
New straighforward approach to teaching quantum mechanics - georgecmu
http://www.scottaaronson.com/democritus/lec9.html
======
karpathy
I took 2 courses on Quantum Mechanics during my undergrad and I'm taking the
Coursera course now as well for fun.

My biggest quibble with how QM is presented is that it is, paradoxically,
never tied into physics. Quantum Physics is really a Linear Algebra +
Statistics but with extension to complex numbers. In my experience it is just
another math course and there is no Physics anywhere.

There's always talk of measuring states, applying Hadamard gates and writing
down their decomposition in the eigenbasis but it's abstract and meaningless.
What does a particle with some particular wave function look like? As it
evolves in time it "smears" out according to the time evolution equations, but
how fast is the process? Does it occur on scale of nanoseconds? Seconds?
Hours? How is it suspended, or operated on? What exactly is an example of a
measurement? There are so many tangible questions that would help with
intuition, but are never addressed.

This would be my approach to teaching quantum mechanics. Connect it to a
concrete physical system, explore in detail going back and forth between
experiment and math. And best, maybe even simulate the system somehow. Some
time ago I made an effort to try to simulate what I learned and this was the
result (as an example):
[http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp](http://www.youtube.com/watch?v=a88GlrUmI9Y&feature=plcp)
I'ts ugly and it's probably wrong, but it's tangible and the best I could do
because finding this kind of Quantum Mechanics, as opposed to a lot of talk
about measuring things is very hard.

~~~
drostie
There is an advantage to the original formulations of QM and they are
precisely these. It's true that in Quantum Field Theory you don't see these
Bell Inequality ideas and I've seen people working in Quantum Information
theory who struggle to prove that you can multiply a wavefunction by an
arbitrary phase factor and it is an unobservable change. QFT has real current
statistics and Lagrangian densities and Feynman diagrams, which give you a
much more tangible feel of what physics you're describing.

 _In my experience it is just another math course and there is no Physics
anywhere._

Heisenberg equations of motion are a good place to start. The original way we
stumbled upon quantum mechanics was due to Heisenberg, who noticed that a lot
of the wavy stuff people wanted to explain could be explained if Hamilton's
equations of motion df/dt = {f, H} + ∂f/∂t were generalized by treating x(t)
and p(t) as matrices and insisting that they do not commute, leaving instead
[x, p] = i ħ as a matrix version of an "uncertainty principle."

The corresponding quantum equation for an observable Â is that dÂ/dt = i [Ĥ/ħ,
Â], which allows you to start (most famously) with a harmonic oscilator Ĥ and
derive the Hamilton equations dx/dt = p/m, dp/dt = - k x, precisely _due to_
the failure of x and p to commute.

So you get this direct connection between known physics equations and the
quantum theory, and often the same thing which is responsible for driving the
uncertainty relation also drives all of classical physics.

 _What does a particle with some particular wave function look like?_

|Psi|^2 in the appropriate basis, I should say.

 _As it evolves in time it "smears" out according to the time evolution
equations, but how fast is the process? Does it occur on scale of nanoseconds?
Seconds? Hours?_

In quantum mechanics there is a discrete energy level spacing ΔE, and the
answer is that _no observable can change_ much faster than ħ / ΔE. For the
harmonic oscillator for example, nothing can change much faster than 1/Ω by
this criterion.

 _What exactly is an example of a measurement?_

This can get a little touchy but a good example is a photon hitting a
photomultiplier tube and generating a click -- especially if you start to play
with the polarization of the photon to generate entanglement and so forth.

Edit: With all that said, I highly recommend watching Feynman's New Zealand
lectures which explain, among other things, why CDs show rainbow patterns (in
a time, sadly, before people had CDs and so Feynman kind of just says "I wish
I could have brought you an example.") <http://vega.org.uk/video/subseries/8>

~~~
Dn_Ab
You're right. And it's well explained despite the subtle jabs at Quantum
Informations theory =P. There are definite advantages to the physics focused
approach. As a physicist (which I am not), in your practice of QM you get
familiarish concepts like spin, momentum , oscillators and Lagrangians.

But despite the wild successes of QFT, when it comes to explaining things it
is extremely hard to do without either using broken analogies or talking about
Feynman diagrams, Lagrangians and Hamiltonians. To realistically depict what
is going on you are replacing one branch of math with another more complex
branch with the main advantage being that someone who has learned the math of
classical mechanics can have a slightly stronger physical intuition.

The advantage of QIT is that because it is relatively simple already, a
slightly simplified form is still easier to understand and more representative
of the real thing than a highly simplified explanation of QFT. The other
advantage of this is that the simplicity allows the raw structure to be
exposed and tackled much more readily.

If I would bet I would say answers to questions like what is the wave function
exactly , how does the macroscopic universe arise from the cloudy quantum
picture, what is really going on in measurement etc will come from QIT. I get
the impression that many physicist don't have much respect for foundations but
these questions are still worth answering and would have a practical effect on
our world right away.

QIT is young yet, there are advantages to being able to take multiple
viewpoints of the same thing. The correct viewpoint can vastly simplify a
problem. To quote Egan: " Everything becomes clearer, once you express it in
the proper language."

~~~
drostie
Oh, I don't mean to demean the field of Quantum Information. Especially, I
find it really useful to run through the double-slit experiment by labelling
one slit as |0> and one slit as |1>, and then going through both slits comes
out as sqrt(1/2)[ |0> \+ |1> ] = |+>, which has certain "off-diagonal terms"
in its "density matrix."

If all that formalism is built up, you can have fun working through when these
off-diagonal terms exist and when they do not, especially in cases where you
take a new qubit as |0> and then entangle it into the system with a CNOT gate,
where you get |00> \+ |11>.

If folks are confused by the above, I have begun trying to explain it here:
[https://github.com/drostie/essay-
seeds/blob/master/physics/d...](https://github.com/drostie/essay-
seeds/blob/master/physics/doubleslit.md)

It's really kind of rough (in particular I'd like to use proper HTML
subscripts rather than Unicode subscripts eventually) but it should be
intelligible to a bright student who wants to know the basic ideas of QM.

------
raylu
This sounds a lot like what I started reading at LessWrong:
<http://lesswrong.com/lw/r5/the_quantum_physics_sequence/>

One line that particularly stuck with me is: "Dragging a modern-day student
through all this may be a historically realistic approach to the subject
matter, but it also ensures the historically realistic outcome of total
bewilderment. Talking to aspiring young physicists about 'wave/particle
duality' is like starting chemistry students on the Four Elements."

~~~
thebooktocome
It's nothing like the quantum physics sequence. Yudkowsky starts out making up
numbers from nowhere and asserts things for purely philosophical reasons,
whereas Aaronson actually has proofs (albeit left to the reader).

~~~
loup-vaillant
Sure, the numbers in "Quantum Explanations" are made up, but (1) the
experiments are real, and (2) everything besides the numbers is accurate (as
far as I know). Plus, the goal of the sequence was never to actually explain
quantum physics. It was to explain why a realist perspective (the wave
function is all there is, and the math says it doesn't collapse, so it really
doesn't) is by far the most probably correct, despite the fact that it makes
no new prediction compared to previous interpretations. That, plus answering
some philosophical question with physics.

Aaronson's explanation definitely is a step in the right direction. I still
have a quibble however: he keeps mentioning "probability" as an analogy to
amplitude. That confuses his explanation in my opinion. I'd rather have a
straight explanation of QM math, _then_ an explanation about its similarities
with probability theory. And the mixed state paragraph seems to conflate
subjective probability and actual distribution of amplitude. Ick.

------
gwern
> Two other perfect examples of "obvious-in-retrospect" theories are evolution
> and special relativity. Admittedly, I don't know if the ancient Greeks,
> sitting around in their togas, could have figured out that these theories
> were true. But certainly -- certainly! -- they could've figured out that
> they were possibly true: that they're powerful principles that would've at
> least been on God's whiteboard when She was brainstorming the world.

Actually, people studying the Pre-socratics like to point out that Anaxagoras
(IIRC) theorizes something eerily close to evolution: that at the beginning,
there were all sorts of random creatures, and only the ones which did well
survived and created more creatures like them. I don't know why noone followed
up on it (in contrast to something like heliocentrism, where we know why the
Greeks abandoned it for what were excellent and unobjectionable reasons at the
time).

------
bobwaycott
I find the references to "God" (e.g., "...why did God choose to do it that way
and not some other way?") to be both distracting and unhelpful working through
the text itself. It signals to my brain (possibly incorrectly) that either (1)
the author is trying to inject an unwarranted religious idea into an otherwise
potentially helpful explanation in a devious way, or (2) the author has sadly
mis-chosen a loaded term that doesn't add anything helpful to the explanation
at all, and instead detracts from working through it because it creates the
nagging question in my head of, "Is he really suggesting that _God_ (whatever
that may be in the reader's mind) _chose_ to do this?".

A better option (for just the chosen example) would be: "... why does the
Universe do it that way and not some other way?".

~~~
thebooktocome
This is exceedingly common language in the mathematical community,
particularly for the generation of Scott Aaronson and the one before him.

~~~
andyjohnson0
Are they referring to a God in the religious sense, or using it as a shorthand
for "nature" or "the universe"?

~~~
thebooktocome
Some of them are no doubt serious; I seem to remember a study showing that
mathematicians are more likely to be theist than atheist.

In my experience, "Why did God pick X?" and the like is shorthand for, "Is
there a classification/uniqueness theorem that says X is the only possible
solution to the problem?", which is how it is used here.

~~~
bobwaycott
Again, thanks for drawing out the shorthand. I personally prefer reading the
"is there a classification/uniqueness theorem that says..." over "why did God
choose to...". It requires zero cycles to process the intent of the first
version, while the second makes me stop each time and wonder which one the
author means.

------
glimcat
I'm sold, although I'd still start by sitting them down with a copy of
Feynman's QED. When doing math about a thing, it helps tremendously if you
already have some sort of intuition about what the math is describing.

Also, the historical stuff is fun and interesting. Physics students should
probably be taught this stuff regardless, but maybe not as part of a class on
theory.

~~~
jasomill
On the other hand, Feynman's most important lesson is thoroughly modern: the
surest way to understand something "intuitively" is to create it.

~~~
Tichy
Cool idea. Screw learning QM - let's rediscover it on our own!

------
pirateking
I find historical narrative to be an effective way to learn, but just reading
names, dates, and important events is boring. You have to pretend like you
have a time machine.

An effective historical lesson takes you back in time and puts you right at
the scene of the development, almost allowing you to have a conversation with
the characters and follow along with their thought process.

That being said, I appreciate this straightforward approach just as much, and
find that having a good understanding of the core concepts first, makes
adjusting the settings on the "time machine" much smoother.

~~~
Retric
IMO, the simple approach is dangerous. Saying here is the theory have at it is
fine if you want to teach someone how to design transistors, but the point is
not that QM is correct it's just the best approximation yet and in no way the
'holy word' on how the universe operates. The single most important thing to
teach science students is to discard existing theory's when reality does not
support them and to require any new theory to also be supported by the body of
existing experimental evidence.

~~~
loup-vaillant
> _the simple approach is dangerous_

Definitely. However, it also happens to be the approach used since
kindergarten up to high school, and in most universities. Whatever the field,
as far as I know, students learn real science only when they start their PhD.
Sometimes even later.

I would love to see a real science course, where students come up with
theories and discard them, again and again, until they come up with something
probably correct (or not). That would teach them to accept being wrong (I
hope). Today, it's hard to do for anything but open questions, which tend to
be pretty difficult.

------
gus_massa
Perhaps this is a good approach to Quantum Mechanics for a course about
Quantum Computers. (Maybe is has too much emphasis is the probability part.)
Quantum Mechanics has a little set of clear rules that are easy to follow and
get the correct result. So if your plan is to have some little black quantum
boxes and combine them to get quantum computations perhaps it is a good idea
to jump directly to the rules and forget about the history part.

The problem arises when you really want to understand how the little black
quantum boxes work, or how this scales to bigger systems and to the real work.
The correspondence principle says that if the system is big enough the quantum
effects should almost disappear, and the system should be correctly
approximated by Newtonian Physic. (There are some macroscopic effects that
depend on the quantum nature of the world, the most clear is superfluity.)

The biggest problem is the correspondence, how to make the connection between
the real world that you see and the quantum physics. The pseudo-historic path
gives in each step a more complex model, but the jumps are smaller. (For
example, usually the Schrödinger's model is explained before the Heisenberg's
model because it is more intuitive, but Heisenberg's work was a few years
before Schrödinger's work.) In this approach, there is no reference to the
method that you should use to transform a classical model to a quantum model.
The main examples are the harmonic oscillator and the hydrogen-like atom that
are used as the base to explain the properties of the matter.

------
geuis
By completely random occurrence, I was watching this video yesterday:

[http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-his...](http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-hist)

Its by the fellow that does Minute Physics on Youtube. Towards the end of his
talk, he _specifically_ talks about this topic.

He makes a comparison between mathematics and physics, wherein mathematics you
are taught the math without a ton of history. No scaffolding. In physics, you
are taught the entire history of physics, i.e. scaffolding, from the ground
up.

He also advocates directly teaching modern theory and leaving most of the
history out. Make learning physics more like learning math.

~~~
nhaehnle
As a PhD student in mathematics, I have to say that history is valuable in
mathematics because it explains why people are interested in certain problems.

When I was a master level student, I always got annoyed by advanced courses in
algebra (things like homological algebra) where it was often unclear _why_ one
should be interested in the problems stated there. Prompting the profs to give
some short historical overview can be very enlightening.

I do believe you can leave the vast majority of the history out. Advances in
notation etc. were made for a reason. But a little bit of history can be quite
important for context.

~~~
btilly
History may be valuable in mathematics, but it is demonstrably skipped. Few
know much about it.

When I was in grad school in math, I made the interesting discovery that if p
and q are polynomials over a commutative ring, any polynomial that is
symmetric in the roots of p and q is actually a polynomial in the original
ring in the coefficients of p and q. (The construction works whether or not
the ring can be embedded in a field where said roots actually exist.) Using
this observation it is trivial, for instance, to write down in fully factored
form a polynomial that has sqrt(2) + cube_root(3) as a root.

This construction was news to various mathematicians that I talked to,
including a combinatorics prof who studied symmetric polynomials and a number
theorist who worked on stuff related to the algebraic integers. Then finally I
talked to a very old mathematician with an interest in history. He told me
that I had rediscovered an old way to do things. At his encouragement I went
to the library, and picked up an algebra book from the 1800s. My construction
was taught, and there was a whole chapter full of problems where students were
expected to use it to come up with polynomials with specific roots.

Furthermore as I looked into it _both_ of the professors that I mentioned
before worked in areas whose history dated back to the observation that I
mentioned. It was used in the original proof that the algebraic integers form
a ring, and that construction was the original reason that people were
interested in symmetric polynomials.

For another demonstration of how little of their own history mathematicians
know, ask anyone why the notation for the second derivative is d^2y/dx^2. Then
ask them where the f' notation comes from. Then ask them what Cauchy was
trying to do that lead to Cauchy sequences. Most will draw a blank on all
three.

Don't read on until you're satisfied that you don't know the answers.

In the original infinitesmal notation, d was an operator. It could be defined
by d(y) = y(x + dx) - y(x). And you'd calculate a slope as dy/dx (drop any
infinitesmal bits). Well when you work out d(dy/dx)/dx it turns out that you
get d(d(y))/(dx * dx) which is more compactly written d^2y/dx^2.

The f' notation was introduced by Lagrange in an attempt to get rid of
infinitesmals by defining differentiation as a formal algebraic operation on
polynomials and power series. This fell apart when Fourier demonstrated that
apparently well-behaved power series could be used to construct pathological
things like step functions.

Cauchy came up with Cauchy sequences while attempting to define infinitesmals
rigorously. His approach fell apart on the seemingly trivial example of how
you rigorously prove the chain rule when the derivative of the inner thing is
0. (He was trying to avoid 0/0, but in that special case you get 0/0 all over
the place.)

