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Harmony
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How do we describe wave in Quantum Mechanics? Is it different from classical physics?
koantum said: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.
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.)harmony said: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.
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.Is it correct to say that when interaction do not occur, particles may possesses any location in a certain region?
Physics = mathematics + experiment + philosophyreilly said:QM is physics, not philosophy, and not mathematics.
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 defy anyone to read this book and say that they do not understand QM and atomic physics.
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.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.
Quantum states are tools for calculating probabilities, not physical waves.how would you describe the asymptotic states if you gave up waves?
Well, if you call Pagels conversant with philosophy…. A couple of quotes from The Cosmic Code: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.
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.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.
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.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.
I never, anywhere, take on established practice. I take on the lousy philosophy that usually goes with it. See the Pagels example above.To take on established practice with any success
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.masudr said: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?
Why do you keep saying things that are either well known or beside the point?reilly said: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...
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.his tomato seed analogy is brilliant -- it is a metaphor, designed to enhance intuition, not to be rigorous.
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.Heinz Pagels was a mensch.
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.And, it's the Hamiltonian that's the generator of time-translations
Actually it'smasudr said:Actually it's
[tex]H|\psi\left(t\right)\rangle=i\hbar\frac{d}{dt}|\psi\left(t\right)\rangle.[/tex]
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.
Dear Reilly,reilly said: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?
No objection.masudr said: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 [itex]i\hbar d/dt[/itex] (thanks to Schrödinger), and I've always used E (with sufficient labels) to represent the energy eigenvalues of some particular system.
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.reilly said: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?
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.To say that in and out states cannot be entangled strikes me as at variance, again, with standard practice.
I guess I would be contradicting myself, insisting as I do that quantum mechanics, at bottom, does nothing but correlate in and out states.Perhaps you might give an example of physics not described by the S-matrix approach.
Wave motion in quantum physics refers to the behavior of particles at the atomic and subatomic level. In this context, particles can exhibit both wave-like and particle-like properties, and their motion is described by wave functions rather than traditional trajectories.
In classical physics, particles are treated as discrete, point-like objects with well-defined positions and velocities. In quantum physics, however, particles are described by wave functions that represent the probability of finding the particle in a particular location. This means that particles can exist in multiple places at once and their motion is not deterministic.
Wave-particle duality is the concept that particles can exhibit both wave-like and particle-like behavior. In the context of wave motion in quantum physics, this means that particles can have a wavelength and exhibit interference patterns, similar to waves, while also behaving like discrete particles with well-defined positions.
In quantum physics, the energy of a particle is related to the frequency of its associated wave function. This is described by the famous equation E=hf, where E is energy, h is Planck's constant, and f is frequency. This relationship is known as the wave-particle duality principle.
While wave motion is most commonly observed at the atomic and subatomic level, there are also macroscopic phenomena that exhibit wave-like behavior, such as sound waves and ocean waves. However, these are still considered classical waves and do not exhibit the unique properties of quantum wave motion.