantization, some basics of wavefunctions, etc. Students don't go through all the gory details of calculating wavefunctions in real situations, but they get some of the flavor.

The idea of sophomore-level "modern physics" is to let students see a little of both the mathematics and the concepts before they get to the real deal, so that they're not doubly blown away when they hit "real" quantum mechanics. If you just throw students into a regular quantum class with no preparation, they go into vapor lock-- they can't handle the math, they don't understand the concepts, and they shut down. By giving them half of the math, and half of the concepts in advance, they've got a little something to cling to when they hit "real" quantum mechanics.

There are, of course, other ways to do this. When I was an undergrad at Williams, the third course in the physics sequence, in the spring semester of the sophomore year, was called "Waves and Optics," but was really a stealth math methods course. It introduced all the key mathematical concepts for quantum mechanics, in a more familiar context. We talked about waves on strings, and used that to introduce wave equations and Fourier analysis. We talked about physical optics, and used that to introduce a little complex analysis, representing light waves as the real part of a complex exponential. We talked about normal modes of oscillators, and used that to set up ideas about basis states, and so on.

I thought that approach worked very well as a way of setting up the math needed for quantum physics in an approachable way. The topics covered had fairly direct and comprehensible applications, so there was relatively little conceptual oddness, leaving time to get our heads around the math. Then, in the quantum class, we could deal with all the gory details a little more easily. It also fit well with the research interests of the faculty at that time, who were very heavily biased toward lasers and optics in general.

There's another purpose served by the "modern physics" class, though, which is to give students who won't necessarily go on to take all the upper-level electives (or who won't be able to, due to limited resources for teaching electives) some idea of the key ideas of modern physics. At Union, we generally manage to offer one or two upper-level electives per year-- classes like "Particle and Nuclear Physics," "Statistical Mechanics," "Modern Physical Optics," "Solid State Physics" and the like. That means that students following the normal course sequenceget a shot at maybe three of these. We require only one for graduation with a Physics major.

A full-blown quantum mechanics class doesn't have time to really get to any of the key applications-- if you're starting with the basic postulates, you just don't have time to get through all the solvable problems and all the solution techniques in time to get to the details of band structure and its implications for solid state physics. That's why the special topics classes exist, after all.

That means that, in the absence of something like a "modern physics" class, students could graduate with a major in physics without ever studying some of the topics that are most closely associated with physics. The end-of-term sprint through applications of quantum mechanics does at least give students a glimpse of these topics-- I spend a couple of classes on band structure, and how diodes and transistors work, for example, and a couple more on basics of nuclear physics and nuclear decays. That way, they have at least a glimmer of an idea of what "beta decay" means, down the line, whether they get the chance to take "Solid State Physics," or "Particle and Nuclear Physics" down the line, or not.

The calculation of costs versus benefits of this approach comes out differently for different departments. There are trade-offs involved in either approach-- what Williams did, back in the day, did a fine job of preparing us for quantum mechanics, atomic physics, or laser physics, but in the time I was there, the department never offered any classes on particle or nuclear physics (and thus, my knowledge of those areas remains pretty sketchy). That worked fine for them, as all the experimental faculty did work in those areas, but the same approach wouldn't work at Union, where we have faculty doing research in a wider range of areas.

I do have some doubts about the utility of the "modern physics" approach, and as a result, I tend to push the "stealth math methods class" aspect of the course a little harder than may be typical. I don't think they can be dismissed out of hand, though-- they do play a useful role in the curriculum, which is why they're common enough to have spawned a whole category of ridiculously expensive textbooks.

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As a non-physics major I really enjoyed my Modern Physics class. It was an introduction primarily to the concepts with no math higher than Calc II necessary, similar to how you describe your class. I was, at the time, a comprehensive science education major. I had to take the two introductory semesters of Biology, Chemistry, Physics, and Geology; an elective in each; and a couple of other electives to make a "concentration". (BTW, the curriculum for sci. ed. majors has completely changed since then.) I choose Modern Physics instead of the suggest elective, Optics, because it interested me more. I couldn't, perhaps naively, see how you could spend a whole semester on mirrors and lenses but Relativity is cool.

One of the odd parts of the US(&Canadian) college curriculum is that it tends to assume the students have never seen calculus and linear algebra. Hence the slow start. But that assumption is increasingly false for well-prepped students.

Would it be significantly easier to build a physics curriculum if the students were assumed to be be coming in with AP Physics, Calculus, and (um) whatever the Linear Algebra course is called?

At the last place I was teaching, they essentially eliminated the modern physics course when they made huge changes in the curriculum a number of years ago.

The final net result was that most of the modern physics course topics like the Planck distribution, deBroglie wavelength, Compton scattering, Bohr model, special relativity, etc ... ended up in a new 3rd semester course in the introductory physics sequence, covered in a manner similar to the presentation in one of those huge freshman textbooks like Halliday & Resnick. This new 3rd semester intro course also covered stuff like geometrical optics, interference, diffraction, etc ... also done in the Halliday & Resnick style.

The new 2nd semester intro course is only electricity + magnetism. In the old intro sequence, electricity + magnetism and the optics topics were originally covered in the 2nd semester course in a somewhat fast neckbreaking manner. (This was done deliberately in the past to kick out as many freshman engineering majors as possible, as a sort of "trial by fire").

The old 1st semester course covered mechanics and thermodynamics, which has not changed much at all other than dropping the section on fluid mechanics if the instructor has run out of time.

Having taught a 'modern physics' course myself, it serves one other nice role (though this may be my 'history of science' bias speaking): teaching a bit more of the 'why we do it this way' instead of just teaching them the 'how we do it.' My undergraduate quantum course pretty much jumped right into the postulates of quantum mechanics without giving much of a justification for how those postulates were developed. In my modern physics course, I try and introduce quantum, and relativity, with more of a historical flavor. My hope is that this makes the postulates of QM seem less like 'magic' and more like a reasonable set of assumptions based on the experimental evidence.

My undergrad university took the same approach as Williams did, offering a "waves and optics" course as a pre-quantum course (QM was a two-semester sequence, normally taken sophomore spring and junior fall). At the time there was no separate modern physics class (they have since added a relativity course to the core). Unlike Williams, this department has (and has long had) extensive particle/nuclear research. Specialized electives were not emphasized; statistical mechanics was considered part of the core curriculum.

As for why some departments favor one or the other approach, the answer is probably along the lines of "Because we've always done it that way."

I took my modern physics and math methods courses concurrently, and it was probably one of the best things I did. I breezed through my "general physics" intro courses (the standard newton and lightweight e&m) without breaking a sweat, or really thinking too hard at all. I don't think I even did more than a single homework assignment in each of the two semesters (they weren't collected). Modern physics did the important job of kicking my rear end, hard. I ended up doing pretty miserably on the first test in both classes, and, ashamed of my performance, I buckled down and really

learnedthe material inside and out as best I could for the rest of the semester. I feel like if I went into upper division classes with the attitude and study habits I went into modern physics with, I would not have been able to recover from my mistakes (in m.p. I managed to do well enough after that first exam to earn an A in the course).So I guess what I'm saying is I think that they also serve a more meta purpose of giving students something tougher to chew on in a "nicer" setting than upper division classes.

I like the idea of one year mathematical methods course first, then having the freedom to concentrate on the physics later. Not coincidentally, this is how I did it. I can see at least two advantages:

1. Preventing duplication: how many times do you get to learn boundary value problems in a typical physics degree? they are the same in electrostatics, optics, quantum mechanics and half a dozen other places. Why not consolidate them into one place?

2. Preventing confusion: there are lots of people, professional physicists included, that think the quantum mechanics is all about Hilbert spaces and discrete eigenvalues etc., whereas those are just mathematical facts relevant for any linear differential equation. Thing is, quantum mechanics course is where most of them acquired that machinery, so they are confusing the math with the physics.

I like the idea of one year mathematical methods course first, then having the freedom to concentrate on the physics later.The failure mode of this approach is the inevitable collisions with students' mental compartmentalization: "But that was in

mathclass-- why do you expect me to remember it in physics?" Of course, maybe it's better to build that in from the start, and get it over with quickly...Not coincidentally, this is how I did it.I think that is probably the explanation of most faculty preferences on this sort of thing.

"But that was in math class-- why do you expect me to remember it in physics?"

In my experience, having seen the math you do in physics in a math class at all was an unusually lucky break, not an expectation. And nobody would even dream of expecting to not have to use whatever math one learned before. But maybe years of grad classes are spoiling my memories.

In any case, thanks for the huge response. It's a good argument, and to some degree I agree. Specifically I can see the use of getting people more prepared mathematically for QM (with a math-methods type class), and including other subjects like relativity, optics, simple solids etc. that most people will not take a dedicated course in. Some history-style things with black body radiation and debroglie wavelength are also probably a nice touch.

However I don't remember much of that in the modern physics class I took. If I remember correctly our class had too much time spent on trying to solve special cases for the hydrogen atom and such with ready-made equations (I think the instructor showed differential equations but didn't actually expect us to solve any). It probably just wasn't a good class.

Still, QM itself just didn't make much sense to me before I took a class dedicated to it. So at least to my mind, it's better to leave the wierdness alone until then.

Or maybe that's just because I learned it that way heh.

If the students need to overcome their "mental compartmentalization", all the better. If nothing else, good thinking habits are a nice side-benefit of a science education.

Chad, I would say that the big failure mode for a math methods course/courses is a turf war with the math and/or applied math departments. As best I can recall, everyone who progressed past part of sophomore year in physics internalized the use of the math we learned elsewhere in the curriculum. It's the non-majors who keep the math and the physics compartmentalized, whereas the majors worked hard to prepare for the battering they saw upper division students would get if their PDE fu was weak. (Though we did have a particularly coherent cohort, including getting someone teaching as a long-term high school sub through quantum so that he could get a full-time job and health insurance for his growing family.)

For my grad work, at a residential school instead of a commuter school, it seemed similar in terms of demand, but that's perhaps not a good discriminant since math methods courses are part of the standard curriculum.

I benefited (spectacularly) from a "quarter" course on basic QM taught from the Berkeley physics book 4 by a really good instructor - a course that followed book 3 on waves, which appears to have been quite similar to the one you took at Williams. We did relativity in the first quarter on mechanics, and applied it in the second on E&M.

The value the course I took was that its focus was on concepts, the big idea, as it were. I developed the intuition in that sophomore course that carried me through everything else. (It also converted me from a math major to a physics major, but that is a different story.)

The "mini PhD" courses normally taught seem to sacrifice some depth in QM for a digression into the specialty areas from condensed matter through particle physics, but a good teacher could make up for that. The real value of the course I took was not the book, it was the professor.

Chad, I would say that the big failure mode for a math methods course/courses is a turf war with the math and/or applied math departments. As best I can recall, everyone who progressed past part of sophomore year in physics internalized the use of the math we learned elsewhere in the curriculum. It's the non-majors who keep the math and the physics compartmentalized, whereas the majors worked hard to prepare for the battering they saw upper division students would get if their PDE fu was weak.Right.

The thing is, the courses I'm talking about take place in the sophomore year, when students haven't yet completely sorted themselves into "majors" and "non-majors." A lot of them are thinking about maybe being physics majors, but haven't quite decided yet, or developed the full set of study habits needed for the major.

(I'm speaking partly from personal experience-- it wasn't until junior year quantum that I really started doing the things that I associate with true physics majors. Prior to that point, I had a couple of badly motivated math classes that I never really connected with physics until later on. This also partly accounts for my lack of understanding of what particle theorists do, but that's another post...)

In the class that I'm teaching this term, we typically get a handful of majors, a handful of maybe-majors, maybe-minors, and a couple of students from other departments (math, chemistry, or engineering, usually) taking it as a science elective. It's not really reasonable to treat the class as if they were all definitely physics majors.

Does Feynman's approach to sophomore QM deserve consideration here, or is it another one of those things that works really well... if you are Feynman?

I think the usefulness of a Modern Physics course depends partly on the institution. In my department we have a very active 3-2 engineering program (students spend three years with us and two with a partner institution and receive two bachelor's degrees out of the deal). About half our students are in the 3-2 program. Some of our partner institutions suggest or even require the engineers take Modern Physics since it at least exposes them to the basic ideas of SR and QM (which are really the goal in those classes).

In addition, it used to be that most texts for GR and QM assumed students had some basic familiarity with the ideas from their Modern Physics classes. So, for instance, many a GR text would skim SR very, very lightly assuming students had received sufficient depth in Modern Physics.

I guess I can see both sides of the issue. Having a Modern Physics course allows me to spend more time on SR, particularly the geometric aspects, and really gives students the conceptual foundation they need for either applications of SR (e.g. in E&M) or for GR. For QM, Modern gives me the opportunity to spend more time on things like the double-slit experiment, the photoelectric effect, and a heuristic derivation of the 1-D time-independent SchrÃ¶dinger equation. This, again, seems to give them a better foundation for the full QM course.

There was no "modern physics" course when I was an undergrad. Our first two and a half years was all classical physics.

The quantum mechanics courses were mainly done in junior and senior year. The first quantum mechanics course required intermediate classical mechanics, wave + physical optics, and partial differential equations courses as mandatory prerequisites.

So by the time we were taking our first quantum mechanics course, we had already finished:

- 2 semesters of intermediate classical mechanics

(including Lagrangian + Hamiltonian)

- 1 semester of classical thermodynamics

(without any statistical mechanics)

- 1 semester of intermediate electromagnetic theory

(including retarded potentials + radiation)

- 1 semester of wave + physical optics

(using the optics textbook by Hecht)

- 1 semester of mathematical physics

(partial differential equations + special functions)

These courses were all taken in sophomore year, and the first semester of junior year. Our first quantum mechanics course was taken in the second semester of junior year, along with a second mathematical physics course (mainly covering complex analysis, Laplace + Fourier transforms, Green's functions, etc ...).

I didn't know it at the time, but the department thought it was largely pointless doing quantum mechanics until closer to senior year, and that a significant background in intermediate classical physics was a better strategy. (One of my previous professors mentioned this years later, when I asked him about the curriculum).

the department thought it was largely pointless doing quantum mechanics until closer to senior year, and that a significant background in intermediate classical physics was a better strategyThere are some advantages to this strategy. For one, you will have encountered most of the math methods you need in QM before you actually take QM, so the professor can spend more time on the physics. Also, having been exposed to Hamiltonians and Poisson brackets (the classical version of commutators) in classical mechanics, you are not subjected to the

deus ex machinaof having these concepts sprung on you in QM, as typically happens in both the Modern Physics and Waves/Optics approaches to the curriculum.The biggest potential downside is that it becomes much harder to change majors into physics after the start of sophomore year, since you would likely need a fifth year (or at least a ninth semester) to fit in your QM class. I don't know how big a problem this is in practice, but I did my undergrad work in a department where someone changing majors into physics during the sophomore year could still (with a bit of overloading on physics courses) expect to graduate after four years. (That everybody, regardless of major, had to take the two-semester freshman physics sequence probably helped, as did the fact that most of the core curriculum courses were offered every term.)

Eric Lund,

At my alma, the hard science and engineering majors were deliberately made to be unattractive to folks who wanted to change their majors beyond freshman year. Essentially the undergraduate programs were structured such that if somebody wanted to change majors in the middle of their sophomore year, they had to repeat sophomore year over again in their new major for the most part.

This was the case even for majors like mathematics. The sophomore year curriculum for math majors, required year long courses (ie. two semesters in length per course) on abstract algebra, real analysis, and theoretical differential equations. The freshman math majors also took year long courses which covered calculus and linear algebra done in a very theoretical manner, with a heavy emphasis on rigorous proofs. It was basically a killer to change into the math major.

In my comment #16 on January 14, 2009 3:13 PM

"(without any statistical mechanics)"

, it should have said

"(without any quantum statistical mechanics)"

Eric Lund,

In the first quantum mechanics course I took, the professor did spend the first few weeks on relevant "modern physics" topics done at a slightly more mathematical level.

For example, the professor actually bothered to go over the details of deriving the Rayleigh-Jeans law starting from electromagnetism and classical thermodynamics. (We were assigned the derivation of the Planck distribution and Wien's law as a homework assignment on the first day).

He also went into some detail in working out the Bohr-Sommerfeld quantization rules for hydrogen. Basically it was a messy problem in classical mechanics for elliptical orbits, for which we were also assigned to do as a homework problem. Considering we already had taken two semesters of intermediate classical mechanics, the professor thought it was a problem which he felt we could do.

Another thing covered was classical radiation, with problems like calculating how long it would take for a classical "atom" to collapse upon itself. (We had already covered retarded potentials and radiation solutions in an electromagnetic theory course the previous semester).

This probably wasn't the easiest way to cover "modern physics" topics, but nevertheless it did assume a lot of previous knowledge of intermediate classical physics for which we already had taken courses on.

For special relativity, we didn't have any single dedicated course covering it. Relativity was first covered in the intermediate classical mechanics courses. The two electromagnetic theory courses covered the more advanced relativity topics like: tensor notation, transformations of EM fields, etc ...

I can't think of any universities today which cover the physics curriculum in the way I went through it. These days, many universities like to push the modern physics and quantum mechanics courses earlier to sophomore year (or even freshman year). Though going back further in time, several older professors I knew mentioned they never did quantum mechanics until they were in graduate school back in the 1960's. (These professors would be over age 70 today and retired). Back then, the undergraduate physics curriculum was all classical physics.