A question about Matrix Mechanics

[naima]

could you jump to Matrix_mechanics#Harmonic_oscillator

I do not see why
"the matrix element Amn is the (m − n)th Fourier coefficient of the classical orbit, the matrix for A is nonzero only on the line just above the diagonal, where it is equal to √2En"

Where can i find this (m-n)th coefficient? in which serie?
thank you.

 

[Simon Bridge]

To understand this, you need to look back at what the matrix is supposed to be describing - how it is derived.
It can get confusing to just work in individual elements ... so work out the $A_{1-4,1-4}$ elements the long way.

 

[naima]

I am reading papers on the origins of QM.
Heisenberg did not know what a matrix was.
He started with frequencies intensities and orbits.
When he has a periodical orbit (Om) he takes its Fourier serie. It has different coefficients and he says that
the (m-n)th is to be placed in a square tableau. It gives information on the transition to the (On) orbit.
This is the point i want to understand. Why does an orbit Om contain information about another orbit On?

 

[Simon Bridge]

You need to know what the matrix is for to understand what it says ... how are the rows and columns determined? What decides which number goes into a particular place?
"He started with frequencies and intensities and orbits" of what exactly?

It looks like he's talking about a transition matrix - the realization of which was arrived at by the back door - so to speak. By understanding the physics involved. It's very difficult to do it that way. We usually teach it the forward way these days.

In general, you set up a matrix by manipulating two sets of numbers.
A particular off-diagonal element will be composed of information about both the original values for that entry because that is how it is calculated. For instance - look at a multiplication grid: the A(n,m)=n*m.
In this case, the mth column is the multiplication table for the number m. (the 2nd for is the "x2 table" etc) and the trace of A is the series of squares. Can you see how this happens?

in QM you can have a wavefunction which can be expanded in a basis - so we can write it as a sum of basis vectors $$\psi = \sum_n a_n\psi_n$$ and we say that $\psi$ represents a superposition of the basis states.

We can, as a shorthand, express the wavefunction $\psi_a$ as a vector $\vec a = (a_1, a_2, \cdots)^t$ against that basis ... and another $\psi_b$ may have a different expansion: $\vec b = (b_1,b_2,\cdots)^t$

We could construct the matrix $A: A_{nm}=a_n^\star b_m$ then it should be clear why the off-diagonal elements could contain information about the "orbits" $\psi_a$ and $\psi_b$. But - without understanding how the matrix was constructed, you cannot understand the relationship each element is describing.

The origins that you are reading are of interest only for the history though, not the physics.

 

[naima]

Are you saying:
Do not read Heisenberg, do not read what Dirac said. they belong to history. Read only Peskin and Srednicki textbook.
I think that theoricists like Witten, Penrose et al. have read everything.

 

[dauto]

Are you saying:
Do not read Heisenberg, do not read what Dirac said. they belong to history. Read only Peskin and Srednicki textbook.
I think that theoricists like Witten, Penrose et al. have read everything.

Have you ever read Newton's Principia? Often times original work is hard to understand. Go ahead and read them if you want to but be aware that there other easier ways to learn theoretical physics.

 

[Bill_K]

I think that theoricists like Witten, Penrose et al. have read everything.

Perhaps, but I strongly doubt it. I know for a fact that this was not Feynman's approach. Paraphrasing, his advice was, reading the old papers will just cause you to make the same mistakes.

 

[Jano L.]

Do not read Heisenberg, do not read what Dirac said. they belong to history. Read only Peskin and Srednicki textbook.
I think that theoricists like Witten, Penrose et al. have read everything.

Reading modern textbooks on the subject is useful, to get some idea about the current majority view on the subject and because textbook authors try to be pedagogical and sometimes they succeed. Perhaps the best way to begin study of the subject is to read few standard textbooks on it first.

One of the best textbooks on wave mechanics, atoms and molecules I know, by Feynman's teacher John Slater, "Quantum theory of atomic structure" contains references to important original papers scattered throughout the book. Undoubtedly he read a lot from them and thought they were helpful and serious student should read them too.

I can say from my own experience that reading old papers is very valuable, especially on modern 20th century physics. I liked especially collection of papers by van der Waerden "Sources of Quantum Mechanics", Schroedinger's papers in "Collected papers on wave mechanics" and Planck's book "Theory of heat radiation".

If you already understand what modern textbooks try to say, reading original papers and books written by Planck, Schroedinger, Einstein, Bohr, Dirac, Heisenberg and others will help you to fill in some of the gaps and help to look at the basic items of the theory from a different point of view than that found in recent textbooks, which can be very useful in improving physical theory.

I know for a fact that this was not Feynman's approach. Paraphrasing, his advice was, reading the old papers will just cause you to make the same mistakes.

That is hard to understand. Feynman read works of past physicists and cited some of them in his papers and books. Could you post a reference or write more about the context in which he said that? Stated this way it sounds silly, since it is obvious that those who do not learn from the errors done in the past are prone to repeat them.

 

[Bill_K]

Perhaps, but I strongly doubt it. I know for a fact that this was not Feynman's approach. Paraphrasing, his advice was, reading the old papers will just cause you to make the same mistakes.

That is hard to understand. Feynman read works of past physicists and cited some of them in his papers and books. ... Stated this way it sounds silly, since it is obvious that those who do not learn from the errors done in the past are prone to repeat them.

Although it was a casual remark at the time, it's completely consistent with his approach to Physics. While it may not apply to us mortals, Feynman felt that the best way to learn was not by listening to others and trying to understand physics through them second hand, but by confronting the physics directly and developing your own understanding of it.

He had great admiration and respect for the great men who preceded him, e.g. Dirac, but it was on a personal level. Once at a Caltech colloquium given by Dirac, Feynman ignored the words completely, and spent the hour making a sketch.

Could you post a reference or write more about the context in which he said that?

It was a remark spoken to me at a student's PhD Oral.

 

[kith]

Do not read Heisenberg, do not read what Dirac said. they belong to history.

If you are interested in history, you should definitely have a look at the original papers. But you have to keep in mind that these papers were cutting edge research at the time they were published. They were written for researchers who knew the problems they were addressing very well and who had much formal training in theoretical physics. Also many of them were written under the paradigm of old quantum theory which is a collection of very different approaches and results without the unifying framework of quantum mechanics yet. Much of the terminolody is dated. In order to learn the physics, textbooks are superior because they are aimed at people who know little about the subject and have less formal training.

Heisenberg's 1925 paper is especially famous for being difficult to access. Aitchison et al. tried to make it more accessible by speculating about how he arrived at his conclusions.

/edit: Here's a quote from Steven Weinberg which is cited by Aitchison et al:

If the reader is mystified at what Heisenberg was doing, he or she is not alone. I have tried several times to read the paper that Heisenberg wrote on returning from Heligoland, and, although I think I understand quantum mechanics, I have never understood Heisenberg’s motivations for the mathematical steps in his paper.

 

[naima]

It is a very good link.

 

[Simon Bridge]

Thank you kith, that was what I had in mind :)
Anyone who wants to learn modern formulations of QM is best advised to use a work that is recent and also designed to teach them modern QM.

Scientific progress is not linear ... we may have got our modern QM from work based in ideas in papers like Heisenberg's but not a lot of the modern formulation actually relies on them much any more.

Science is not like a religion where you can keep going back to some older Original Text or refer to some Greater Authority to get at the Fundamental Truth or something. It's like building a house where you keep going back and relaying the foundations while the rest of the house is constantly being added to or bits torn down.

Feynman's comment reflects on how scientific progress is made: by learning from mistakes.
His (shared) Nobel Prize-winning paper on QED still contains useful stuff but it isn't much use for learning QED.

The path trodden by previous explorers is a twisty one, with lots of dead ends. If you follow them, you will find yourself meeting the same dead ends. You have to read some, and exploring twisty passages can be fun, but, to get anywhere further you have to take a shortcut.

There's a rule of diminishing returns about historical papers ... too far back and they can be worse than useless - except for history buffs perhaps.

 

[Pollock]

I am reading papers on the origins of QM.
Heisenberg did not know what a matrix was.
He started with frequencies intensities and orbits.
When he has a periodical orbit (Om) he takes its Fourier serie. It has different coefficients and he says that
the (m-n)th is to be placed in a square tableau. It gives information on the transition to the (On) orbit.
This is the point i want to understand. Why does an orbit Om contain information about another orbit On?

Please give me references to the papers you refer to.I want to know how Heisenberg assembled the arrays of numbers which Born realized were non-commuting matrices.I raised this question last week in a post,but none of the replies came close to answering it.From your post,you have clearly got a lot of information which might help me ; I would probably find Heisenberg's original papers too difficult.

 

[Pollock]

If you are interested in history, you should definitely have a look at the original papers. But you have to keep in mind that these papers were cutting edge research at the time they were published. They were written for researchers who knew the problems they were addressing very well and who had much formal training in theoretical physics. Also many of them were written under the paradigm of old quantum theory which is a collection of very different approaches and results without the unifying framework of quantum mechanics yet. Much of the terminolody is dated. In order to learn the physics, textbooks are superior because they are aimed at people who know little about the subject and have less formal training.

Heisenberg's 1925 paper is especially famous for being difficult to access. Aitchison et al. tried to make it more accessible by speculating about how he arrived at his conclusions.

/edit: Here's a quote from Steven Weinberg which is cited by Aitchison et al:

Can you please give me a reference to the Aitchison paper?.

 

[naima]

The book i was reading is (1932):
http://en.wikipedia.org/wiki/The_Physical_Principles_of_the_Quantum_Theory

Kith gave you a very good link (Aitchinson)
http://arxiv.org/abs/quant-ph/0404009
it contains all the calculation details and the way Heisenberg assembled all that.

 

[naima]

I want to know how Heisenberg assembled the arrays of numbers which Born realized were non-commuting matrices..

When Heisenberg has a system of orbits (wave functions $\phi_i$) governed by an hamiltonian H, he notes
$X_{m n}(t)$ = $\int [e^{ (iHt)} \phi_m]^* (x).x. e^{(iHt)} \phi_n (x) dx$

 

[naima]

there was a typo (minus sign)

$X_{m n}(t)$ = $\int [e^{ (-iHt)} \phi_m]^* (x).x. e^{(-iHt)} \phi_n (x) dx$

So if $\phi_i>$ are eigenvectors of H we have for a Dirac operator O
$O_{m n}(t) = <\phi_m|O|\phi_n> e^{i(E_m - E_n)t/hbar}$
Of course Heisenberg did not start from the Dirac operators. He used Fourier series.

you can read the 1925 paper:
http://fisica.ciens.ucv.ve/~svincenz/SQM261.pdf