Wave Motion in Quantum Physics

[Harmony]

How do we describe wave in Quantum Mechanics? Is it different from classical physics?

 

[sneez]

they have similarities by the nature of them being 'waves' as well as differences by the nature of their application. Can you be more specific?

 

[masudr]

They satisfy different wave equations (the most important difference would be the inclusion of the unit imaginary on one side) and they have different interpretations of what the values of the wave means (e.g. Born interpretation for QM or electric field strength in an EM wave)

 

[koantum]

A classical wave exists in physical space. A quantum-mechanical wave "exists" in a space that has as many dimensions as the system with which it is associated (as a probability algorithm) has degrees of freedom.

 

[Harmony]

How do Quantum Mechanics describes wave characteristic of particles such as electrons? Surely they don't oscilate up and down as described by classical physics. Is it correct to say that when interaction do not occur, particles may possesses any location in a certain region?

 

[vanesch]

A classical wave exists in physical space. A quantum-mechanical wave "exists" in a space that has as many dimensions as the system with which it is associated (as a probability algorithm) has degrees of freedom.

This is the whole idea ! The term "wave" is unfortunate in quantum theory - but is now so much incorporated into the technical language that it is hopeless to change it. There is only a ressemblence between the quantum state and a wave for a one-featureless-point-particle system.

 

[dextercioby]

Suggestion for the OP: Read about coherent states for a quantized em field.


Daniel.

 

[koantum]

How do Quantum Mechanics describes wave characteristic of particles such as electrons? Surely they don't oscillate up and down as described by classical physics.

There are (at least) "Nine formulations of quantum mechanics" (title of an article by Styer, et al., in American Journal of Physics, March 2002, p. 288). The answer to your question depends to some extent on which one you use. I prefer the path integral formulation (introduced by Feynman), where you calculate the probability of a particle going from spacetime point A to spacetime point B by taking the absolute square of a complex number ("amplitude") <B|A>, and you calculate this amplitude as a sum over all continuous spacetime paths leading from A to B. (In the relativistic theory the paths can zigzag also in the time direction, the turning points representing pair creations or annihilations. I'm omitting complications due to the electron's spin etc.)
Each path contributes a complex number (also called "amplitude"), and if you deal with a stable particle such as the electron, all paths contribute with the same (irrelevant) absolute value. So only the phases of the amplitudes associated with the paths matter (correction: only the differences between these phases).
But if what you are doing is essentially adding phase factors (complex numbers of unit absolute value), then the probability of electron detection, considered as a function of B, will exhibit the same kind of interference as waves in a ripple tank.
So forget about physical oscillations. What does the "waving" and "interfering" is the probability of detecting the electron as a function of B (assuming, as always, that a detector is at the right place at the right time).

Is it correct to say that when interaction do not occur, particles may possesses any location in a certain region?

Yes. If all you can infer from measurement outcomes is that a particle is in a region R at the time t, then the particle isn’t inside any smaller region contained in R (nor is it absent from any such region!) but there is a probability of finding it inside any smaller region in R if the appropriate measurement is made.

 

[reilly]

Let's go back to basics. One of the first indications that electrons, neutrons, ... were kind'a strange was finding that beams of these particles going through crystals produced wave-like diffraction and interference patterns. So, with all due respect, the attribution of wave-like properties to "elementary' particles was a very natural thing to do -- even if the phenomena was mind-blowingly strange, totally unexpected. Not only that, but as understanding increased it became clear that the observed patterns could be explained in much the same fashion as interference in E&M. So, the ascription of wave-like properties is, for most of us who focus on applications of QM, see this ascription as very sound -- IT'S BASED ON EXPERIMENTS -- and, it helped with the invention and design of the electron microscope -- which works very nicely. This is all explained in detail in most any book on atomic physics. And, atomic physics is extraordinarily powerful and successful -- it was so in the 1930s, see Condon and Shortley's, The Theory of Atomic Spectra -- it was still a bible when I was a student in the 1950s and early 1960s. I defy anyone to read this book and say that they do not understand QM and atomic physics. The next task. could be to study Linus Pauling's books on the Chemical Bond, and his General Chemistry. He's very intuitive at times, and it's clear that his understanding -- which somehow he's not supposed to have -- at a very basic level guides much of his work and writing. The point is, ontology or not, many people do understand QM -- the proof is in the pudding, scattering theory, atomic and nuclear physics, the huge field of solid state physics, quantum optics, QED, the Standard Model, and on and on. I say flatly that if you do not have at least a basic knowledge of most these fields of application, then you do not know much about QM. (If understanding QM obeys different rules than understanding classical physics; that's physics and so be it.

Again, I keep saying until I'm blue in the face: at the core, QM is physics, not philosophy, and not mathematics . It's physics. It's physics, and thus is based on experiment. All the peculiar aspects of QM are fundamentally tied to empirical observations, and of a world that we cannot directly perceive. Small wonder that the phenomena do not fit well into our perceptual and conceptual schemes, the evolution of our brains never had to deal with interfering electrons. QM, in it's normal Born probability form as is comonly used -- see Landau and Lifschitz and Dirac for clear expositions. Just for the record, my sense is that many of the mysteries of QM will become less mysterious as we increase our understanding of the brain and mind -- like signal processing and all that.

History is exceedingly important.
Regards, Reilly Atkinson

PS Standard scattering theory and QFT are based on 'waves', and have been for years. That is, free particles are generally described as momentum states. In configuration space such a wave function is

exp (ikx-iEt)

I fail to see how waves are unfortunate in the context of scattering theory -- how would you describe the asymptotic states if you gave up waves?

 

[koantum]

QM is physics, not philosophy, and not mathematics.

Physics = mathematics + experiment + philosophy

I defy anyone to read this book and say that they do not understand QM and atomic physics.

There are different ways of understanding. When Feynman says that nobody understands quantum mechanics or Penrose says that quantum mechanics makes absolutely no sense, they are talking about the philosophical way of understanding. The other way is to understand the maths, the experimental tools, and the relation between these two, which only requires what Redhead calls the "minimal instrumentalist interpretation." All these subjects are too complex to be mastered by a single person. I urge you to read http://www.nyas.org/snc/updatePrint.asp?updateID=41" "A Crisis in Fundamental Physics". Here are some excerpts:

I think the problem… goes deeper, to a whole methodology and style of research. The great physicists of the beginning of the 20th century—Einstein, Bohr, Mach, Boltzmann, Poincare, Schrodinger, Heisenberg—thought of theoretical physics as a philosophical endeavor. They were motivated by philosophical problems, and they often discussed their scientific problems in the light of a philosophical tradition in which they were at home. For them, calculations were secondary to a deepening of their conceptual understanding of nature.
After the success of quantum mechanics in the 1920s, this philosophical way of doing theoretical physics gradually lost out to a more pragmatic, hard-nosed style of research. This is not because all the philosophical problems were solved: to the contrary, quantum theory introduced new philosophical issues, and the resulting controversy has yet to be settled. But the fact that no amount of philosophical argument settled the debate about quantum theory went some way to discrediting the philosophical thinkers. It was felt that while a philosophical approach may have been necessary to invent quantum theory and relativity, thereafter the need was for physicists who could work pragmatically, ignore the foundational problems, accept quantum mechanics as given, and go on to use it. Those who either had no misgivings about quantum theory or were able to put their misgivings to one side were able in the next decades to make many advances all over physics, chemistry, and astronomy….
By the time I studied physics in the 1970s, the transition was complete. When we students raised questions about foundational issues, we were told that no one understood them, but it was not productive to think about that. "Shut up and calculate," was the mantra….
after a lot of thought I've come to the conclusion that the pragmatic style of research is failing…. Perhaps the problems of unification and quantum gravity are entangled with the foundational problems of quantum theory, as Roger Penrose and Gerard t'Hooft think. If they are right, thousands of theorists who ignore the foundational problems have been wasting their time…. I suspect that the crisis is a result of having ignored foundational issues.
​

the ascription of wave-like properties is, for most of us who focus on applications of QM, see this ascription as very sound -- IT'S BASED ON EXPERIMENTS -- and, it helped with the invention and design of the electron microscope -- which works very nicely.

I must agree with vanesch that this way of thinking, however fruitful as a heuristic visual aid, is seriously misleading philosophically inasmuch as it misses the point that quantum-mechanical waves "propagate" not in real 3D space but in configuration space.

how would you describe the asymptotic states if you gave up waves?

Quantum states are tools for calculating probabilities, not physical waves.

 

[reilly]

Dr. Smolin is a brilliant man. He is entitled to his opinion, with which I pretty much disagree. A historical perspective suggests to me that the primary, but not only, push behind innovation in physics is experimental data. So, my take is that the next big advances in theory will come from new data.

Now, in regard to philosophy. Certainly there was a huge philosophical buzz in the 1960s, with S-Matrix/Bootstrap/Dispersion-Chew, Mandlestam, , Goldberger, Watson, ...-- approach vs the field theory and current algebras of Schwinger, Gell Man , Weinberg, Pais, Glashow, and so on. Let's face it, there is always a philosophical context for physics -- is it to be fixed, anchored in the 1900s, or should it adapt, as it invariably has in the past.

And, of course, there's the whole debate and discussions centering around Bell and his famous Thrm. Unless things have changed greatly, grad students in virtually every physics dept. are heatedly debating the nature of physics, and the nature of the universe, and which professors spell big trouble for oral exams. I was fortunate to have such debates with Heinz Pagels when we were graduate students -- he went on to write numerous books, the Cosmic Code is my favorite (he does a great job with Bell's Thrm) which touches very much on philosophical aspects of physics. There are many physicists who do the Born thin., and who are very conversant with philosophy.

I think Einstein and Newton tie for the head of the class. But I think Bohr was the most innovative -- he did not help his cause with his tortured writing. But getting the H atom more-or-less right was and is an astonishing achievement.

Something to think about: QM takes place in normal space, the one in which we live. By assumption, the eigenvalues of the position operator are equivalent to all the points in space. (And, for those who take umbrage with the position operator, note first, that it is conjugate to momentum, and is constructed to have the same continuous spectrum as momentum -- appropriately labeled of course. Second, all doubts about the legitimacy of the position operator can be erased by looking at Hille and Phillips, Functional Analysis and Semi-Groups, Chapter XIX, "Translations and Powers"

If the idea of waves is incorrect, where do we go wrong in particle physics, or quantum optics? Without evidence your claim is empty. To take on established practice with any success, you must provide a better way. What is wrong with waves? (You'll have to take on Weinberg's four volumes on field theory, you will have to take on the entire standard model, not to mention lasers, superconductivity... Where are the mistakes?

The crisis as I see it is: the math is too hard, experiments are too hard. Right now, nobody is smart enough or clever enought to make a breakthrough. At some point something will give.

Regards,
Reilly Atkinson

 

[koantum]

I was fortunate to have such debates with Heinz Pagels when we were graduate students -- he went on to write numerous books, the Cosmic Code is my favorite (he does a great job with Bell's Thrm) which touches very much on philosophical aspects of physics. There are many physicists who do the Born thin., and who are very conversant with philosophy.

Well, if you call Pagels conversant with philosophy…. A couple of quotes from The Cosmic Code:

• "The visible world is neither matter nor spirit but the invisible organization of energy." – What a naïve philosophy! Energy is the generator of time translations. Or (my favorite take) it is the rate at which a particle ticks as it travels along one of its myriad paths from A to B in Feynman's path-integral calculation of the propagator <B|A>.
• "I have always thought that wet seed from a fresh tomato illustrate the Heisenberg relation. If you look at a tomato seed on your plate, you may think that you have established both its position and the fact that it is at rest. But if you try to measure the location of the seed by pressing your finger or a spoon on it the seed will slip away. As soon as you measure its position, it begins to move. A similar kind of slipperiness for real quantum particles is expressed mathematically by the Heisenberg uncertainty relations." – Philosophy? I have always thought of this as a world record of silliness in the context of quantum mechanics. ("Everything should be made as simple as possible, but not simpler." – Einstein)

I think Einstein and Newton tie for the head of the class. But I think Bohr was the most innovative -- he did not help his cause with his tortured writing. But getting the H atom more-or-less right was and is an astonishing achievement.

Complete agreement. But in retrospect the achievement is a very logical step. Planck had already quantized the emission of energy by introducing a new constant of nature, which is measured in units of angular momentum. What would be more obvious, then, than to quantize the angular momentum of the electron in atomic hydrogen? The rest is high school stuff.

Something to think about: QM takes place in normal space, the one in which we live. By assumption, the eigenvalues of the position operator are equivalent to all the points in space.

Indeed. Which is why |psi(x₁,y₁,z₁,x₂,y₂,z₂,t)|² is the joint probability density for finding particle 1 at (x₁,y₁,z₁) and particle 2 at (x₂,y₂,z₂) if their positions are measured at the time t. The two particles are in "normal" space. But the wave function isn’t. If you reify it (something I don’t do) then it exists in a 6+1 dimensional configuration spacetime.

To take on established practice with any success

I never, anywhere, take on established practice. I take on the lousy philosophy that usually goes with it. See the Pagels example above.

It is true that many scientists are not philosophically minded and have hitherto shown much skill and ingenuity but little wisdom.

Max Born​

 

[masudr]

koantum i agree with you on some of the stuff you say, so don't automatically go on the defensive.

but do you have a problem with classical hamiltonian phase space where things exist in 2N dimensions in the same way you have a problem with quantum mechanical phase space?

 

[reilly]

I'm short on time right now, but I sugest, koantum, that you review generalized coordinates, Contact transformations and Hamilton-Jacoby Theory (See, in particular, Lanczos, The Variational Principles of Mechanics,) all of which use an extended phase space. An N particle system can be equally well described by 3N coordinates in ordinary configuation space, or by a point in a space of 3N dimensions. In fact, a classical system of arbitrary numbers of particles can be represented by a point in any segment of the real axis -- the mapping to accomplish this will, in general not be continuous -- and not very useful either.

You are, of course, entitled to your opinion of Dr. Pagels. But it looks like to me that you have cherry-picked a couple of quotes to diss him, gently to be sure. And, I can assure that Heinz Pagels was not naive, and was enormously knowledgeable about philosophy. Quite frankly, I think his tomato seed analogy is brilliant -- it is a metaphor, designed to enhance intuition, not to be rigorous. His untimely death -- he was in his late 40s or early 50s -- was a great loss to physics. Jeremy Bernstein, the high powered Harvard trained theoretician turned New Yorker writer, wrote a very touching farewell to Pagels(in the Journal of the New York Academy of Sciences 10+ years ago.) who was, in fact the chairman of that organization at the time of his death. He was held in very high esteem by many in the physics community, none of whom considered Heinz naive, and some of whom might even tend to agree with some of your ideas. Heinz Pagels was a mensch.

And, it's the Hamiltonian that's the generator of time-translations -- cf Goldstein and/or Lanczos. (Or, it's the "time momentum" see Chapters 8 and 9 in Goldstein's Classical Mechanics.)

Regards,
Reilly Atkinson

 

[koantum]

do you have a problem with classical hamiltonian phase space where things exist in 2N dimensions in the same way you have a problem with quantum mechanical phase space?

Where did I say I have a problem with quantum mechanical phase space? I certainly have no problem with Sec. 10-4 (Quantum Mechanics in Phase Space) of Asher Peres' Concepts and Methods.

 

[koantum]

An N particle system can be equally well described by 3N coordinates in ordinary configuation space, or by a point in a space of 3N dimensions...

Why do you keep saying things that are either well known or beside the point?

his tomato seed analogy is brilliant -- it is a metaphor, designed to enhance intuition, not to be rigorous.

If that's what you call brilliant… A brilliantly misleading intuition I'd call it, and one of the reasons why still nobody understands quantum mechanics, to quote Feynman.

Heinz Pagels was a mensch.

Do you read me as saying anything against Pagels as a mensch? I have very good friends with whom I heartily disagree on a variety of matters.

And, it's the Hamiltonian that's the generator of time-translations

According to Schrödinger, E \psi = H \psi. The Hamiltonian is on the right hand side, the generator of time translations is on the left. Being equal is not quite the same as being the same.

Regards,
Ulrich Mohrhoff

 

[masudr]

Actually it's

$$H|\psi\left(t\right)\rangle=i\hbar\frac{d}{dt}|\psi\left(t\right)\rangle.$$

It's H that's the operator, not E. E is just some number that is an eigenvalue of H. It doesn't do anything to the state vector but multiply it by a number. H affords some change to the state vector, which happens to be the infinitesimal change over time.

 

[koantum]

Actually it's
$$H|\psi\left(t\right)\rangle=i\hbar\frac{d}{dt}|\psi\left(t\right)\rangle.$$
It's H that's the operator, not E. E is just some number that is an eigenvalue of H. It doesn't do anything to the state vector but multiply it by a number. H affords some change to the state vector, which happens to be the infinitesimal change over time.

Actually it's
$$E|\psi\left(t\right)\rangle=H|\psi\left(t\right)\rangle$$
with
$$E =i\hbar\frac{d}{dt}.$$

 

[masudr]

OK, so we use different notation. I've always used H as the Hamiltonian, which varies from system to system. But we can always identify the action of H on the state ket as $i\hbar d/dt$ (thanks to Schrödinger), and I've always used E (with sufficient labels) to represent the energy eigenvalues of some particular system.

 

[koantum]

Standard scattering theory and QFT are based on 'waves', and have been for years. That is, free particles are generally described as momentum states. In configuration space such a wave function is exp (ikx-iEt). I fail to see how waves are unfortunate in the context of scattering theory -- how would you describe the asymptotic states if you gave up waves?

Dear Reilly,
Standard scattering theory and QFT represent in and out states as free individual particles, with given (on shell) energies and momenta. Such particles can of course be associated with amplitudes proportional to exp (ikx-iEt). They all exist in the 3 dimensional space of the experimenter. And since the question of entanglement does not arise in this context, we may think of them (heuristically, not ontologically) as waves in "ordinary" space. I reiterate that I never said anything about this being unfortunate.
Where I begin to jump up and down (to quote you) is when I am told (by A. Zee, to give a specific example) that "quantum field theory can arguably be regarded as the pinnacle of human thought." A theory whose almost only contact with the real world is transformations of free in particle states into free out particle states? There is so much more, even in physics, that cannot be treated in this way.
You are offering me a list of must read books dealing with such transformations and the like. I could provide you with a list not so much of books as of academic articles on foundational issues, which use a somewhat different formalism and language, suited to this particular field of research. Once again, there is no implication that these different methodologies contradict each other. They simply serve different purposes.

 

[koantum]

OK, so we use different notation. I've always used H as the Hamiltonian, which varies from system to system. But we can always identify the action of H on the state ket as $i\hbar d/dt$ (thanks to Schrödinger), and I've always used E (with sufficient labels) to represent the energy eigenvalues of some particular system.

No objection.

 

[reilly]

koantum --

Check Weinberg's Quantum Theory of Fields, for a discussion of the Poincare Algebra, in which the Hamilton is taken to be the infitesimal generator of time displacements, just as 3-momentum is the generator of spatial translations, the angular momentum tensor generates rotations and Lorentz boosts. Among the representations used in many parts of physics, the so-called interaction representation is often used, in which the Hamiltonian is not the energy, but does generate translations in time. This more generalized situation is also discussed by Goldstein.

The ascription of the Hamiltonian as the generator of temporal displacements is certainly well known in much of physics, and, perhaps, is at least, minimally relevant to the point. Is there something wrong with common practice?

To say that in and out states cannot be entangled strikes me as at variance, again, with standard practice. The S-Matrix approach you describe above works nicely for EPR experiments -- certainly in the photon and electron versions, the "particles" are out states, while the in state is the source generating the particles. Indeed the conservation of angular momentum creates substantial entanglement between the two emitted particles. And conservation laws applied to scattering and decay problems create all manner of entanglements.

Perhaps you might give an example of physics not described by the S-matrix approach.

Re "mensch" -- just another case of "well known"

Dr. Zee is stating an opinion, If I'm not mistaken, there are. from time-to-time, opinions stated in this forum. And, anyway, it's Zee's book -- you can always skip the opinions.

Regards,
Reilly Atkinson

 

[koantum]

Check Weinberg's Quantum Theory of Fields, for a discussion of the Poincare Algebra, in which the Hamilton is taken to be the infitesimal generator of time displacements, just as 3-momentum is the generator of spatial translations, the angular momentum tensor generates rotations and Lorentz boosts…. The ascription of the Hamiltonian as the generator of temporal displacements is certainly well known in much of physics…. Is there something wrong with common practice?

Just happened to read "Quantum Field Theory and Representation Theory: A Sketch" (hep-th/0206135) by Peter Woit (Department of Mathematics, Columbia University). He writes: "The Hilbert space of quantum mechanics is a (projective) unitary representation of the symmetries of the classical mechanical system being quantized. The fundamental observables of quantum mechanics correspond to the infinitesimal generators of these symmetries (energy corresponds to time translations, momentum to space translations, angular momentum to rotations, charge to phase changes)." No mention of the Hamiltonian, and for a good reason.

Consider the dispersion relation for a free scalar particle in one dimension: E=p²/2m or (natural units: c=1) E²=m²+p². Substitute

$$i\frac{\delta}{\delta t}$$​

for E and

$$-i\frac{\delta}{\delta x}$$​

for p (natural units: $$\hbar=1$$), apply both sides of these equations to a wave function, and get the corresponding wave equation (Schrödinger or Klein-Gordon). What is the physical significance of these relations?

If I think of the propagator for this particle as a sum over paths, then the particle's rest energy or (rest) mass is the rate at which it "ticks" in its momentary rest frame as it travels along any path, energy is the rate at which it "ticks" in an inertial frame (that is, the number of phase cycles per time unit associated with time translations), and momentum is the number of phase cycles per space unit associated with space translations – may I call it the number of "marks"?

To my way of thinking, spacetime doesn’t come with an inbuilt metric. The metric properties of the world are rooted in the behavior of particles, which is described by laws that correlate measurement outcomes. To identify or construct the so-called "metric properties of the world", we use the behavior of particles (that is, the quantum-mechanical correlation laws). Which time coordinates are uniform? Those relative to which free particles "tick" at constant rates. Which space coordinates are homogeneous? Those in which free particles associate equal numbers of cycles ("marks") with equal distances.

But a particle does not separately "tick" and "create marks". It just ticks in its momentary rest frame (as it travels…). Therefore we need a rule that tells us how to decompose the proper ticks (as in "proper time") into inertial ticks and marks. And the relativistic dispersion relation tells us how to do this. (The non-relativistic relation is simply its low-energy limit.)

How can we describe effects (due to whatever) on the motion of a scalar particle? By modifying the rates at which it ticks and creates marks. Non-relativistically but now in 3 dimensions:

$$\left(i\hbar\frac{\delta}{\delta t}-V\right)\psi=\frac{1}{2m}\left(\frac{\hbar}{i}\nabla-\textbf{A}\right)^2\psi$$​

The bracket on the left-hand side contains the kinetic energy, that on the right-hand side contains the kinetic momentum. If you take the potential energy term over to the right-hand side you get the familiar

$$i\hbar\frac{\delta\psi}{\delta t}=\frac{1}{2m}\left[\left(\frac{\hbar}{i}\nabla-\textbf{A}\right)^2+V\right]\psi$$​

On the left-hand side you now have the (total) energy, whereas on the right-hand side you have a hotchpotch called "Hamiltonian". I prefer to keep the two conceptually apart. If the opposite is common practice then, yes, I think there is something wrong with common practice.

To say that in and out states cannot be entangled strikes me as at variance, again, with standard practice.

To my way of thinking (which is indeed at variance with the standard way of thinking) in states are "prepared" and out states are detected. Detected states cannot be entangled, but I admit my mistake. Ensembles of detected out states (given the same in states) do exhibit correlations.

Perhaps you might give an example of physics not described by the S-matrix approach.

I guess I would be contradicting myself, insisting as I do that quantum mechanics, at bottom, does nothing but correlate in and out states.

Regards - koantum