Feeds:
Posts
Comments

Archive for the ‘Many-worlds Interpretation’ Category

mcfly

“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.”

Read Full Post »

Schrodinger’s cat

I am hesitant, sometimes, to expound upon the many-worlds interpretation of quantum mechanics, for fear of something I call the Deepak Chopra effect.  (I won’t give you a hyperlink to the guy, because I don’t want to increase any traffic to any website he’s associated with.)  The Deepak Chopra effect is this:

If you talk about some weird aspect of quantum mechanics, they will come.

Who are “they”?

They are the Deepak Chopras of the world: people who make money by peddling vague new age philosophies.

Suppose you’ve made up some sort of new religion.  You want followers, people to buy your books and watch your DVD’s and attend your seminars and drink your Flavor Ade and buy your T-shirts.  (Yes, Deepak Chopra sells T-shirts.)  What better way to attract attention, to give your puerile ideas a veneer of respectability than to cloak them in the mystique of quantum mechanics?  Quantum mechanics is weird—everyone knows that—but almost no one really knows the details.  PhD physicists don’t grow on trees, after all.  Therefore, if you appeal to quantum mechanics to cover up the stench of your ideology, you will most likely get away with it.

I’m tempted to write some computer code that invents Chopra-esque prose.  The output would look like this:

Your [mind] and [eternal light] have been exquisitely formed by [the cosmic warmth] to help you fulfill [your potential matrix] and your [soul capability].  This is because [wave-particle duality] and [the principle of decoherence] prove that your [neo-human consciousness] transcends the [body-mind paradox] to inhabit the [weak nuclear force] under the auspices of [string theory].

Easy, right?  Yet Deepak Chopra is the one worth $80 million dollars.  Sigh.

So back to the many-worlds interpretation.  What could someone like Chopra ever do with such an idea?  How could he co-opt the multiverse to scratch out a few more ducats?  I shudder to think on it.

People have been using the strange ideas of physics for a long time now, with predictable results.  Take this garbage:  “What the #$*! Do We Know!?”  I wish I had been blogging in 2004 when this farce came out; my review of the movie would have been six words: “Nothing about physics, that’s for sure.”  Yet part of the blame rests with physicists themselves: they bandy about strange ideas amongst themselves, with nary a thought about how the public at large will perceive said ideas.

Consider Schrödinger’s cat, for example, the popular notion of which is as follows: a cat can be alive and dead at the same time!  Weird!  And yet when Schrödinger first proposed this “paradox” his intent was to attack the Copenhagen interpretation of quantum mechanics, by pointing out an obvious absurdity.  I know of no physicist on the planet who seriously believes a cat can be alive and dead at the same time.  And yet Chopra and others like him point to such quantum weirdness and use it to excuse all manner of hooey.

But what about many worlds?  Isn’t it just as crazy, just as loony, as anything Chopra peddles to the masses?

Well, no.  It’s weird, sure.  But it is based in peer-reviewed science, and is an active topic of investigation to this day.  (I doubt anyone’s in a lab somewhere, trying to verify Chopra’s claims.)  Many worlds is an interesting mathematical structure to explain our universe, but it doesn’t really affect anyone’s life.  It’s not even relevant to how anyone should live their life.  It should certainly never be used to prop up a shaky religion.

My advice to you, Dr. Chopra, is to quit using physics to bolster your claims.  After all, I don’t use your only field of expertise (endocrinology) to support my idea that Matt Damon is really a cyborg, do I?  Then again, maybe I could start a religion—

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

(Photo credit: http://en.wikipedia.org/wiki/File:Schrodingers_cat.svg)

Read Full Post »

Read Full Post »

It may seem odd that this blog is called “Many Worlds Theory” and yet I have not yet blogged about, well, many worlds theory.  Well, I am doing so now.

The arguments that lead a sizable portion of theoretical physicists to postulate a multiverse are subtle and complicated.  I hope to eventually cover these arguments, but I’d like to start with a discussion of what many worlds is not.  Therefore, with much fanfare, I present to you:

The Top 5 Misconceptions About the Many-worlds Interpretation (MWI) of Quantum Mechanics

1.      Every time you do a quantum experiment, the universe branches into multiple universes.

This is a popular notion, as seen in TV shows like Nova and as presented by Martin Gardner.  Unfortunately, it is the wrong way to look at things.  It is much better to imagine that all of the possible universes already exist, and that doing an experiment just tells you (the experimenter) which universe you happen to be in.

Suppose you’re watching Star Wars.  You have no idea whether you are watching the original, or the retconned 1997 version.  You finally realize which version you’re watching when you see Han shoot and kill Greedo without Greedo ever getting a shot off.  You conclude you’re watching the original version.

Of course, up until that point, the movies are exactly the same. (Rather, let’s just say they are the same.  I haven’t actually checked this.) You wouldn’t conclude, when you got to that scene, that two movies were created from one, would you?  There were two movies all along; watching Han shoot first just told you which movie you were watching.  A physicist who performs a Stern-Gerlach experiment doesn’t split the universe in two; doesn’t create a whole universe; instead she has gained some new information: “Oh, so that’s what universe I am in.”  No new physics of universe-creation is needed, and we need not violate conservation of energy.

2.      The existence of a multiverse is a postulate of a strange kind of quantum mechanics.

There is a formulation of quantum mechanics often called universally valid quantum mechanics, which was first described by Hugh Everett III in 1957.  It involves (see this for details) just one postulate: isolated systems evolve according to the Schrödinger equation.  That’s it.  A multiverse is a prediction of this postulate, not a postulate in and of itself.  So if you believe that isolated systems evolve according to the Schrödinger equation, you will be led to the MWI, unless you invent new postulates to make yourself feel better (see #3).

3.      The Many-worlds Interpretation has a lot of baggage.

This obviously depends upon what you mean by baggage, but the claim is often made that the MWI is horribly antithetical to Occam’s razor.  That is, how could anyone seriously believe that countless billions upon billions of universes exist, when believing in one universe is much, much simpler?

If you feel this way, I have two responses.

One: how can you believe that countless billions upon billions of stars exist, when believing in just one star is much, much simpler?  Shouldn’t you be Earth-centric, and call for Galileo’s excommunication?  Or what about the integers?  Mathematicians claim that there are an infinite number of them, but infinity is too hard to fathom, so why don’t you just say that there are a lot of integers, but that there is only a finite number of them?  There.  I bet you feel better.

Two: like it or not, Occam’s razor cuts both ways, and can be used to defend MWI.  The idea is whether Occam’s razor applies to the number of universes, or the number of postulates in your physical theory.  As Max Tegmark pointed out, universally valid quantum mechanics leads to a multiverse as a consequence.  In order to get the Copenhagen interpretation (for years, the most popular flavor of quantum mechanics) and rid yourself of those pesky many worlds, you have to take Everett’s quantum mechanics and add one additional postulate: that wave functions collapse according to random and ultimately unknowable criteria.  That is, the MWI is simpler in the number of postulates required.  As Tegmark put it, which way you use Occam’s razor depends upon whether you prefer many worlds, or many words.

4.      The Many-worlds Interpretation is not falsifiable and therefore not science.

The jury’s still out on this one, but many people (including David Deutsch) think that the MWI is misnamed: that it is actually a theory in and of itself, and that it is falsifiable.  I haven’t made up my mind on this.

I tend towards Tegmark’s view that MWI is an untestable prediction of quantum mechanics, which is testable.  Because we take quantum mechanics seriously, we have to take one of its children (MWI) seriously.  As Tegmark says, it’s like black holes.  We can never see inside a black hole, so what goes on in there is never falsifiable; yet we take black holes seriously and call black holes “science” because general relativity (the theory that predicts black holes) is so successful.

5.      The Many-worlds Interpretation is fringe science and only believed by kooks.

These kooks include Stephen Hawking (who said the MWI was “trivially true”), David Deutsch, Bryce DeWitt, and Max Tegmark, among others.  They also include a sizable number of theoretical physicists working today.  In 1995 one poll (published in the French periodical Sciences et Avenir in January 1998) showed that 58% favored MWI; see also this informal 1997 poll.

Maybe we are all kooks.  But there are a lot of us, and the number is growing.

[Note: my book Why Is There Anything? is now available for download on the Kindle!  This book examines the many-worlds interpretation from a philosophical perspective.]

Read Full Post »

« Newer Posts