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The history of mathematics in QM


geordief
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I think it was @joigus who may have  pointed out recently  that for an understanding of quantum effects one must rely on the mathematics  rather than any physical demonstration.(or words to that effect?)

Since,in my case it seems unlikely I will at any time soon gain such mathematical  understandings could I ask instead maybe for a  general description of what those mathematical tools were and how it came about that they were seen to be necessary to address the problem?

 

Was it a gradual process of mathematical progress or were there one or two breakthrough moments ?

 

 

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1 minute ago, geordief said:

I think it was @joigus who may have  pointed out recently  that for an understanding of quantum effects one must rely on the mathematics  rather than any physical demonstration.(or words to that effect?)

 

This is a bit too strong of a statement for me to subscribe it, but once the mathematical basis is well established, yes, we should trust it. But experiments are the final deciders of everything in science.

What I may have said is that arguing with just words is dangerous, because sometimes words carry hidden assumptions with them.

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8 minutes ago, joigus said:

This is a bit too strong of a statement for me to subscribe it, but once the mathematical basis is well established, yes, we should trust it. But experiments are the final deciders of everything in science.

What I may have said is that arguing with just words is dangerous, because sometimes words carry hidden assumptions with them.

Maybe it was someone else (or my memory has wind blowing through the attic)

No matter.

Is there a "father" of the maths in QM?

(like Minkowski for GR I think)

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6 hours ago, geordief said:

Maybe it was someone else (or my memory has wind blowing through the attic)

No matter.

Is there a "father" of the maths in QM?

(like Minkowski for GR I think)

Heisenberg? I think it was he that established the operator:observable formalism and the use of matrices. But the development of QM was very much a collective effort: more so than relativity.

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11 hours ago, geordief said:

I think it was @joigus who may have  pointed out recently  that for an understanding of quantum effects one must rely on the mathematics  rather than any physical demonstration.(or words to that effect?)

Since,in my case it seems unlikely I will at any time soon gain such mathematical  understandings could I ask instead maybe for a  general description of what those mathematical tools were and how it came about that they were seen to be necessary to address the problem?

 

Was it a gradual process of mathematical progress or were there one or two breakthrough moments ?

 

 

 

There were many breakthrough moments and much infilling in between.
QM has always also been intimately bound up with particle physics.

Quantum theory started in 1900 when Max Planck announced a mathematical solution to the mathematical problem of the 'ultraviolet catastrophe'.

Einsten came next using this quantum idea to mathematically describe the photoelectric effect, in 1904.

1913 brought the Bohr atom which tried to describe electron orbits in terms of classical electro-mechanics, whilst introducing a quantisation of the energy levels.
This is called the old quantum theory.

Quietly Max Planck was busy during this time and introduced 'zero point energy' in 1911.

This led to the old quantum theory being modified to include this phenomenon.

At this point quantum theory quantum theory provided specific energy levels using 3 'quantum numbers' to describe transitions between them.
This was enough for the develoipment of orbit(al) mechanics a la Schrodinger and Heisenberg.
In turn this provided chemists and spectroscopists with mathematically based formulae describing their observations.

However there was blurring of the spectral lines, originally observed by Zeeman in 1896, and this phenomenon was re-examined.
This led to the introduction of a fourth quantum number the spin quantum number which is non classical in its physical manifestation.
Pauli introduced his exclusion principle (1925) and spin matrices (1927.

By this time researchers were beginning to uncover a whole new catalogue of particles.

The rest of the 20th century saw the relationship between QM and particle physics develop symbiotically as one feed on and influenced the other.

So we had Quantum Field Theory (QFT) in 1927 and Quantum Chromodynamics (QCD) in 1973.

and so on.

I suggest you look at this book in your local library or even buy a S/H copy.

Quote

A brilliant populariser and award-winning writer John Gribbin tells the whole storyof the micro-world, and the people who made the discoveries. An essential complement to Gribbin's Companion to the Cosmos, it is about the inner structure of everything- a quest which, like the quest for the understanding of the Universe at large goes back to the ancient Greeks and touches on all of scientific and philosophic thought since then.

https://books.google.co.uk/books/about/Q_is_for_Quantum_Particle_Physics_from_A.html?id=rS_8BUE7eN8C&source=kp_book_description&redir_esc=y

 

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13 hours ago, geordief said:

Maybe it was someone else (or my memory has wind blowing through the attic)

No matter.

Is there a "father" of the maths in QM?

(like Minkowski for GR I think)

The mathematical "engineer" (I prefer that to "father") of GR was Bernhard Riemann. And the mathematical "engineer" of QM is David Hilbert. By that I mean the people who introduced the "mathematical scaffolding" that later accomodated the physical theory.

But I don't think either one of them would have come up with the respective physical theories without experimental or theoretical physics input. In fact, when the essential ideas of both theories were formulated, the physicists that did it couldn't imagine the mathematical tools were there already. That realisation, as always, came later.

I think there's always a cycle that goes something like --example: electromagnetism--,

1) Induction: Observation of patterns, or "crude" observation: Lenz, Biot-Savart, etc.

2) Inference of a mathematical or pre-mathematical simple relations: Faraday.

3) The big picture in mathematical terms: Maxwell

4) Experimental confirmation of further predictions: Hertz

Something like that. The history of the development of electromagnetism is a great example of how this works. But, of course, it's more complicated than just that. The different "branches" feed each other in a complicated way.

Once we get the mathematically-closed form of the laws, the great generalisation, it's a matter of pushing and pushing the mathematical model until we find where it contradicts the experiments. It's also a matter of doing more and more refined experiments to check everything's OK.

In the case of quantum mechanics:

1) Wien, Stefan, the spectroscopists (Lymann...), etc.

2) Planck, Bohr, Einstein

3) Heisenberg, Schrödinger, Dirac, etc. find out about a previously-existing mathematical scaffolding -> matrix algebra, Hilbert spaces, Poisson's formulation of mechanics...

4) Anderson finding positrons, which is a prediction of the relativistic version...

Etc.

Sometimes it goes the other way. We find a puzzling experimental discovery, and the theorists must rack their brains, within the mathematical scheme we already trust, in order to understand the unexpected result. If it doesn't, the mathematical scheme must be generalised minimally, ie, in such a way that the treasure of previous results is preserved. Example: discovery of the neutrino. So it's complicated.

We may differ a little bit in what stage is what, but I think we agree in general terms.

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1 hour ago, MigL said:

then again, he was supposedly high on cocaine on Helgoland Island, in the North Sea.

 

Interesting. I thought he was just suffering from hay fever and went to Helgoland, where he went through an epiphany... That's what he says in his book Encounters with Einstein, and Other Essays on People, Places and Particles.

 

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12 hours ago, MigL said:

IIRC, W Heisenberg developed Matrix Mechanics based on spectral line emission experiments; then again, he was supposedly high on cocaine on Helgoland Island, in the North Sea.

11 hours ago, joigus said:

Interesting. I thought he was just suffering from hay fever and went to Helgoland, where he went through an epiphany...

Yes, what Joigus says here is also what I read everywhere. The landlady of the apartment on Helgoland thought Heisenberg had been in a fight or something, so bad he looked in his face. I've been on Helgoland: it is a small rocky island, very different from the main land around it, and I can't even remember that I saw one single tree. Pity enough I was there before I knew about Heisenberg having been there, and discovered his version of QM.

And MigL, just for fun, Google 'Heisenberg Cocain', and you probably realise where the story comes from... It was good for a laugh at least.

Oh, and a fat +1 for Joigus, of course.

Edited by Eise
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No, that Heisenberg ( Walter White ) cooked crystal meth.

Werner Heisenberg went to Helgoland, with its sparse vegetation, because he had a bad case of 'hayfever' ( allergies ), and he used aspirin and cocaine as a remedy.

He was also the head of nuclear weapon research for the NAZIS, and was close to being targeted for assassination, as a result.

See here       Werner Heisenberg - Wikipedia

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+1, studiot

Also second mentions by exchemist, and the point towards Heisenberg's wiki article by MigL. I'd point to the section, "Matrix mechanics and the Nobel Prize", where Born and Jordan get honorable mention for formalizing Heisenberg's work.

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