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Fundamental Quantum Mechanical relations


Sriman Dutta
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Hi,

Planck while explaining the black-body radiation postulated that photon energy is quantised, that is, E=hf, f is frequency. 

Similarly to explain the matter waves, de Brogile proposed that p=h/y, y is wavelength. 

Using this two relations, the whole theory of QM has been developed. 

Is there any derivation of these results ? Or are they accepted to be fundamental relations of nature? 

 

Thanks !

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Planck's and deBroglie's work would be in their papers on these topics. 

In addition, Schrödinger developed the wave equation.

AFAIK, these basics were proposed, not derived. Subsequent work was derived, but not these building blocks.

 

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12 minutes ago, swansont said:

Planck's and deBroglie's work would be in their papers on these topics. 

In addition, Schrödinger developed the wave equation.

AFAIK, these basics were proposed, not derived. Subsequent work was derived, but not these building blocks.

 

Then these are the basic or the most fundamental equations ?

I might say then that they are axioms. 

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Historically, they started as axioms, as said by Swansont. Those Einstein-DeBroglie axioms helped Schrödinger guess his equation, but he took a further step, because he involved the potential energy, which plays no role in the DeBroglie, Einstein, Bohr, etc. set of old quantum rules. Heisenberg used a more algebraic approach (matrix mechanics). Dirac proved that Schrödinger and Heisenberg's formulations are equivalent. But it was all guesswork.

But in the modern formulation, you can deduce them by using the postulates. In particular, the canonical commutation relations.

\[ \left[ X, P_x \right] = i \hbar I \]

as well as the correspondence principle.

Even today quantization of fields rests on the correspondence principle, which relies heavily on guesswork, because there is no unique way in general to postulate a quantum operator for a classical observable.

Edited by joigus
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18 hours ago, joigus said:

Historically, they started as axioms, as said by Swansont. Those Einstein-DeBroglie axioms helped Schrödinger guess his equation, but he took a further step, because he involved the potential energy, which plays no role in the DeBroglie, Einstein, Bohr, etc. set of old quantum rules. Heisenberg used a more algebraic approach (matrix mechanics). Dirac proved that Schrödinger and Heisenberg's formulations are equivalent. But it was all guesswork.

But in the modern formulation, you can deduce them by using the postulates. In particular, the canonical commutation relations.

 

[X,Px]=iI

 

as well as the correspondence principle.

Even today quantization of fields rests on the correspondence principle, which relies heavily on guesswork, because there is no unique way in general to postulate a quantum operator for a classical observable.

Okay. So pretty much the fundamentals are a guesswork. And yet the entire formulation that relies heavily on those axioms works out so well. 

Hence there isn't any independent deduction of the Planck-Einstein relations, as I infer. 

This is unlike other branches. In relativity the main postulate that speed of light is constant for all observers has a well-reasoned deduction feom Mawell equations solutions on the form of plane waves. The rest part of relativity like length contraction, time dilation, Lorentz transformation can be deduced mathematically from that postulate. 

 

 

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2 hours ago, Sriman Dutta said:

In relativity the main postulate that speed of light is constant for all observers has a well-reasoned deduction feom Mawell equations solutions on the form of plane waves.

I would argue here that it is in fact the other way around - the form that solutions of Maxwell’s equations can take depends on the geometry of the underlying spacetime. You can see this most clearly when you write down Maxwell’s equations in their most general form, using differential forms; you will find the Hodge dual appearing in them, which explicitly depends on the metric. Unless you use a metric that has the right form (i.e. metric signature), the invariance of c isn’t guaranteed.

I am also not sure whether the invariance of c alone is enough to derive the full set of Lorentz transformations - I don’t think it is. I think you need to postulate the invariance of the spacetime interval, which is a stronger condition; it implies the invariance of c, but maybe not the other way around. I’m actually not completely sure about this.

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23 hours ago, Sriman Dutta said:

Hi,

Planck while explaining the black-body radiation postulated that photon energy is quantised, that is, E=hf, f is frequency. 

Similarly to explain the matter waves, de Brogile proposed that p=h/y, y is wavelength. 

Using this two relations, the whole theory of QM has been developed. 

Is there any derivation of these results ? Or are they accepted to be fundamental relations of nature? 

 

Thanks !

There has been a great deal of mathematical development in axiomatic QM since the workers you mention.

You should look up the work of your countryman, V.S.Varadarajan, who died last year.

He proved one of the 10 Mathematical axioms.

https://en.wikipedia.org/wiki/Veeravalli_S._Varadarajan

Edited by studiot
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23 hours ago, Sriman Dutta said:

Planck while explaining the black-body radiation postulated that photon energy is quantised, that is, E=hf, f is frequency. 

Similarly to explain the matter waves, de Brogile proposed that p=h/y, y is wavelength. 

Using this two relations, the whole theory of QM has been developed. 

Is there any derivation of these results ? Or are they accepted to be fundamental relations of nature? 

In general, the main thing in QM is the uncertainty principle formulated by Heisenberg. It means the rejection of complete determinism in the laws of physics. And this is correct, because if there were only determinism in the laws of physics, that the fate of each person would be predetermined even at the time of BB.

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11 hours ago, Markus Hanke said:

I would argue here that it is in fact the other way around - the form that solutions of Maxwell’s equations can take depends on the geometry of the underlying spacetime.

But spacetime came later. This may be the situation now, but that wasn’t the historical path.

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On 12/14/2020 at 12:47 PM, joigus said:

the potential energy, which plays no role in the DeBroglie, Einstein, Bohr, etc. set of old quantum rules.

I didn't mean Bohr here. Bohr did formulate his quantisation principle with the potential energy very much in mind. I meant the Sommerfeld-Wilson-Ishiwara quantisation principle, which doesn't explicitly mention it, but it implies it.

6 hours ago, Sriman Dutta said:

Hence there isn't any independent deduction of the Planck-Einstein relations, as I infer. 

(My emphasis.)

I think the word "deduction" points to a very interesting feature of how new principles come about, and why it's anything but easy to get at them. The immediate temptation we all have is that we must deduce the new underlying principle --it is perhaps a consequence of our overridingly-deductive education--, when actually what time and again proves to be the essential step is an inductive reasoning. Something along the lines of,

What simple assumption must I adopt so that all these facts can be an immediate consequence of it?

It's what Markus, in other post, called the overarching principle.

It's what Einstein was a master at.

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9 hours ago, Markus Hanke said:

I would argue here that it is in fact the other way around - the form that solutions of Maxwell’s equations can take depends on the geometry of the underlying spacetime. You can see this most clearly when you write down Maxwell’s equations in their most general form, using differential forms; you will find the Hodge dual appearing in them, which explicitly depends on the metric. Unless you use a metric that has the right form (i.e. metric signature), the invariance of c isn’t guaranteed.

I am also not sure whether the invariance of c alone is enough to derive the full set of Lorentz transformations - I don’t think it is. I think you need to postulate the invariance of the spacetime interval, which is a stronger condition; it implies the invariance of c, but maybe not the other way around. I’m actually not completely sure about this.

Yaa. There were three main postulates: the constant speed of light for all observers, the same laws of physics for all inertial observers and the homogeneity of space. 

 

7 hours ago, SergUpstart said:

In general, the main thing in QM is the uncertainty principle formulated by Heisenberg. It means the rejection of complete determinism in the laws of physics. And this is correct, because if there were only determinism in the laws of physics, that the fate of each person would be predetermined even at the time of BB.

Well, Heisenberg's uncertainty principle is a special case of a nore general result in mathematics, which is the uncertainty principle for two Fourier conjugate variables. If g(t) be a function of t, and it's Fourier transform be G(f) in the conjugate domain f, then the uncertainty principle is equally valid for them. 

In QM x and p are conjugate variables and gence there exists an uncertainty principle between them. But what relates the two is that p=hk, h is Planck's constant divided by 2*pi and k is the wavenumber( also called the wavevector in 3d commonly). Hence as you see the more fundamental thing is p=hk and the uncertainty principle directly follows from that.

 

5 hours ago, joigus said:

 

I think the word "deduction" points to a very interesting feature of how new principles come about, and why it's anything but easy to get at them. The immediate temptation we all have is that we must deduce the new underlying principle --it is perhaps a consequence of our overridingly-deductive education--, when actually what time and again proves to be the essential step is an inductive reasoning. Something along the lines of,

What simple assumption must I adopt so that all these facts can be an immediate consequence of it?

It's what Markus, in other post, called the overarching principle.

It's what Einstein was a master at.

That's pretty much interesting. Of course physics is an inductive study. Mathematics is more of a deductive study where results are deducted from some logical axioms or intuitively satisfying axioms.

Though the actual test of physics is obviously experiments, the great enthusiasm in theoretical developments in the last century has perhaps popularized the tendency to derive everything, even the most fundamental aspects of nature. Physicists are trying to understand now why the universal constants have those specific values. They could have taken any possible value but out of all random numbers, Nature assigned them those numbers. Is it completely arbitrar or has deeper meaning- that's the question.

8 hours ago, studiot said:

There has been a great deal of mathematical development in axiomatic QM since the workers you mention.

You should look up the work of your countryman, V.S.Varadarajan, who died last year.

He proved one of the 10 Mathematical axioms.

https://en.wikipedia.org/wiki/Veeravalli_S._Varadarajan

Yes. Most Modern QM books begin with study of Hilbert space and linear algebra theorems and then into topics of operators, eigenvalue equations and wave mechanics. Although this is a more methodical study, the actual historic proess of development was different. I remember in the first sem QM lectures, there was introduction to experimental observations which proved the failure of classical physics. Observations took discrete values. Black body radiation, Compton effect, Bohr's model and Davisson-Germer experiment were the stepping stones to modern QM. 

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1 hour ago, Sriman Dutta said:

Yes. Most Modern QM books begin with study of Hilbert space and linear algebra theorems and then into topics of operators, eigenvalue equations and wave mechanics. Although this is a more methodical study, the actual historic proess of development was different. I remember in the first sem QM lectures, there was introduction to experimental observations which proved the failure of classical physics. Observations took discrete values. Black body radiation, Compton effect, Bohr's model and Davisson-Germer experiment were the stepping stones to modern QM. 

Modern ?

Black body 1860  :  Compton 1923  :  Bohr  1913  Davisson 1921  a century to a century and a half ago.

Here is something more modern, at least into the second half of the 20th century.

It is set in terms of more modern maths than Schrodinger,, Hilbert or Heisenberg had.

Mackey1.thumb.jpg.18d277fe25bc86ae19ba3bbbc2d13263.jpg

 

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M Planck's successful Black Body Radiation law, was derived in 1900, after many unsuccessful ( UV catastrophe ) ones ( since 1860 ? ).
But I see where you are going with this.
The teaching of QM has always been approached historically, and always starts with M Planck's desperate 'guess' at a minimum discrete energy for a charged oscillator in a cavity, after many previous unsuccessful attempts. It is then followed by the Bohr atom, Shrodinger, Heisenberg and maybe some Dirac. Perhaps it is time for a change, so the student doesn't pick up unnecessary baggage along the way, such as the Bohr atom.

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10 hours ago, studiot said:

Modern ?

Black body 1860  :  Compton 1923  :  Bohr  1913  Davisson 1921  a century to a century and a half ago.

Here is something more modern, at least into the second half of the 20th century.

It is set in terms of more modern maths than Schrodinger,, Hilbert or Heisenberg had.

Mackey1.thumb.jpg.18d277fe25bc86ae19ba3bbbc2d13263.jpg

 

I actually meant to say that modern QM begins with Hilbert space definition.

However in introductory college lessons, they first teach you the phenomenon that raised doubt in classical physics. As an example, I told about my first sem lectures in QM. 

 

 

on a side note, can you tell me the name of the book? It looks intriguing. 

Edited by Sriman Dutta
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11 hours ago, MigL said:

M Planck's successful Black Body Radiation law, was derived in 1900, after many unsuccessful ( UV catastrophe ) ones ( since 1860 ? ).
But I see where you are going with this.
The teaching of QM has always been approached historically, and always starts with M Planck's desperate 'guess' at a minimum discrete energy for a charged oscillator in a cavity, after many previous unsuccessful attempts. It is then followed by the Bohr atom, Shrodinger, Heisenberg and maybe some Dirac. Perhaps it is time for a change, so the student doesn't pick up unnecessary baggage along the way, such as the Bohr atom.

You forgot

 

Black body definition Kirchoff  1860.

Stefan's law 1879

:-)

 

3 hours ago, Sriman Dutta said:

on a side note, can you tell me the name of the book? It looks intriguing. 

It's one of the Princeton Mathematical Physics series of  Monongraphs

Mathematical Foundations of Quantum Mechanics

G W Mackey   1963

(Professor of Mathematics at Harvard)

I know it's now over half a century old itself, but I don't have any newer ones from an underlying  pure Mathematics point of view, all my later stuff is via Physics or Chemistry.

For example

The mathematical Principles of Quantum Mechanics

By D F Lawden (Professor of Maths Univesity of Canterbury New Zealand)

leans much more to the applied side, although it does include much more detail about quantum angular momentum,which of course is not included in Schrodinger.

 

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14 hours ago, studiot said:

Modern ?

Black body 1860  :  Compton 1923  :  Bohr  1913  Davisson 1921  a century to a century and a half ago.

Here is something more modern, at least into the second half of the 20th century.

It is set in terms of more modern maths than Schrodinger,, Hilbert or Heisenberg had.

Mackey1.thumb.jpg.18d277fe25bc86ae19ba3bbbc2d13263.jpg

 

John Von Neumann.

There is no other book that has been cited more times by people who haven't read it.

After it was published, QM would never be the same.

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On 12/16/2020 at 10:29 AM, joigus said:

John Von Neumann.

There is no other book that has been cited more times by people who haven't read it.

After it was published, QM would never be the same.

Ah yes Von Neumann  +1

Here is a series of 35 short lectures on Von Neumann Algebras.

http://people.math.harvard.edu/~lurie/261ynotes/lecture1.pdfhttp://people.math.harvard.edu/~lurie/261ynotes/lecture1.pdf

The link is to the first one, change the number to get the rest.

This is an excellent series in its range and content and in the way it breaks the subject up into small digestible packages.

enjoy.

Note a Borel algebra is the same as a Sigma algebra

 

Edited by studiot
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On ‎12‎/‎16‎/‎2020 at 4:30 AM, studiot said:

You forgot
Black body definition Kirchoff  1860.
Stefan's law 1879

Ahh, but Kirchoff, and Weins, just had a set of rules.
And Stefan-Boltzmann, as well as Raleigh-Jeans, didn't work, and headed for infinity at high frequencies ( UV catastrophe ).
Planck was the first to accurately describe Black Body radiation … in 1900 :) .

I wonder...
The fact that we all know QM to some degree, leads us to recommend textbooks which are fairly advanced.
But to a noob, a lot of material is left out ( or taken for granted as common knowledge ), leading to the confusion that prevails among people new to the subject , or the general population.
While the historical approach includes knowledge which is later discarded, a fully modern view leaves a lot of gaps.
Maybe a textbook which starts from basic principles, and gives a theoretical ( not historical ) foundation, before tackling advanced material is the best choice.

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3 hours ago, MigL said:

Ahh, but Kirchoff, and Weins, just had a set of rules.

Kirchoff's 1860 law stated (states - it is still 100% true today)

The power radiated by a body in thermal equilibrium equals the power absorbed by that body.

 

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10 hours ago, MigL said:

I wonder...
The fact that we all know QM to some degree, leads us to recommend textbooks which are fairly advanced.
But to a noob, a lot of material is left out ( or taken for granted as common knowledge ), leading to the confusion that prevails among people new to the subject , or the general population.
While the historical approach includes knowledge which is later discarded, a fully modern view leaves a lot of gaps.
Maybe a textbook which starts from basic principles, and gives a theoretical ( not historical ) foundation, before tackling advanced material is the best choice.

I agree. A very good primer to the formalism without historical perspective (requiring thermodynamics) is Gillespie:

https://books.google.es/books/about/A_Quantum_Mechanics_Primer.html

It concentrates just on the mathematical formalism, but it's very clear and you can read it in no time.

The more physical counterpart to me would be Quantum Mechanics of Atoms, Molecules and Photons (by Avery, a chemist):

https://www.amazon.com/Quantum-Molecules-Photons-European-chemistry/dp/0070941785

A very unorthodox book the latter. Both books can be read very fast and really trick you into believing that the subject is easier than it really is. They remind me of Tony Zee's "in a nutshell" series. 

Edited by joigus
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On 12/17/2020 at 7:10 PM, MigL said:

Ahh, but Kirchoff, and Weins, just had a set of rules.
And Stefan-Boltzmann, as well as Raleigh-Jeans, didn't work, and headed for infinity at high frequencies ( UV catastrophe ).
Planck was the first to accurately describe Black Body radiation … in 1900 :) .

I wonder...
The fact that we all know QM to some degree, leads us to recommend textbooks which are fairly advanced.
But to a noob, a lot of material is left out ( or taken for granted as common knowledge ), leading to the confusion that prevails among people new to the subject , or the general population.
While the historical approach includes knowledge which is later discarded, a fully modern view leaves a lot of gaps.
Maybe a textbook which starts from basic principles, and gives a theoretical ( not historical ) foundation, before tackling advanced material is the best choice.

I agree. I personally followed Griffiths Quantum Mechanics and McGraw Hill's Demystifying Quantum Mechanics as introductory textbooks, later supplemented by JJ Sakurai's Modern Quantum Mechanics as I got interested in the concepts of angular momentum. 

 

 

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This was the original English language text, and is still definitive on the development of angular momentum in QM.

Prior to offering this I was asked for, and provided texts which offered a mathematical background and setting within mathematics,

angmom1.jpg.dad9e061277a71b55fb458c541ceb399.jpgnot physic books which obviously use mathemtics, but within a physics setting.

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