Sorry, not in this post, but upcoming. I’m scheduled to do another episode of Bloggingheads.tv with David Albert, and we’ve decided to spend the whole hour talking about quantum mechanics. Start with the basics, try to explain this crazy theory and some of its outlandish consequences in ways that anyone can understand, and then dig into some of the mysteries of measurement, superposition, and reality.
So — what do you want to know? What are the really interesting questions about QM that we should be talking about?
One thing I don’t think we science-explainers get as clear as we could is the idea of the Wave Function of the Universe. It sounds scary and/or pretentious — an older colleague of mine at MIT once said “I’m too young to talk about the wave function of the universe.” But it’s a crucial fact of quantum mechanics (arguably the crucial fact) that, unlike in classical mechanics, when you consider two electrons you don’t just have a separate state for each electron. You have a single wave function that describes the two-electron system. And that’s true for any number of particles — when you consider a bigger system, you don’t “add more wavefunctions,” you beef up your single wave function so that it describes more particles. There is only ever one wave function, and you can call it “of the universe” if you like. Deep, man.
Here is another thing: in quantum mechanics, you can “add two states together,” or “take their average.” (Hilbert space is a vector space with an inner product.) In classical mechanics, you can’t. (Phase space is not a vector space at all.) How big a deal is that? Is there some nice way we can explain what that means in terms your grandmother could understand, even if your grandmother is not a physicist or a mathematician?
(See also Dave Bacon’s discussion of teaching quantum mechanics as a particular version of probability theory. There are many different ways of answering the question “What is quantum mechanics?”)
Andy S said…
“Question 3: In a delayed choice two-slit experiment, a particle knows when it’s emitted whether its path is going to be through one of two slits or a superposition of both paths based on how it’s going to be measured, even if the measurement happens 50,000,000 years later. HOW THE HELL … ahem. Excuse me. How does it GOD DAMN KNOW HOW … ahem excuse me again.
If that damn thing can somehow look 50,000,000 years into the future and see the laboratory it’s going to wind up in and see the scientist with his finger on a button and it makes its decision on how to propagate based on that, then the universe is rigidly deterministic to an extent that makes me want to just go and slit my wrists.”
…all of which ties in with Seans brief comments on the wave function of the universe!
This is going to be a very interesting discussion. I get the impression we are already in a postion to draw some (at least) preliminary conclusions!
QM, like SR and GR, is awesome!
Sean has plenty of material for the upcoming discussion!
How about giant magnetoresistance, without which no hard-disk iPod would be complete? Although arguably many bits of modern technology rely on quantum mechanics, GMR is a nice one to use as an exemplar, because it is a blatantly quantum-mechanical phenomenon that went astoundingly rapidly from discovery (1988) to universal industrial adoption and Nobel prize (2007)
It is also a good one to wave under the noses of those idiot politicians who think all money should go on “applicable” science rather than “useless” blue skies science (MRI is another good one there, co-invented in my department on the basis of an entirely pointless experiment, and now worth a billion or so a year).
Well, if the movie “What the Bleep…” is any indication, you can use it to sink free throws and to make yourself a great photographer or something.
wow! more than 50 comments! Usually the science stuff get 10 or 15, tops. Good going, Sean.
Re: the wave function of the universe. Wouldn’t this idea only make sense from the perspective of an observer standing outside the universe (which doesn’t make sense)?
Thanks.
Re: Debbie #38
That’s always bugged me, too. To date, I’ve got 4 Intro level Quantum texts in my library. Not one of them derives the equation.
The only one who makes a stab at it, is –you guessed it– Feynman in one of his Lectures on physics volumes.
It’s something along these lines: (note – there’s no preview, so my Latex’d equations might not be legible).
(also note: I’m only a B.S., so my equations might also be gibberish).
$latex |Psi(t^prime)rangle = U(t, t^prime) |Psi(t)rangle $
meaning the wave function at a future t’ is some operator times the wave function at t – the operator depends on t and t’.
Obviously,
$latex U(t, t) = I $
So in the infinitesimal region around t, U is probably something like,
$latex U(t, t+delta t) = I + delta t Omega + (delta t)^2 (whatever) $
That makes
$latex |Psi(t+delta t)rangle = (I + delta t Omega + (delta t)^2(whatever) ) |Psi(t)rangle $
Getting the derivative gives you:
$latex |dot{Psi}rangle = Omega |Psirangle $
Big whoop right? We had some operator U that we didn’t know anything about, now we have some operator $latex Omega$ that we don’t know anything about. But hit it on the left with $latex langle Psi |$ and you get:
$latex langle Psi |dot{Psi}rangle = langle Psi |Omega |Psirangle $
Adding the complex conjugate,
$latex langle Psi |dot{Psi}rangle + langle dot{Psi} |Psirangle = frac{partial}{partial t}langle Psi |Psirangle = 0 = langle Psi |Omega +Omega^dagger|Psirangle $
Which implies that $latex Omega^dagger = -Omega$.
Cool. So $latex Omega$ is anti-hermitian.
But we like Hermitian operators, so let’s multiply both sides by i. This gets us
$latex i|dot{Psi}rangle = i Omega |Psirangle $
That makes the operator on the right side Hermitian. That’s where the “i” in the Schroedinger equation comes from.
Next is the units. $latex |Psirangle $ always has some weird units like length^-1/2, and naturally $latex |dot{Psi}rangle$ would be length^-1/2 s^-1. That means that $latex Omega$ will have units of s^-1 or frequency.
Lucky we called it $latex Omega$ then.
So $latex iOmega$ is Hermitian and has units of frequency, so it must be some kind of frequency observable. Well, the energy levels of atoms are always associated with absorption and emission frequencies so we can get the operator to have energy units by multiplying both sides by $latex hbar$ to get:
$latex ihbar|dot{Psi}rangle = i hbarOmega |Psirangle $
Which is just the Schroedinger equation, with $latex ihbarOmega$ identified with the Hamiltonian.
That’s a great example, thanks! I really wasn’t aware of that, I assumed the modern computer was a product of classical physics (except for the laser in the optical drive), amazing that something as crucial as the hard drive depends on this “mysterious” science!
Basically, everytime you access a database, anywhere in the world, the results returned require Quantum Mechanics to be correct.
http://www.research.ibm.com/research/gmr.html
Just glancing through some of the comments here, Sean, I would recommend that you take at least a moment during BloggingHeads to mention that the Periodic Table of the Elements is QM at its finest. Everybody seems so interested in the off-the-deep-end questions that nobody has looked closer to home–how do we know the shape of water or CO2, or one of a hundred thousand important biomolecules? Basic QM and the Periodic table gave us all the necessary tools. I say this because the Periodic Table is something that nearly every Layman will have passing familiarity with and because it’s one of the greatest, most impactful, most tangible victories of quantum theory. Moreover, while I know it’s not as sexy as Quantum information theory and entanglement, nor as hardcore as QED or QCD or the Standard Model, it is accessible.
andy.s,
What you describe is interesting as far as it goes, but there are some problems in taking it as an explanation of the Schrödinger equation:
– Schrödinger developed his theory in the context of solutions to partial differential equations, not state vectors in Hilbert space. What you are describing seems to be closer to Dirac’s presentation of QM.
– The logical predecessor to the Schrödinger equation was of course Heisenberg’s theory. My one-minute summary of that invention was that Heisenberg was thinking about focusing on the Fourier components of the dynamical variables, and got the idea of limiting the frequencies in the spectrum of components to the allowed frequencies in the energy spectrum. This is described in Max Born’s book, The Dynamical Theory of Crystal Lattices, based on lectures given very shortly after Heisenberg’s invention. (The first part of the book is on lattices, the second is on Heisenberg’s brand-new QM.)
So why doesn’t any textbook ever derive it?
I’ve got several introductory texts and they all talk around it in different ways. It gets kind of annoying after a while.
@49 This one is actually not that hard! You’d have no chemistry at all, because the coulomb solutions don’t have stable orbits. I believe without a stable classical orbit the quantum business still couldn’t conspire to give you any chemistry.
I’m also not a quantum chemist, but I’m not sure you really need one here.
Following up on Doug’s answer (60), I think I also remember from partial diff-e, that any linear extra dimensions making the total number of linear dimensions an even number (4, in the case of the original question) results in every event echoing into infinity. Or something like that. I couldn’t even recognize a differential equation to save my life any more.
I’m guessing that if someone has a fairly clear-cut idea of what Hilbert Space is they are not in your target audience. I like #39 Cecil’s idea of using the two-slit experiment as a point of departure (a la Feynman). The explication thereof would give you the opptunity to explore any number of streams to any desired depth. I’d also second the suggestions those who have asked for some discussion of practical applications.
andy s.,
Are you equally bothered that textbooks don’t derive Maxwell’s equations? Or F=ma?
Can someone please explain bra-ket notation to me? The explanation on wikipedia just goews around in circles. I get as far as “Every key has a dual bra”, and follow the link on dual, and it’s off into chaosland.
How do you get from bra-ket to actual numbers? I see lost of psis and thetas, but at what point do you plug actual numbers into these things and get some sort of result?
Those spherical harmonics that are the orbitals of the hydrogen atom – does that apply to other atoms? What about molecules? I mean … as I understand it, the state of an electron is basically a complex number field which extends through all space, except that it’s very nearly zero almost everywhere except where the “location” of the electron is. If you have two electrons, then you multiply their wave functions together and integrate it, and the square modulus of the result gives you the probablility that they will interact. If they interact (exchange a virtual photon), the the momentum transfer means that they move apart, and that’s what electrical repulsion is. Or something. I’m still not sure how this interacts with time – the probability that they interact has got to be a “chance per second”, and all this stuff has got to be symmetrical WRT relativistic boosts.
Now, was that kind of right? Or is the field some sort of thing where at each “point” in space there is actually a matrix of complex numbers?
If an electron hits an antielectron and is anihalated, does the wave equation field thing dissapear everywhere simultaneously? Or does the dissapearance of it sort of propagate outward at the speed of light?
Anyway.
How did they figure out that Buckminsterfullerene was going to be yellow? I mean – what numbers to you plug in, and where do you plug them? In order to come up with that result, it couldn’t all have been algebra – they’d have to have dome some adding and multiplying to come up with the absorption spectrum. Sorry to carp on about it, but from the wikipedia page on bra-keyt notation, I can’t see how any of this stuff gets from math to reality.
Why does a water molecule have that mickey-mouse shape? How do you get from “The hydrogens have one electron, the oxygen has 8” to that particular shape?
That’ll do.
48. TimG:
Why does [measurement] leave the system in an eigenstate? That’s an empirical observation: if you measure a particular observable, the state is changed so that further measurements yield the same value for the observable. Why that happens is a deep question, but we can see that it does happen and the math has to reflect that.
Yes, we see it does happen, but that is all from examples of traditional “one shot” measurements where a single interaction results in either an eigenstate left over, or destruction of the particle (e.g., photon either passing or being absorbed by a polarizing filter. But that is not what happens when a photon passes through a half-wave plate! (Or, similar birefringent element.) Like I said, the photon gets its RH and LH bases swapped, but not collapsed. Of course, one pass through a HWP doesn’t measure anything anyway. Yet many passes should build up detectable angular momentum along a range, not just binary results, as I explained above. I can’t say for sure and must be humble about something that pushes the envelope, just asking for it to be considered.
weichi
My copy of Reitz and Milford on E&M has a quite extensive discussion on Maxwell’s equations and I think most students would be uncomfortable with a text that stated them right off the bat with no discussion of their roots in Coulomb’s law, Ampere’s law, etc.
As for F=ma it is, considerably more intuitive than $latex -frac{hbar^2}{2m}nabla^2Psi + VPsi = EPsi$.
Intuition is a fairly useful faculty. It certainly leads me to suspect that you don’t have a lot of friends.
I think this is probably off-topic, because I guess it probably has more to do with relativistic theories than with QM per se. But I have yet to read a really good explanation of how time dimension(s) and spatial dimensions are different. The whole “time as a fourth dimension” thing has made intuitive sense to me since I was a little kid. I can picture the whole of spacetime as the 4-D analogue of a 3-D block with world-lines running through it, a block that one can slice on different angles into different relativistic viewpoints on the world. But I’ve never quite understood, okay, time is a dimension like the left-right axis, but then why does it seem so different?
One explanation I could make sense of would just say, well it’s only different in that the universe had this very low-entropy state at one end (the beginning) of its time dimension, so we have the “arrow of time” and thus the notion of experience, the “course” of our lives, etc. But I’ve also read in several places that, no, it’s not that easy, temporal and spatial dimensions really are just different.
I don’t expect solid answers; I imagine this might be an area where different theories (general relativity, string theory, QM?) have different things to say (whether or not their answers are mutually contradictory). But is there a good discussion of “time as a dimension, but not a spatial dimension” anywhere?
It certainly leads me to suspect that you don’t have a lot of friends.
Sometimes I wish that I could enjoy the whole game of internet put-downs and flame-throwing, because it seems like something that would be a lot of fun if you were the kind of person who could enjoy it. Other times, I’m glad that it just seems ridiculous.
Talk about the classical analog of the Heisenberg Uncertainty Principle in classical waves- it makes sense when you’re thinking of water and not electrons, and the HUP is one of the things that makes QM so mysterious to the layman.
On the wave function of the universe I love Gell-Mann’s comment to Jim Hartle about the Hartle-Hawking wavefunction. “Hey Jim, if you know the wavefunction of the universe how come you’re not rich ?” or words to that effect.
I watched Jim Hartle’s talk at Oxford for the Everett 50 years of MW meeting on the web. It was very interesting, I can’t remember the URL of the back of my head but its well worth a view together with the accompanying slides.
Anyway as a chemist my interests in QM are more immediate and practical. Yesterday I compiled GAMESS (General Atomic and Molecular Electronic Structure System) on my system. So I guess I should shut up and calculate.
As a non-physicist, I’d add a vote for David Albert’s book as a nice introduction to QM.
andy s,
I apologize if I struck a nerve – it wasn’t my intention.
Are you really looking for a derivation – i.e., logical deduction from some “more fundamental” foundation? I don’t think you are going to find it – Schrod eqn is just the god-given way that the world works (*). Sure, you can offer plausibility arguments like in your post 55 (and I agree they are helpful!) but that’s not what I think of as a “derivation”.
In particular, I don’t think it’s correct at all to think of Maxwell’s equations as derived from Coulomb & Ampere laws; in fact I prefer to think of Coulomb’s law as a consequence of Maxwell eqns (in the electrostatic case). The roots of maxwell in coulomb and ampere are historical, not logical.
Anyway, some places to look:
* have you looked at Shankar’s path integrals chapter? 8.6 in particular shows how path integral approach is equivalent to schrod eqn, and maybe you’ll find path integral a more intuitive starting point?
* Landau & Lifschitz QM (section 17) shows that the classical limit of Schrod is the Hamilton-Jacobi eqn.
* Schrodinger’s paper of 1926 “Quantization as a problem of proper values” (reproduced in the book below) explains how he come up with time-independent version. He starts with H-J eqn.
* The book “Probability and Schrodinger’s Mechanics” by David Cook promise a “variational derivation” of Schrod Eqn in 8.1. I think it’s an expansion of schrod’s original argument, but haven’t read it, so not sure.
Anyway, my view is that the Schrod eqn is not something that can be derived – it simply is.
(*) I guess you can derive schrod eqn from quantum field theory, but to me that just begs the question of how do you derive quantum field theory.
1 What does the many-worlds interpretation really mean ? And why do so many physicists believe it’s correct ?
2 How is many-worlds different from parallel universes or the multiverse ?
3 Given that quantum mechanics is the basic theory that explains how the world works, why don’t we see its bizarre aspects in everyday life ? (I think this gets at the heart of many laypeople’s interest in QM) Could you discuss decoherence vs the theory of Zeillinger’s group (referenced in the link in comment # 50 above)
4 The work of Vedral and Zeilinger’s group on entanglement in many-body systems or bulk materials (this gets at beginning to see one of QM bizarre aspects
in (closer to) everyday life situations)
Why do you have to limit yourself to one episode ? Sean, you do a great service by attempting to explain physics to non-professionals, and in the age of the Internet the ability (bandwidth) to communicate has been immensely increased.
After all, you did inherit Feynman’s desk 🙂
This is the beauty of decoherence: it doesn’t have to be put in by hand at all. A really simplistic way of looking at it is as follows. Imagine that we start with a two-state system:
$latex midpsirangle = mid 1rangle + mid 2rangle$
(normalization ignored for clarity)
Now, as we progress this state forward in time, the two states will oscillate between one another dependent upon the difference in energy. But what happens when we observe the state? Observation of the state is provided by an interaction with some other system, a system that happens to be vastly more complex (let’s call the other system $latex midphirangle$:
$latex midpsiranglemidphirangle = left(mid 1rangle + mid 2rangleright)midphirangle$
After this interaction, what happens is that the oscillation time between states 1 and 2 becomes huge. If the state $latex |midphirangle$ is complex enough, the time to oscillate can become effectively infinite. Due to the fact that we are complex, then, every time we interact with such systems we end up preventing further interference: the wave function decoheres, and only one result is ever visible. Nothing put in by hand, just simple wave function evolution.
P.S. I hope the tex came out okay.
I think you’re misunderstanding what is meant by energy. Energy and matter are not different sots of thing. You don’t convert from one to another. Instead, energy is a property of matter. Now, you can convert from one sort of energy into another. For example, if I collide an electron and a positron together, the two can annihilate to form a pair of photons. This isn’t a conversion between matter and energy: photons are still a form of matter. What happens, however, is that the mass-energy of the electrons gets converted into kinetic energy of the photons.
So far as we know, these interactions are point interactions. But this will probably be amended once we understand more about high energy physics.