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Showing posts with label Harmonic Oscillator. Show all posts
Showing posts with label Harmonic Oscillator. Show all posts

Saturday, June 15, 2013

Musical Acoustics


A recipe for a violin

Chladni patterns show the geometry of the different types of vibration of violin plates. This site has an introductory explanation of modes of vibration and a library of photographs of the Chladni patterns of the bellies and backplates of two different violins (one mass-produced and one hand-made). It also has photographs of plates with regular geometries which assist in understanding the violin modes. For some related history, see Chladni's law. For some Chladni patterns on metal plates, with sound files, see Acoustics of bell plates. To make your own Chladni patters, try this site.








See Also:

Tuesday, September 04, 2012

The Quantum Harmonic Oscillator

Quantum Harmonic Oscillator

 


Here are a series of written Blog entries by Matt Strassler from his Blog, Of Particular Significance.
  1. Ball on a Spring (Classical)
  2. Ball on a Spring (Quantum)
  3. Waves (Classical Form)
  4. Waves (Classical Equation of Motion)
  5. Waves (Quantum) 
  6. Fields
  7.  Particles are Quanta
  8.  How fields and particles interact with each other 
  9.  How the Higgs Field Works



Given a preceding map  by Proffessor Strassler according to what has been gain in finality views requires this updating in order to proceed correctly in the views shared currently in science. So that lineage of thought is important to me.

Probability Distributions for the Quantum Oscillator



At the same time one cannot be held back from looking further and seeing where theoretical views have been taken beyond the constraints applied to the science mind.:)





So what is the theory, then?

Pythagoras could be called the first known string theorist. Pythagoras, an excellent lyre player, figured out the first known string physics -- the harmonic relationship. Pythagoras realized that vibrating Lyre strings of equal tensions but different lengths would produce harmonious notesratio of the lengths of the two strings were a whole number. (i.e. middle C and high C) if the......

   Pythagoras discovered this by looking and listening. Today that information is more precisely encoded into mathematics, namely the wave equation for a string with a tension T and a mass per unit length m. If the string is described in coordinates as in the drawing below, where x is the distance along the string and y is the height of the string, as the string oscillates in time t, 


See: Official String Theory Web Site


Sunday, June 26, 2011

A General Guide to Harmonic Analysis and Beyond


Thanks to Clifford of Asymptotia for the Link too, and from Good Vibration

Some of us do look toward these analogies as signs of Complexity science, so as to apply this thinking to the life they lead. How such implementations allow them to look at this life very differently.

***

Each time the operator tuned to a new frequency, the wave was very simple and repetitive, just as above. This wave can be expressed by only two qualities: frequency and amplitude, and yet each new frequency created a new and many times surprisingly different result than it’s neighbor.

This phenomenon is analogous to all aspects of Complexity Science. And just like the simple rules set by the speaker in the video, an economy can achieve surprisingly complex results with simple rule sets.
Harmonic Science
 ***

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x:
 F = -k x \,
where k is a positive constant.
If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).
If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Depending on the friction coefficient, the system can:
  • Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator).
  • Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator).
If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.

Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
Simple Harmonic Motion Frequency
The frequency of simple harmonic motion like a mass on a spring is determined by the mass m and the stiffness of the spring expressed in terms of a spring constant k

The Landscape “avant la lettre” by A.N. Schellekens

The lowest harmonics correspond to the particles of the Standard Model, plus perhaps a few new particles. The higher harmonics correspond to an infinite series of particles that we can never observe, unless we can build a Planck Energy accelerator


***
See Also:

The Sound The Universe Makes

Cymatics and the Heart Song

Thursday, October 16, 2008

Fear and Ignorance

This is a very significant physical result because it tells us that the energy of a system described by a harmonic oscillator potential cannot have zero energy. Physical systems such as atoms in a solid lattice or in polyatomic molecules in a gas cannot have zero energy even at absolute zero temperature. The energy of the ground vibrational state is often referred to as "zero point vibration". The zero point energy is sufficient to prevent liquid helium-4 from freezing at atmospheric pressure, no matter how low the temperature.
See:Quantum Harmonic Oscillator: Energy Minimum from Uncertainty Principle

It would be hard here to explain the way I see these things. In the way one can shift perspective, to think, that this measure of any "systemic reason" would ask that one consider the state of equilibrium?

It would be foolish to me for any science process to discount the value on how one can measure storms in space not to think that "such resonances" could have not found suitable actions as being represented in sociological correspondence.

Let us see how these great physicists used harmonic oscillators to establish beachheads to new physics.

Albert Einstein used harmonic oscillators to understand specific heats of solids and found that energy levels are quantized. This formed one of the key bridges between classical and quantum mechanics.

Werner Heisenberg and Erwin Schrödinger formulated quantum mechanics. The role of harmonic oscillators in this process is well known.

Paul A. M. Dirac was quite fond of harmonic oscillators. He used oscillator states to construct Fock space. He was the first one to consider harmonic oscillator wave functions normalizable in the time variable. In 1963, Dirac used coupled harmonic oscillators to construct a representation of the O(3,2) de Sitter group which is the basic scientific language for two-mode squeezed states.

Hediki Yukawa was the first one to consider a Lorentz-invariant differential equation, with momentum-dependent solutions which are Lorentz-covariant but not Lorentz-invariant. He proposed harmonic oscillators for relativistic extended particles five years before Hofstadter observed that protons are not point particles in 1955. Some people say he invented a string-model approach to particle physics.

Richard Feynman was also fond of harmonic oscillators. When he gave a talk at the 1970 Washington meeting of the American Physical Society, he stunned the audience by telling us not to use Feynman diagrams, but harmonic oscillators for quantum bound states. This figure illustrates what he said in 1970.


We are still allowed to use Feynman diagrams for running waves. Feynman diagrams applicable to running waves in Einstein's Lorentz-covariant world. Are Feynman's oscillators Lorentz-covariant? Yes in spirit, but there are many technical problems. Then can those problems be fixed. This is the question. You may be interested in reading about this subject: Lorentz group in Feynman's world.

Can harmonic oscillators serve as a bridge between quantum mechanics and special relativity
?


To consider such geometrical form "as the sphere," to have encouraged collapse, and find a resulting behaviour as signalling a change overwrought by influences that will insight idealizations to division and the idea that "no" global consideration is present.

While one may debate the idea of the classification of democracies in the 167 countries around the world, a consensus to quality control of information is insinuated. So now moving beyond "the border" to lesser degrees of, while there is no offering of what the idea of democratic institutions in the free countries of the world could be related too. It's measure in the degrees thereof.

While I had offered "in bold" the understanding, they( should I offer by name?) are quick to point out in rebuttal, by an offering to discount the very source of this consideration. I am all for further dialogue, but it looks like that won't happen.

See:Central Theme is the Sun

So how does it look, as a spherical realization, that SOHO measure in terms of predicting an "outcome of weather" could not have found "early warnings" to possible outcomes in the evolution of the planet it's electrical grids and power usage, telecommunications, and the events thereof?

You had to know that Plato "saw further" by understanding the examples of the sun, as a source of "seeing beyond the shadows of the cave." Of knowing, that one could be "free of the chains that bind."

No where does this say it is easy to overcome. The sociological and psychological behaviour that evolves in that "spherical engage", but that it is always the life struggle to get back to the light. One had to be fully aware of the topological translation of the relationship between the inner/outer and the reductionist move to what is self evident. There is no way for one to be aware of the analysis and the final outcome without knowing the way in which one could move to such a result, without knowing the wider perspective that is held about life.

Thursday, July 13, 2006

GRand Quantum Conjecture



My continued looked into the "fluids dynamics" had me wonder about the superfluid anomalies. How would the "sphere look" if it collapsed and allowed information to travel through it, based on what has been given here for perspective, when the "state of equillibrum" is arrived at?

In regards to 3, let's just say the assumption is from a theoretcial standpoint, that microstate blackholes "are created." They are created in "cosmic particle collisions" as well?

This is the premise from which I work, and how I gave "how particles are created," a beginning(dimensional referencing), and a basis from which all science becomes "evidentary" in the particle creations.

Exotic physics finds black holes could be most 'perfect,' low-viscosity fluidVince Stricherz, University of Washington

Son and two colleagues used a string theory method called the gauge/gravity duality to determine that a black hole in 10 dimensions -- or the holographic image of a black hole, a quark-gluon plasma, in three spatial dimensions -- behaves as if it has a viscosity near zero, the lowest yet measured.

It is easy to see the difference in viscosity between a jar of honey or molasses at room temperature and a glass of water. The honey is much thicker and more viscous, and it pours very slowly compared with the water.

Using string theory as a measuring tool, Son and colleagues Pavlo Kovtun of the University of California, Santa Barbara, and Andrei Starinets of the Perimeter Institute for Theoretical Physics in Waterloo, Ontario, have found that water is 400 times more viscous than black hole fluid having the same number of particles per cubic inch.


Your points in conclusion,I, II, III

I-yes
II-yes
III- from my conclusions as well.

Again in above quote, I am defining the leading perspective on blackholes as they are being theoretically defined now, and will be subject to experimentation soon?:)

Now again "backreaction in the laval nozzle" is up for inspection here as we delve deeper into the nature of the blackhole.

Nature in Analog Models

Analogue models of (and for) gravity have a long and distinguished history dating back to the earliest years of general relativity. In this review article we will discuss the history, aims, results, and future prospects for the various analogue models. We start the discussion by presenting a particularly simple example of an analogue model, before exploring the rich history and complex tapestry of models discussed in the literature. The last decade in particular has seen a remarkable and sustained development of analogue gravity ideas, leading to some hundreds of published articles, a workshop, two books, and this review article. Future prospects for the analogue gravity programme also look promising, both on the experimental front (where technology is rapidly advancing) and on the theoretical front (where variants of analogue models can be used as a springboard for radical attacks on the problem of quantum gravity).


"Analogistical behaviors" help to push perspective, where before, our theoretical explorations had ran dry?

Q:
These wormhole like 'blackholes' do not lead to other pocket universes, unless we choose to call another sector of space a pocket universe, like Europeans first called the land across the Atlantic the 'New World' or Australia 'Another World' yet still clearly part of this World we call Planet Earth.


If we are to think that the overall context can be apllied to this universe, then such evidence "should be obtainable" as to the nature of such a beginning? But even still, to your point and aspect within this universe, we are looking for accontable methods to such dark energy creation?

Plato:
Every picture held in mind is a link to other pictures


Each event in regards to gravitational collapse would be indicative of what can be "put back into this universe" and sustain it?

Lubos Motl:
The mechanism behind sonoluminiscence remains a bit controversial. Claiming that a thermonuclear fusion occurs during sonoluminiscence is among the more conservative explanations. The physicist Claudia Eberlein argued that the correct explanation is that the imploding bubbles create sonic black holes and the flashes are the counterpart of Hawking radiation as the sonic black hole evaporates. You should not think that this is an example of a very, very low energy quantum gravity because the sonic black holes have no connection with the scales of gravity. It is not a supercollider in a glass of beer. But let me admit that as an undergrad, I was excited by this proposal, at least for a few minutes, but I apparently forgot the details of that encounter.


So by developing this picture of the "bubble collapse in sonofusion", and let's forget about the energy produced from such bubbles and focus on the geometrics of such a collapse. That's my point.

Lubos Motl:
Janice Granhardt has pointed out a press release that is two days old and arguably much more serious and potentially far-reaching than the news about "sonofusion" we described yesterday.


That is part of my conjecture as well as the "unification factor" in my GRand Quantum perceptions.:)That if you remember Kip thorne's plate 27 you will understand that information from the collapse had to be sent over a great distance for us to make sense of the geometrical dynamics that are unfolding from that time and place.

So you look for the gravitational waves that Webber initiated, and Kip Thorne encouraged in our measures of what is actually being transmitted. Kip Thorne is the father of the LIGO program?

You must remember gravitational waves have not yet been verified, yet the theory of GR implicitly tells and is about gravity. It was thus taken further in my conclusions having understood that the creation of this infomration would allow one someday "to map" this very collapse in terms of the gravitonic information left in the bulk?

This is "Dimensional orientated" from a beginning(11dimensional view?), from which evidence is "the 3+1."

That's outside the box thinking? :)Cosmologists work "inside," as Clifford of Cosmic Variance once said?

How then is such a gravitational heat generated from the boundary conditions(blackhole), which grows ever smaller in that collapse, and our energy valuations go higher to supersymmetical realizations? The present volume calculated in the extension of our universe would have to be in concert with the volume before such a collapse was to be expected?

This "total energy value," assuming the universe is flat teeter's on the brink of ?:)

Total dark energy would have to account for this and supernova events contributing as well as, particle collisions that go on all the time?

So if space is not really empty, then what is it supposed to be filled with? Quantum harmonic oscillator and zeropoint?

See:
  • Charlatan's Who Use Graviton
  • Monday, June 12, 2006

    Harmonics will Color Your World?



    If you are a active participator of the very world around you, how is it, the makeup of high energy particle creations could not have included the standard model make up "harmonically described" does it not also apply to our "very thinking and conscious mind?" :)

    The Landscape “avant la lettre” by A.N. Schellekens

    The lowest harmonics correspond to the particles of the Standard Model, plus perhaps a few new particles. The higher harmonics correspond to an infinite series of particles that we can never observe, unless we can build a Planck Energy accelerator


    So of course the very basis of the thinking was drawn in my mind to the very subject enlisted by the minds of our predeccessors, to wonder, how this associative function could have ever been at the basis of how we may look at the World?

    Lee Smolin:
    In case it is not obvious, let me emphasize that harmonic oscillators are not relevent here, and can play no role in a background independent quantum theory, precisely because the division of a field into harmonic modes requires a fixed background metric. Thus, the physics of the problem REQUIRES an alternative quantization


    Of course it is never easy for me to understand what is going on while we have the issues of the background, versus, the non background, and this brings up the ole debates about positions and adoptives stances scientists have taken in regards to the "duality" of science's "quantum gravity" issues?



    Do I have a complete grasp of it. Absolutely not, while it forces me back to the issues, as to what is the basis of this "difference of opinion?"

    THE ANTHROPIC PRINCIPLE


    Leonnard Susskind and Lee Smolin


    While this is a conversation written by physicists for physicists, it should nonetheless be of interest for Edge readers as it's in the context of previous Edge features with the authors, it's instructive as to how science is done, and it's a debate that clarifies, not detracts.

    So by historically looking back, this is a reminder, about the ways in which science people are still looking at things, while holding their positions of thought today?

    SEan's meeting at PI, is a very interesting one becuase what it does is take the teacher and student scenario, and manifests the circumstance of clarification as to positions, while providing for a intuitive surge to present itself in the minds of it's participnats.

    So this debate then was held and it's relationship rememebered within this blog as to the basis of determination, about how we see the universe and all that is in it.

    Lee Smolin:

    The aim of this paper is to explain carefully the arguments behind the assertion that the correct quantum theory of gravity must be background independent. We begin by recounting how the debate over whether quantum gravity must be background independent is a continuation of a long-standing argument in the history of physics and philosophy over whether space and time are relational or absolute. This leads to a careful statement of what physicists mean when we speak of background independence. Given this we can characterize the precise sense in which general relativity is a background independent theory. The leading background independent approaches to quantum gravity are then discussed, including causal set models, loop quantum gravity and dynamical triangulations and their main achievements are summarized along with the problems that remain open. Some first attempts to cast string/M theory into a background independent formulation are also mentioned.

    The relational/absolute debate has implications also for other issues such as unification and how the parameters of the standard models of physics and cosmology are to be explained. The recent issues concerning the string theory landscape are reviewed and it is argued that they can only be resolved within the context of a background independent formulation. Finally, we review some recent proposals to make quantum theory more relational.


    So if someone saids that space is empty, I have a really hard time with it.

    See:

  • Quantum Harmonic Oscillator
  • Tuesday, November 01, 2005

    Harmonic Oscillation

    This "math sense" has to become part of one's makeup? An inductive process. Experimentally challenged. Deductive.

    If such a idea is held from weak to strong idealizations in terms of comological views, then you get this sense of "energy valuations" as well. If you calculate when the binary pulsar distances around each other, the value of that information has been released in the bulk. This information should become weaker, as the orbits get closer?


    The theory of relativity predicts that, as it orbits the Sun, Mercury does not exactly retrace the same path each time, but rather swings around over time. We say therefore that the perihelion -- the point on its orbit when Mercury is closest to the Sun -- advances.



    I would think this penduum exercise would make a deeper impression if held in concert with the way one might have look at Mercuries orbit.

    Or, binary pulsar PSR 1913+16 of Taylor and Hulse. These are macroscopic valutions in what the pendulum means. Would this not be true?

    Part of the Randall/Sundrum picture Sean supplied of the brane world perspectives needed for how we look at that bulk view. If you are to asume that space is not indeed empty, then what is it filled with? Gravitonic perception would make this idea of the quantum harmonic oscillator intriguing to me in the sense that "zero point", would be flat space time. Any curvature parameters would have indeed signalled simple harmonic initiations?

    Omega valutions in regard to the what state the universe is in, would have been defined in relation to a triangulation.

    The quantum harmonic oscillator has implications far beyond the simple diatomic molecule. It is the foundation for the understanding of complex modes of vibration in larger molecules, the motion of atoms in a solid lattice, the theory of heat capacity, etc. In real systems, energy spacings are equal only for the lowest levels where the potential is a good approximation of the "mass on a spring" type harmonic potential. The anharmonic terms which appear in the potential for a diatomic molecule are useful for mapping the detailed potential of such systems.


    But indeed while we understand this large oscillatory factor in our orbits, does it not make sense to wonder how simple that harmonic oscillator can become when we are looking for extra dimensions?

    I had a picture the other day of a music instrument of a wire stretched, and weights being applied respectfully. The string when strummed gave certain frequencies accordingly to different mass valuations. This is the early pythagorean instrument I had see a few years ago, that would have similarities with "gourds of water" as weight and levels changed.



    Here we seen a torsion pendulum. The way the wire twists and it's resulting valuation.



    So you see how simple experimental processes help to correct our views on the way we see things.

    From a historical perspective views of scientists with this explanation support the harmonic oscillators as follows:



    Let us see how these great physicists used harmonic oscillators to establish beachheads to new physics.

    Albert Einstein used harmonic oscillators to understand specific heats of solids and found that energy levels are quantized. This formed one of the key bridges between classical and quantum mechanics.

    Werner Heisenberg and Erwin Schrödinger formulated quantum mechanics. The role of harmonic oscillators in this process is well known.

    Paul A. M. Dirac was quite fond of harmonic oscillators. He used oscillator states to construct Fock space. He was the first one to consider harmonic oscillator wave functions normalizable in the time variable. In 1963, Dirac used coupled harmonic oscillators to construct a representation of the O(3,2) de Sitter group which is the basic scientific language for two-mode squeezed states.

    Hediki Yukawa was the first one to consider a Lorentz-invariant differential equation, with momentum-dependent solutions which are Lorentz-covariant but not Lorentz-invariant. He proposed harmonic oscillators for relativistic extended particles five years before Hofstadter observed that protons are not point particles in 1955. Some people say he invented a string-model approach to particle physics.

    Richard Feynman was also fond of harmonic oscillators. When he gave a talk at the 1970 Washington meeting of the American Physical Society, he stunned the audience by telling us not to use Feynman diagrams, but harmonic oscillators for quantum bound states. This figure illustrates what he said in 1970.

    We are still allowed to use Feynman diagrams for running waves. Feynman diagrams applicable to running waves in Einstein's Lorentz-covariant world. Are Feynman's oscillators Lorentz-covariant? Yes in spirit, but there are many technical problems. Then can those problems be fixed. This is the question. You may be interested in reading about this subject: Lorentz group in Feynman's world.

    Can harmonic oscillators serve as a bridge between quantum mechanics and special relativity?


    Lee Smolin saids no to this?

    Wednesday, June 15, 2005

    Could "Chaos" have been Implied in the Quantum Harmonic Oscillator, as Supersymmetry?

    See:Quantum Harmonic Oscillators


    Fortunately being the junior here in knowledge comprehension, the benefits, as of my having wrongly thought a certain way, allows "fantasy a great journey." I have wondered, could I have ever attained such insights that would awaken even the most determined individual to the quest of, to finally rest easy and a warm satisfaction?

    Let's hold the Trigger in suspension. What is it's potential, had such symmetrical realizations made the axis of this process move according to some plan not seen? Are There methods seen in the matrices, that revolve around this "i" that woud have given the photon dispersement some relevance to a interactive phase within context? How about gluon perceptions? Tipler's lightcones also comes to mind here, as such a process?

    I am so afraid to trip on my tongue, that such words listed, would have provided the necessary crackpot label. I wondered whether I should continue. But a motivation is borne of it's own accord, I couldn't help see some relevance to the "uncertainty in langrange points."

    Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.


    Could have exemplified such unequilibrium, that at first, it made no sense to me that such points could have ever accomplished anything. Was there a lesson from lorentz and his model butterfly that such models had found the transfer from one state to another, could reveal such supersymmetrical acts of consideration in having found the anti- anything, and soon learnt, that it was the other side of the equation? The 720 degree turn, that brought the history of this action into a complete whole, and accord.



    I would like to say this is nice to see that Lee Smolin has taken the time to help set things straight. This caught my eye.

    This is linked to a Post written by Peter Woit called, " Why No New "Einstein"?

    Lee Smolin:
    6) With regard to the non-standard quantization, in which holonomies, but not local field operators are well defined, it is of course true that when applied to standard systems this leads to inequivalent results. “This apparently leads to unphysical consequences, such as an unbounded spectrum for the harmonic oscillator.” But, give me a break, do you really think someone is proposing to replace the standard quantization of the harmonic oscillator with the alternative one? What is being proposed is that the quantization used in LQG is well suited to the quantization of diffeo invariant gauge theories.

    In case it is not obvious, let me emphasize that harmonic oscillators are not relevent here, and can play no role in a background independent quantum theory, precisely because the division of a field into harmonic modes requires a fixed background metric. Thus, the physics of the problem REQUIRES an alternative quantization.




    I have to read things over quite a few times before it seems to sink in, but reference to three sphere and the understanding of Poincare's model did not reduce my interest in seeing such correlations in how we saw such langrange points develope into this suspended state for the trigger to make itself known?

    Many physicists find extra dimensions a distasteful notion. In remarks to an American Physical Society newsletter, physicist Frank Wilczek of MIT called the black hole study a sound way to test an unattractive idea.

    "There's no question that the Auger observatory will be sensitive to this signal, if it exists," says Penn State's Stéphane Coutu, a member of the international Auger Observatory team. "We'll definitely look."



    All of us are on the same wavelenght right? Blackhole determinations, set the stage, for what we hope to percieve is, "the basis of this trigger?"

    How wonderful this rubber band that is slipped over the sphere, and as it expands from this point, "the loop" now becomes a exemplifier of all those things manifested from the spherical gaze of the harmonics?

    What image is given from the blackhole to realize, that the gravitonic messager sent from the daisy of Taylor, would have found the penduluminaric thoughts in Greg's Egans's "animations of Lissajous" and a coordinated frame?

    Wednesday, November 03, 2004

    Quantum Harmonic Oscillator



    Let us see how these great physicists used harmonic oscillators to establish beachheads to new physics.

    Albert Einstein used harmonic oscillators to understand specific heats of solids and found that energy levels are quantized. This formed one of the key bridges between classical and quantum mechanics.

    Werner Heisenberg and Erwin Schrödinger formulated quantum mechanics. The role of harmonic oscillators in this process is well known.

    Paul A. M. Dirac was quite fond of harmonic oscillators. He used oscillator states to construct Fock space. He was the first one to consider harmonic oscillator wave functions normalizable in the time variable. In 1963, Dirac used coupled harmonic oscillators to construct a representation of the O(3,2) de Sitter group which is the basic scientific language for two-mode squeezed states.

    Hediki Yukawa was the first one to consider a Lorentz-invariant differential equation, with momentum-dependent solutions which are Lorentz-covariant but not Lorentz-invariant. He proposed harmonic oscillators for relativistic extended particles five years before Hofstadter observed that protons are not point particles in 1955. Some people say he invented a string-model approach to particle physics.

    Richard Feynman was also fond of harmonic oscillators. When he gave a talk at the 1970 Washington meeting of the American Physical Society, he stunned the audience by telling us not to use Feynman diagrams, but harmonic oscillators for quantum bound states. This figure illustrates what he said in 1970.


    We are still allowed to use Feynman diagrams for running waves. Feynman diagrams applicable to running waves in Einstein's Lorentz-covariant world. Are Feynman's oscillators Lorentz-covariant? Yes in spirit, but there are many technical problems. Then can those problems be fixed. This is the question. You may be interested in reading about this subject: Lorentz group in Feynman's world.

    Can harmonic oscillators serve as a bridge between quantum mechanics and special relativity
    ?