The energy operator ih∂/∂t and the momentum operator ihΔ or ih∂/∂x play a crucial role in the derivation of the Schrödinger equation, the Klein-Gordon equation, the Dirac equation, and other physics arguments.
The energy and momentum operators are not differential operators in the general sense; they do play a role in the derivation of the equations for the definition of energy and momentum.
However, we do not find any reasonable arguments or justifications for the use of such operators, and even their meaning can only be speculated from their names. It is used without explanation in textbooks.
The clues we found are:
1) In the literature [ Brown, L. M., A. Pais and B. Poppard (1995). Twentieth Centure Physics (I), Science Press.], "In March 1926, Schrödinger noticed that replacing the classical Hamiltonian function with a quantum mechanical operator, i.e., replacing the momentum p by a partial differentiation of h/2πi with position coordinates q and acting on the wave function, one also obtains the wave equation."
2) Gordon considered that the energy and momentum operators are the same in relativity and in non-relativism and therefore used in his relativistic wave equation (Gordon 1926).
(3) Dirac also used the energy and momentum operators in the relativistic equations with electron spins (Dirac 1928). Dirac called it the "Schrödinger representation", a self-adjoint differential operator or Hermitian operator (Dick 2012). (D).
Our questions are:
Why can this be used? Why is it possible to represent energy by time differential for wave functions and momentum by spatial differential for wave functions? Has this been historically argued or not?
Keywords: quantum mechanics, quantum field theory, quantum mechanical operators, energy operators, momentum operators, Schrödinger equation, Dirac equation.
The fundamental property of these operators is that they describe one step translations in time and space respectively.
Energy conservation expresses invariance under time translations and momentum conservation expresses invariance under spatial translations.
exp(τd/dt)f(t)=f(t+τ) and exp(ad/dx)f(x)=f(x+a), by the definition of the exponential and Taylor’s theorem.
Now note that τd/dt = -iτ(id/dt) and ad/dx = -ia(id/dx).
(It's useful to choose units such that hbar=1.)
Therefore exp(ad/dx)=exp(iap), if p = -id/dx, for example.
Dear Chian Fan
You ask: "Why can this be used? Why is it possible to represent energy by time differential for wave functions and momentum by spatial differential for wave functions? Has this been historically argued or not?"
From what I understand, it is because Schrödinger grounded his wave equation on de Broglie's 1924 historical equation that revealed that one orbit of the electron in the ground state of the Bohr atom corresponds exactly to the energy of Planck's action constant. Consequently, the number of such orbits provides in 1 second (the time element), the same frequency (6.580495968E15 Hz) as that provided by the energy calculated with the Coulomb equation for the distance between the Bohr orbit and the proton, that is, the radius of this orbit (the space element).
This amount of energy is the actual relativistic energy induced at this distance from the proton, that can be calculated by both methods.
Analyzed in part from Equation (1) in this reference:
Article The Last Challenge of Modern Physics: Perspective to concept...
Best Regards, André
Dear Stam Nicolis
"The fundamental property of these operators is that they describe one step translations in time and space respectively." This is a very valuable explanation. However, there may be differences between differential and translational operations. Is this the case?
Best Regard！
Chian
“What are the roots of the energy operator ih∂/∂t and the momentum operator ih∂/∂x?”
- the operators above are postulated in QM, the roots of the operators above are the same as that is at formulating of any other postulates in any other mainstream physics theories, since in the mainstream all really fundamental phenomena/notions, first of all in this case “Matter”– and so everything in Matter, i.e. “particles”, “fields”, etc., “Consciousness”, “Space”, “Time”, “Energy”, “Information”, are fundamentally completely transcendent/uncertain/irrational:
- i.e. really physicists don’t understand what they study, what they themselves are, and why sometimes at their for some transcendent/mystic reasons and by some mystic ways happening “experimental observations” of some , for some transcendent/mystic reasons and by some mystic ways existent in the observed in some systems of some mystic “particles”, “bodies”, etc., some mystic interactions between the particles, bodies, etc., some universal logical links/ constants are observed,
- while the observers, basing only on the experimental data, completely ad hoc postulate that these universal logical links/ constants are really always exist/act in really mystic Matter, composing/developing the theories that follow from the postulates, and so fit with experimental data.
The same, of course, happened also at formulation and development of QM – in 1905 Rutherford experimentally discovered that atoms are constituted from extremely small nuclei, around of which on much bigger orbits electrons rotate, what contradicts with what was known from experiments on macro scale earlier, where accelerated electrons principally radiate photons and so lose energy, while the atoms are stable, and so electrons in atoms for some mystic reason don’t radiate.
To solve this problem in 1913 Niels Bohr postulated that for some mystic way in [firstly Hydrogen] atoms [briefly, more the QM history see, say, in https://en.wikipedia.org/wiki/Bohr_model] :
- the electron is able to revolve in certain stable orbits around the nucleus without radiating any energy;
- the stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant;
- electrons can only gain and lose energy by jumping from one allowed orbit to another.
Further de Brogile again ad hoc suggested that electrons on the orbits for some mystic reasons and by some mystic ways exist as some mystic “standing waves” that have lengths λ=ћ/p, p is electron’s momentum; and, in certain sense finally, Schrodinger formulated corresponding ad hoc equation for these waves, where the operators of – further “observables”, XYZ space coordinates [the coordinates themselves], particle momentums -ih∂/∂x, and – for non-stationary systems – energy ih∂/∂t were ad hoc introduced;
- while at the application of the equation for description of the Hydrogen atom the solutions of the equation, i.e. values of radiuses of orbits, of electron’s energies on the orbits, etc., turned out to be in well accordance with the experimental data. Further …
Again – by this scheme all happened/happens at formulating and development of all other physical [and all other sciences’] theories
– which so, again, have the one common “roots” – the experimental data and some scientists, which were/are able to formulate fundamentally new and mostly contradicted with older theories’ versions, ad hoc postulates, which in some universal enough conditions are adequate to the reality.
That is another thing, that mainstream physics, because of the transcendence above, really is based on eventually fundamentally non-adequate postulates, and, while the existent theories are well applicable at everyday physical practice and at applications in technologies, the really fundamental physics development is principally possible only if the fundamental phenomena/notions above are really scientifically defined, what is possible, and done only in framework of the 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see
https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed
The post is rather long, so now
Cheers
So let us to continue. In the SS post above it is pointed that really all mainstream physical theories, because of the fundamental transcendence in the mainstream of all really fundamental phenomena/notions principally cannot, and so haven’t any real fundamental roots – all theories are fundamentally inevitably logically are based only on ad hoc postulates that for/by some completely transcendent reasons and ways fit the theories with experiments,
- despite that most of postulates are either really fundamentally wrong illusory interpretations of experimental data, as that, say, are postulates in the SR/GR, or use some really mystic mathematical tricks, as, say, that are QFT theories, where some transcendent “virtual” particles and fields are postulated, and to fit these postulates with experiments inevitably it was/is necessary to introduce in the theories again some transcendent “renormalization” techniques, etc.;
- and so really scientific roots can be only provided that the fundamental phenomena/notions [see the yesterday SS post] are really scientifically defined, what is possible , and done, only in framework of the SS&VT “The Information as Absolute” conception.
That allowed to develop the informational physical model which is based on the conception, the couple of recent main papers are
https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics, and
https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics, where more 30 really fundamental problems are either solved or essentially clarified, including, say, to develop the 2007 initial model of Gravity and Electric Forces, see
https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces
and 2023 initial model of Nuclear Force in
https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force
- where fundamentally by no means some GR “spacetime’s physical effect”, but, of course, only non-existent in the mainstream some fundamental Nature Gravity force, and existent in mainstream Electric and Nuclear Forces, act using quite real mediators; and. at that, it turns out to be that all these Forces act by the one scheme, etc.
Including in the model [see the 2-nd link, more in section “Conclusion”] the problem of the QM momentum and energy operators, where energy quite evidently strangely has two differential operators - ih∂/∂t and Hamiltonians in Schrödinger and Dirac equations, while ih∂/∂t looks quite naturally as some momentum operator,
- what, including, resulted in that Dirac to ground the equation for electron with negative mass/energy assumed real existence of some really transcendent and non-existent “sea of negative energies”, while by the very nice mathematical way really wrote, instead Hamiltonian operator, a version of whole momentum operator; etc.
A couple of SS posts on the page 4 in https://www.researchgate.net/post/How_does_a_wave_function_collapse_upon_observation_How_does_a_wave_function_know_to_collapse_when_its_observed/4 are relevant to this thread question.
Cheers
Chian Fan
the energy operator ih∂/∂t and the momentum operator ih∂/∂x and the others dependant upon them thru various technicalities of Mathematization processualization of physics , are ALL mere abstractions with NO equivalent functionality in perceptible objective world of reality around us ..........
This is sheer Einsteinization { handling the world Einstein-like } of non-Einsteinian { = Empirical } physics , , , ,
Chian Fan
Which means you yourself do not know of any other instances ??
Dear Reza Sanaye
I am sorry.
The information I got is that the energy operator, momentum operator was first used by Schrödinger in 1926 (Brown et al. 1995; Dirac 1981); the Dirac equation (Dirac 2017; Manoukian 2020), Hideki Yukawa to calculate the medium particle mass (Yukawa 1935), etc. all use .
CN. Yang emphasized that one of the important points in quantum mechanics was that the momentum Px becomes a differential operator ih∂x. And in 1927, Fock and London independently used it in Weyl's theory (Yang 2013).
"It also turns out that time and energy are theoretically Fourier transforms, though the details of a time operator is problematic in realistic settings."(KHAN 2013).
No one has argued for the universality of the energy and momentum operator.
Best Regards, Fan
“…Thank you for your attention and comments！
They are not ordinary operators. Does anyone know of examples of using operator ih∂/∂t and operator ih∂/∂x on functions other than wave functions?…..”
- again, the QM theory, including the operators formalism, including the operators of momentum and [in time dependent evolving/interacting quantum systems] operator of energy, was/is developed only as a mathematical tool that is based on postulated ad hoc initial premises, which are consistent with experimental data, which provides in many cases adequate to the reality description of states of material objects/structures on the QM scale,
- as that happens at developing of any other mainstream natural sciences theory, since in the mainstream really everything is completely transcendent, and so the mainstream has no any knowledge from which, say, it could follow any rational scientifically answer to – why the postulates of a theory are as they are? Including really operator ih∂/∂t and operator ih∂/∂x are equally “ordinary” in QM as the other QM operators are, and if these operators are used in some theories beyond QM, they will be in such theories also “ordinary” operators.
Again, here is no principal problems, what are real QM problems - see the couple of SS posts on page 1.
Cheers
Yes
As Sergey Shevchenko is saying
As I told Chian earlier ,
these operators are mere constructs of mathematization
Those who applied them [ in addition to those who apply similar operators now ] have NO objective equivalent for them in the Empirical Physics world around us ...........
All those mathematical constructs , for example ,that have been devised to prove perforce dark matter and dark energy exist . .. All such mathematical constructs : Who is out there from amongst Academia to protest against them or to see thru their falsification ??
Dear @all
I am sorry, we know that differential operators, Laplace operators, Dirac operators... , are operators, but it is still difficult to understand that the energy operator and the momentum operator are just the same kind of symbols that can perform various operations.
Best Regards, Fan
Dear Chian-Fan,
I am sorry in advance for my simple answer on your question "Why is it possible to represent energy by time differential for wave functions and momentum by spatial differential for wave functions?" It will be based on a physical not mathematical representation.
The answer is related to an idea that any discrete element of the surrounding (electromagnetic) space is represented by a photon with h=Eτ and h=Pλ, where τ - its characteristic time or period and λ - wavelength. Any moving particle with a mass m has its momentum P and kinetic energy E, which due to de Broglie concept are related to the action quantum h in the same manner as photon. But the most important thing is that this particle takes the place of photon in the surrounding space with the same E and P. The Schrödinger equation, through a mathematical construction in the form of wave function, simply describes the transfer of particle as a change in its energy in time and periodic behavior in space [Freeman Dyson's lecture in Moscow: https://doi.org/10.3367/UFNe.0180.201008f.0859]. Therefore, in the stationary case, the equation has no time derivative, and the motion of particle is described as a standing wave.
Sincerely yours, Dulin Mikhail.
“…I am sorry, we know that differential operators, Laplace operators, Dirac operators... , are operators, but it is still difficult to understand that the energy operator and the momentum operator are just the same kind of symbols that can perform various operations.….”
- again – see the SS posts above – everything in mainstream physics is based only on some purely ad hoc, i.e. that have no any rational grounds besides that the postulates statements are adequate to experimental data, postulates; including the QM postulates are as they are just for this reason and by this way. Quite equally you can, say, ask – why in equation F=m*a, the letters “F” and “a” are linked by mathematical multiplication operation sign “*”? – for what really is necessary to understand what these letters mean, but Newton and till now the mainstream really fundamentally cannot answer to this question.
Again, that is in the mainstream since in the mainstream all fundamental phenomena/notions, including the phenomenon “Matter”, are fundamentally transcendent/uncertain/irrational, and so really in the mainstream everything in Matter, including “particles”, bodies, fields, both energies [“kinematical”/“ordinary” and “potential”], “forces”, etc., are completely transcendent also; including the mainstream doesn’t know – why on QM scale QM theories are applicable, while their “classical” sisters aren’t applicable?
At that in the mainstream the tenet that mathematics is so extremely effectively describes what exists and happens in Matter – what is again always confirmed at experiments, that so everything in Matter follows from mathematics; and so – Don’t ask, say, what are QM operators ? – Shut up and calculate.
However real physics and mathematics are fundamentally different sciences, and real understanding of what exists and happens in Matter is able only provided that all fundamental phenomena/notions are really scientifically defined, what is possible, and essentially done, only in framework of the Shevchenko-Tokarevsky’s “The Information as Absolute” conception and “informational physical model” that is based on the conception,
- where, say, it is explained why QM is adequate to the reality, and why and where isn’t adequate, essentially what are “particles”, “momentum”, both energies, what is “physical action”, what are “fundamental Nature forces” and the Forces “mediators”, and so what are the Forces “fields” that really are flows of the mediators, etc.; more see the couple of SS posts on page 1, where the thread question is clearly answered.
To understand what are the conception and model it is useful to read SS posts in rather short discussion in https://www.researchgate.net/post/What_is_time11#view=63c940bafa2ccd380d0b4943/6 , at least pages 4,5,6.
Cheers
Dear Mikhail N Dulin
I believe you have provided a very important reference, Freeman Dyson's lecture in Moscow. unfortunately, I do not have access to the original text at this time. I expect to be able to read it soon.
If it is as you say, then is it possible that the wave function itself implicitly expresses the electromagnetic field?
Best Regards, Fan
If it is as you say, then is it possible that the wave function itself implicitly expresses the electromagnetic field?
Dear Fan, I'm afraid I can't exactly express the words of Freeman Dyson, written here from memory. But myself I'm absolutely sure that the wave function itself implicitly expresses the electromagnetic field, as you wrote.
The thing is that it is the electromagnetic space that determines the wave properties of the particle. These properties, for example, of proton are not inherent in it initially. In atomic nucleus, the proton behaves quite differently. And only outside the nucleus, the proton should become like a photon, the place of which it occupies during its movement.
I am very glad that in this respect my opinion coincides with yours, as it is presented in you preprint published here.
Mikhail.
“…The Schrödinger equation, through a mathematical construction in the form of wave function, simply describes the transfer of particle as a change in its energy in time and periodic behavior in space [Freeman Dyson's lecture in Moscow: https://doi.org/10.3367/UFNe.0180.201008f.0859]. ….” → “…I do not have access to the original text at this time..”→“…here you are https://www.bu.edu/pardee/dyson/ (not the 2022 Moscow lecture, but probably a close content) ….”
- the last link has no any relation to the thread question and to QM at all;
- the first link, since the author was a mainstream physicist, and so really fundamentally didn’t understand – why mathematics is extremely effective tool at description and analysis of what exists and happen in Matter [why that was/is in the mainstream so? – see the SS posts above], so principally could not explain really – why QM is the effective tool at description and analysis of what exists and happen on QM scale, including, of course – why QM is as it is; and why QM exists at all, though.
In this case, say, as more reasonable one is an other popular paper in the mainstream – the E. Wigner’s https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf , which is quite frankly, and quite correctly is titled “The unreasonable effectiveness of mathematics in the natural sciences”.
Again, the really scientific answers to the thread question, starting from the completely obligatory first question why does QM exists at all? are here only in the SS posts above, what is possible since the answers are based on the really scientific 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, and the informational physical model that is based on the conception, where everything in Matter, which is fundamentally transcendent and so unreasonable in the mainstream, obtains completely scientific rational grounds.
More see the SS posts on pages 1, 2, and links in the posts.
Cheers
Sergey Shevchenko
Many thanks for your constantly offering very helpful links on various RG threads ...............
YOURS :
reza
To Sergey Shevchenko and all non-believers in Freeman Dyson's lecture in Moscow.
Here are the specific words of Freeman Dyson, printed in the English version [Physics - Uspekhi 53 (8) 825 - 834 (2010)] of the corresponding physical Russian magazine:
"Schrodinger had to invent wave mechanics to complete the unification. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrodinger's idea seemed to be going nowhere. But then came the surprise. Schrodinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation."
The main thing that Freeman said in this piece of text:
1) The equation looked like the equation of conduction of heat in a continuous medium;
2) Schrodinger put the square root of minus one into the equation;
3) Suddenly it became a wave equation instead of a heat conduction equation.
That is, the Schrödinger equation is essentially the heat equation for a certain mathematical construct called the wave function. This is exactly what I was talking about earlier in my comment. Adding that the wave function is periodic in Planck's constant - the quantum of action.
Sincerely, Dulin Mikhail.
Mr Dulin Mikhail :
Other trickery , paradoxes , and Charlatanry in Quanta physics three are ................
for instance :
Spacetime itself is comprised of some foliation of sub-manifolds in which case spacetime turns out to be an entity of stationary nature which can be known only on those Locales where conservation of energy would no longer be deemed sacrosanct. The Many-Worlds interpretation on the energy flow within a certain universe links Noether (local) symmetries to explicit data flux that is acting equivalent to energy contributions. For instance, as of a spinning particle, we may “measure” a certain position or spin. However, measuring things in systems causes changes in wavefunctions. The measured spin mentioned above gets into ANOTHER amplitude of probability. As a consequence, the concept of energy in quantum physics differs relatively significantly from the ordinary one we generally bear in mind. This spells that there is likelihood of foliating spacetime with a time parameter corresponding to surfaces where energy is not conserved.
It is impossible to come to collapse of mathematical superpositions on energy levels where no prediction can be made of the number of times we may possibly decide to do measurements. And you certainly know that measurements do not corroborate each other.
respectfully :
REZA
“…..To Sergey Shevchenko and all non-believers in Freeman Dyson's lecture in Moscow.
Here are the specific words of Freeman Dyson, …
"Schrodinger had to invent wave mechanics to complete the unification. Starting from wave optics as a model, he wrote down a differential equation for a mechanical particle, but the equation made no sense. The equation looked like the equation of conduction of heat in a continuous medium. Heat conduction has no visible relevance to particle mechanics. Schrodinger's idea seemed to be going nowhere. But then came the surprise. Schrodinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation."….”
- though that is rather vague version of how Schrodinger invented his, basic till now in “non-relativistic” QM, equation (more real information see, say, https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation#cite_ref-33 , section “History”) but that – and what is written in Wiki – nothing changes in what is written in the SS posts above
– really, since in mainstream physics everything was/is fundamentally transcendent for all/every mainstream physicists, all/every theories, including QM, completely logically inevitably are based on some completely transcendent ad hoc postulates, which are introduced in the theories only basing on experimental data aimed at to fit the theory with experiments.
Including Dyson, who seems was excited by the fact that introducing of square root of minus one into the equation has made the equation be able , say, to compute Hydrogen atom’s spectral lines energies,
- but Dyson principally by no means could – and so doesn’t - explain - for what/ really mystic for him – and for all mainstream physicists till now, though, - introducing of square root of minus one into the equation makes the equation be adequate to the reality? Though seems Dyson – in contrast to Wigner – seems even didn’t suspect that this quite natural for any normal human question exists.
Moreover, yet till now in the mainstream there exist quite legitimately a number of interpretations even what the wavefunction is at all - though, of course, before to answer to this question it is necessary to answer to the question “why QM exists at all”, which again cannot be answered in the mainstream, etc.; more see the SS posts above.
Cheers
Howdy Chian Fan ,
As ever, Stam Nicolis answer is fine mathematics, clear and helpful, as far as it goes. I had struggled with this issue and the issue of your No.6 question on the use of imaginary in physics. The exchanges, with many fixes of my curiosity, are on my discussion, What is a linearized representation of an essentially quadratic phenomenon? My answer, which is how it seems to work out, is quoted here. That slash-h is used in order to have a radian measure of Planck's Constant for use in the exponent of an exponential wave equation was new to me, and welcome. That the operators enable convenient use of the Hamiltonian in dealing with part of the problem, a linearized representation of an essentially quadratic phenomenon, and then making the answer real with its complex conjugate seems clear now. It may be of value to others.
"I've been away reading "The Dirac Operator and Its Properties." It was a great review and I learned a number of things, like realizing that i and slash-h are included because one is preparing an exponent for an exponential wave equation. Radian measure was needed, but Planck had considered cycles of harmonic oscillators, thus h/2pi, and of course i is required for an exponential wave equation exponent. That the Dirac approach evolved in a few years to QED, and that Dirac was involved in its development was new, since I had gotten my QED from Feynman's lectures.
As noted in earlier posts: "I have come to think that that linearization isolates the quality of the Hamiltonian/Energy in each of two relative reference systems so they may be treated individually in Lorentz-covariance." and "It now seems to me that in order to do Lorentz-covariance with a single electron, H must be written linear. In order to calculate when H is linear, matrices are necessary to carry the . . . load of . . . linear representations." And from home notes: Incidentally, there are a large number of zeros and ones in these celebrated matrices implying that their function is to select relationships more than to modify them. and The Hamiltonian expression of energy in its linear operator persona turns out to be an expression with a complex conjugate. It may used to calculate, but the calculations produce a tentative result that must be actualized by multiplying the resulting Hamiltonian operator expression by its complex conjugate. Finally, as posted earlier: I remain somewhat concerned about the natural state of the linearized representations, but that is a mistake on my part, because they are not natural. The linearized representations "exist" only within the calculation as carriers of information . . .. They are tools, not part of the phenomena. The discussion question appears to be answered.
"I shall explore the issues further, including the velocity problem in Dirac theory that you note, and I have designs on a more formal treatment, but this picture is satisfying for the moment. Thank you for your good wishes and help; I do believe I have found the sense of that for which I was looking."
Happy Trails, Len
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