Archive for December, 2012


“I am your probability density”

In an earlier post I discussed my philosophy of teaching special relativity.  My main idea was that physics professors should keep the “weird stuff” at bay, and start with non-controversial statements; once students are on board, you can push under the grass and show them the seething Lynchian bugs beneath.

Well, what about quantum mechanics?  Does the same philosophy apply?

My answer is yes, of course.  Don’t start with Schrödinger’s cat.  Don’t mention the Heisenberg uncertainty principle, or wave collapse, or the EPR experiment, or Bell’s theorem, or the double slit experiment, or quantum teleportation, or many worlds, or Einstein’s dice.  Start with the problems of physics, circa 1900, and how those problems were gradually solved.  In working out how physicists were gradually led to quantum mechanics, students will build up the same mental framework for understanding quantum mechanics.  At least, that’s how it works in theory.

Now, my perspective is from the point of view of a professor who teaches only undergraduates.  I only get to teach quantum mechanics once a year: in a course called Modern Physics, which is sort of a survey course of 20th century physics.  (If I were to teach quantum mechanics to graduate students, my approach would be different; I’d probably start with linear algebra and the eigenvalue problem, but that’s a post for another day.)  As it is, my approach is historical, and it seems to work just fine.  I talk about the evidence for quantized matter (i.e. atoms), such as Dalton’s law of multiple proportions, Faraday’s demonstration in 1833 that charge is quantized, Thomson’s experiment, Millikan’s experiment, and so on.  Then I explain the ultraviolet catastrophe, show how Planck was able to “fix” the problem by quantizing energy, and how Einstein “solved” the problematic photoelectric effect with a Planckian argument.  Next is the Compton effect, then the Bohr model and an explanation of the Balmer rule for hydrogen spectra…

We’re not doing quantum mechanics yet.  We’re just setting the stage; teaching the student all the physics that a physicist would know up until, say, 1925.  The big breakthrough from about 1825-1925 is that things are quantized.  Things come in lumps.  Matter is quantized.  Energy is quantized.

The big breakthrough of 1925-1935 is, strangely, the opposite: things are waves.  Matter is waves.  Energy is waves.  Everything is a wave.

So then, quantum mechanics.  You should explain what a wave is (something that is periodic in both space and time, simultaneously).  Here, you will need to teach a little math: partial derivatives, dispersion relations, etc.  And then comes the most important step of all: you will show what happens when two (classical!) wave functions are “averaged”:

ψ1 = cos(k1x – ω1t)

ψ2 = cos(k2x – ω2t)

Ψ(x,t) = (1/2) cos(k1x – ω1t)  + (1/2) cos(k2x – ω2t)

Ψ(x,t) = cos(Δk·x – Δω·t) · cos(k·x – ω·t)

where Δk ≡ (k1 – k2)/2, k ≡ (k1 + k2)/2, etc.

[Here I have skipped some simple algebra.]

This entirely classical result is crucial to understanding quantum mechanics. In words, I would say this: “Real-life waves are usually combinations of waves of different frequencies or wavelengths.  But such ‘combination waves’ can be written simply as the product of two wave functions: one which represents ‘large-scale’ or global oscillations (i.e. cos(Δk·x – Δω·t)) and one which represents ‘small-scale’ or local oscillations (i.e. cos(k·x – ω·t)).

This way of looking at wave functions (remember, we haven’t introduced Schrödinger’s equation yet, nor should we!) makes it much easier to introduce the concept of group velocity vs. phase velocity: group velocity is just the speed of the large-scale wave groups, whereas phase velocity is the speed of an individual wave peak.  They are not necessarily the same.

It is also easy at this point to show that if you combine more and more wave functions, you get something that looks more and more like a wave “packet”.  In the limit as the number of wave functions goes to infinity, the packet becomes localized in space.  And then it’s simple to introduce the classical uncertainty principle: Δk·Δx > ½.  It’s not simple to prove, but it’s simple to make plausible.  And that’s all we want at this point.

We’re still not doing quantum mechanics, but we’re almost there.  Instead, we’ve shown how waves behave, and how uncertainty is inherent in anything with a wave-like nature.  Of course now is the time to strike, while the iron is hot.

What if matter is really made from waves?  What would be the consequences of that?  [Enter de Broglie, stage right]  One immediately gets the Heisenberg relations (really, this is like one line of algebra at the most, starting from the de Broglie relations) and suddenly you’re doing quantum mechanics!  The advantage of this approach is that “uncertainty” seems completely natural, just a consequence of being wave-like.

And whence Schrödinger’s equation?  I make no attempt to “prove” it in any rigorous way in an undergraduate course.  Instead, I just make it imminently plausible, by performing the following trick.  First, introduce complex variables, and how to write wave functions in terms of them.  Next, make it clear that a partial derivative with respect to x or t can be “re-written” in terms of multiplication:

d ψ /dx  →  ik ψ

d ψ /dt  →  –iω ψ

Then “proving” Schrödinger’s equation in a non-rigorous way takes 4 lines of simple algebra:

E = p2/2m

E ψ = (p2/2m)ψ

Now use the de Broglie relations E = ħω and p = ħk…

ħw ψ = (ħ2k 2/2m) ψ

iħ(∂ψ/∂t) = (–ħ2/2m) ∂2ψ/∂x2

There’s time enough for weirdness later.  Right now, armed with the Schrödinger equation, the student will have their hands full doing infinite well problems, learning about superposition, arguing about probability densities.  As George McFly said, “I am your density.”  And as Schrodinger said, probably apocryphally, “Don’t mention my cat till you see the whites of their eyes.”


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If you don’t go sideways,
you will return to Earth

Why is it that astronauts “float around” in space?

If you were to ask Bill O’Reilly or the Insane Clown Posse, the answer would be that it is a mystery.  If you were to ask the average person on the street, the answer would be that there’s no gravity in space.  Both answers are ridiculous, of course.  It’s not a mystery; we have a very firm working knowledge of the physics of orbits.  And there’s plenty of gravity in space: at 230 miles up, where the International Space Station is, the acceleration due to gravity is about 8.8 m/s2, which is only 10% less than its value at sea level.

So why does the general public still not understand this whole “floating astronaut” thing?

I submit that some of us physics professors are teaching it poorly.  Here’s an explanation from a physics book on my desk:

“All objects in the vicinity of, say, the space station are in free fall with the same acceleration, and so, absent nongravitational forces, they remain at rest relative to each other and their freely falling reference frame.” [Rex and Wolfson, Essential College Physics (2010) p. 215]

I don’t find this very helpful.  And many physics instructors teach “weightlessness” in the same non-helpful way: by hand waving and saying that astronauts are in free fall, and that they are only apparently weightless.  Unfortunately, to the novice this brings up a host of new questions: what’s the difference between apparent weightlessness and actual weightlessness?  More importantly, if you’re in free fall, why don’t you crash into the Earth?

Another book on my desk does a better job:

“Why don’t planets crash into the Sun [if they truly are in free fall]?  They don’t because of their tangential velocities.  What would happen if their tangential velocities were reduced to zero?  The answer is simple enough: their falls would be straight toward the Sun, and they would indeed crash into it.”  [Hewitt, Conceptual Physics, 10th edition (2006), p. 193]

Newton himself also got it right:

“We may therefore suppose the velocity to be increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at the Earth, till at last, exceeding the limits of the Earth, it should pass into space without touching it.”  [Isaac Newton, The System of the World, Section 3, translated by Motte, edited by Cajori (1946)]  [Note Isaac’s use of the word “till”!]

The key idea which we physics professors should emphasize is that astronauts are in free fall, but they don’t hit the Earth because they are moving very, very fast horizontally.  That’s it.  That’s the secret.  They are going so fast that they fall “around the curve of the Earth” so to speak.  I don’t think this horizontal motion is emphasized enough.  I’ll say it again: you need to go sideways to get into orbit.  The next time you’re piloting a spaceship, remember the old adage: that which goes (straight) up will surely come back down (unless you reach escape speed).  So don’t aim for infinity and beyond—aim for the horizon.

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’til is not a word


For I ne’er saw true beauty till this night…

’til is not a word.

Please, use either until, or till.  Some people think that ’til is an abbreviation of until, but this is folk etymology.  “Till” is the older word by far, going back to at least Shakespeare’s day.  For example:

John 21:22 (KJV)     Jesus saith unto him, If I will that he tarry till I come, what to thee?  Follow thou me.

Romeo and Juliet: I, v     For I ne’er saw true beauty till this night.

Also note that “Till death us depart”, from the marriage liturgy in the Book of Common Prayer, dates back to 1549!  It became “Till death us do part” in 1662.

Seeing advertisements for the old Fox show ’til Death always grated on me like fingernails on a blackboard.  Luckily, like most Fox sitcoms, that show departed quite a while ago.

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