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Quantum mechanics for mathematicians

scottaaronson.com

34 pointshhm12 comments

I had to use Griffith's book, and he certainly didn't follow the historical approach. He writes out Schrodinger's equation on the first page, before moving on to "simple" examples (the infinite well). The hydrogen atom is covered after a 100 pages or so.

http://www.amazon.com/Introduction-Quantum-Mechanics-David-G...

I love Griffith's informal style, and I'd recommend the book. Familiarity with PDEs and harmonic oscillation would be the only necessary prerequisites I can think of.

Very few textbooks follow the historical approach as far as I can tell. The author gives a false account of the way Quantum Mechanics is typically taught.

No, I think Schroedinger's equation is exactly what Prof. Aaronson is complaining about. It's part of "the complicated patchwork of ideas invented between 1920 and 1926". (The equation was published in 1926.) Teaching the Schroedinger equation first is the historical approach.

Consider this sentence from Wikipedia:

"The Schroedinger equation defines the behaviour of \psi\;, but does not interpret what \psi\; is. Schroedinger tried unsuccessfully to interpret it as a charge density. In 1926 Max Born, just a few days after Schroedinger's fourth and final paper was published, successfully interpreted \psi\; as a probability amplitude, although Schroedinger was never reconciled to this statistical or probabilistic approach."

What this tells us is that Schroedinger's equation was a hack. It gave Erwin the right answer, but he didn't know what it meant. And, lo and behold, when I learned quantum mechanics in school I didn't know what the equation meant, either -- I was so busy trying to solve the differential equation, and learning about the relationship between the quantum Hamiltonian and the classical Hamiltonian, and looking up complicated Bessel functions, and being confused by angular momentum quantum numbers that I never really understood what the hell the wave function actually represented.

Aaronson's point is that we now know that quantum mechanics is about probability theory. Once you understand that -- complete with bras, kets, state vectors and matrices -- you are better equipped to understand the Schroedinger equation as one particular application of quantum theory.

I wish I'd known this guy back when I was a first-year physics grad student.

I don't see how one can interpret the first paragraph as referring to the Schroedinger equation:

"The first way -- which for most physicists today is still the only way -- follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then you learn about the "blackbody paradox" and various strange experimental results, and the great crisis these things posed for physics. Next you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you're lucky, after years of study you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex."

It's hardly possible to even write the Schroedinger's equation on the blackboard without revealing the "central conceptual point" that complex numbers are involved.

The "grueling differential equations" phrase refers to classical mechanics and electrodynamics. This part is actually true, you typically learn some classical mechanics and electrodynamics before you start QM.

He describes my experience in undergrad physics (and graduate engineering) pretty well. You get some examples to motivate a new theory, and then they jump right to the Schrodinger equation and expect you to fall back on what you know about turning the crank in boundary value problems. I didn't hear the word 'Hermitian' for a long time.

I think it's possible to start with Shroedinger's equation as an axiom, and then quickly move on to probabilities. Griffith's table of content goes:

1.1 Shroedinger's equation

1.2 The statistical interpretation

1.3 Probabilities

1.4 Normalization

...

2.1 Stationary states

2.2 The infinite square well

2.3 The Harmonic oscillator

2.4 The Free Particle

...

3. Formalism

3.1 Linear algebra

3.2 Function spaces

...

4. Quantum mechanics in 3 dimensions

4.2 The hydrogen atom

The problem with spin states is that it's too abstract for a lot of student, even though the math is simpler. Both approaches have merit, IMO. I find the square well + free particle to be much more interesting.

That was delicious. I've always avoided studying physics because it seems like very hard work to just satisfy my curiosity... so far my copy of the Feynman Lectures has only served an ornamental purpose.

"Quantum Information and Quantum Computation" by Nielsen and Huang is also a fascinating book if you want to look at Quantum Mechanics from a distinctly mathematical and CS perspective.

Awesome, thanks!

I'm Michael Nielsen (the first author). I'm very glad you liked our book! A little correction: my co-author is Ike Chuang.

(I'm not grepping my name. I've taken an extended, possibly permanent, break from quantum computing, and will likely start a web-based company or foundation. YC News is a great resource, which I check regularly.)

Out of curiosity, will you be applying any principals from the book to the startup?

The startup won't be quantum related, if that's what you're asking. Of course, there is a lot that I learnt in the course of writing the book that will be useful - techniques related to skills like abstraction, communication, synthesis, and managing large projects.

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