The problem is not the number of particles, it's that the QM theory is inadequate at distances smaller then the associated wavelength.

JES

Stellan Gustafsson

You cannot change the question. If you want to discuss another issue, open a suitable thread.

Coherent state is a superposition of N-particle state normalized to unity, which does not have a definite number of particles. But it has definite average number of particles. Henceforth, you can treat it as a psi-function in most applications but, technically it does not belong to the projective space of an original Hilbert space.

When we talk about coherent state of an atomic ensemble, we talk about the quantum state of particles' motion not the particles themselves.Certainly, a combined momentum of the particles can fluctuate but the number of atoms themselves is conserved.

It means that for the given state the number of particles is undefined. Just as position of momentum are undefined if the state is not an eigenstate of position or momentum. If you measure the particle number, then you will get a definite value. And if the particle number operator commutes with the Hamiltonian, that number will remain the same after measurement.

Now I don't know much about coherent states of atoms. Do they arise for Bose condensates primarily? In principle, there are two possibilities. If the description as a coherent state is only an approximation, the true state may still have a fixed particle number and then measurement will just reveal it, not create it. On the other hand, with Bose condensates or superconductivity, you typically have a state containing an arbitrary numbers of particles -- which may be atoms in a condensate or Cooper pairs. The number of these can be indefinite, because one is close to a phase transition where atoms can go from a reservoir to the condensate and back or electrons can form Cooper pairs and Cooper pairs can split up into electrons. So you have large fluctuations of the particle number about its mean value.

For elementary particles, each of which has a field, there should be no conceptual problem with states not containing any fixed number of particles. The constituents of reality are fields and particles are only their excited states. But a state that is a superposition of vacuum and a one-particle state is equally possible. We just cannot imagine it easily, if we think only in terms of the particle concept. As to atoms, it is obviously possible to construct their states as sums of products of the base states of the underlying fields. So a hydrogen atom will be an entangled state of several product states each made from an electron and three quark fields.

K. Kassner

Dear Klaus,

The expression of the coherent state is

|α> = exp(-|α|2/2) Σαn |n>/n!

the sum being from 0 to infinity. This is the typical state of the light emitted by lasers. I also know that today coherent beams are produced with atoms too, but I don't know details about the preparation procedure.

Anyway, as you see, the coherent state is a superposition of Fock states. It's not an approximation. About a coherent state of photons I can imagine that in each trial of an experiment the number of photons may be undefined, since photons may pop up from the vacuum or return to it. But I can't imagine such a process with atoms.

With kind regards

Dear Sofia,

about cohenrece states and coherence domains you can find some interesting insights from the paper of Marian Apostol. He revised some of the work of Giuliano Preparata (Coherence in matter) a very good Italian Physicist who dealt with coherent quantum electrodynamics which is quite a difficult discipline.

Article Coherence domains in matter interacting with radiation

Stefano Quattrini

Thanks, my lovely friend. I will read. You see, it's an isuue already discussed in the past, but partially.

In each trial and trial of the experiment we produce an identical coherent beam, and measure the number of particles. The number we find is bound to vary from trial to trial.

If the particles are photons, our explanation of the different number that we find in each trial is that actually, in each copy of the beam the number of the photons is not fixed. Photons pop up from the vacuum or return to it.

However, there exist already coherent beams of atoms. I discussed once with Klaus, he said that he does not believe that the vacuum contains atoms. I agree with his view, the vacuum contains only elementary particles.

Stefano Quattrini

My friend, I read the article. But I don't see that it speaks of the emission of a coherent beam of atoms. I expect to see a coherent beams of atoms emitted from an atom laser.

With kind regards

Dear Sofia,

that is about the role of the coherence in matter. For sure atoms do not emerge from vacuum but coherent states of them do... A laser is just a form of super radiance which was first found by Dicke : LASER = light amplification by stimulated emission of radiation.. there are forms of collective polynuclear superradiance also at nuclear level for gamma rays.

You cannot emit atoms from a laser but just photons, it does not make sense to emit atoms...the laser is a consequence of Quantum coherence.

Stefano Quattrini

Please read this:

https://en.wikipedia.org/wiki/Atom_laser

KInd regards

Dear Sofia,

Technically, the last paragraph in the above K. Kassner`s reply properly describes the Coherent State.

However, concerning your conceptual problem, in my humble opinion, the origin of your imagination problem is in your (mainstream, yet naive) believe in the existence of "particles", "atoms" and other debries of "matter". You, just like most of people, inherited these concepts from Classical Mechanics` worldview. However, Orthodox QM denies their existence. In OQM there ontologically are only abstract entities: "Tendencies" to have various perceptions, "Space", "Time", a mathematical rule for the temporal evolution obeyed by these tendencies (called the Schrodinger`s equation) a "Mind" capable to pose questions to "Nature", "Nature" (or God if you wish) that supplies an answer to the posed question, and our "Perception" of that answer that feeds the "Mind".

The Coherent State shows it clearly. In this state there are Tendencies to get various numbers of quanta (strictly not "particles", which is a misleading concept). If the Mind asks Nature how many quanta there are (by an experimental set up, of course) Nature supplies an definite number. Our Mind just perceives that Nature answer. In this respect there is nothing special about atoms. Atoms are just entangled Tendencies (that form bound states). As such what you call "atoms" are merely tendencies in their true nature.

Sofia, then how can you explain the charge of electron measurement by Robert Millikan

( Nobel prize) and it’s mass derivation from m/e ratio?

Can a wave packet have a mass?

I‘m not agaInst what you suggested- just trying to understand you.

Natalia S Duxbury

Natalia, I don't see the connection you make between the mass deviation that you speak of, and my question. By the way I wasn't aware of such deviation - thank you for telling me. Can you give me some reference explaining the topic?

About a wave-packet having (or not) a mass, I think that I can answer precisely. Imagine what we call a single-particle wave-function, e.g. of a proton. Let the wave-function have more than one wave-packet, e.g.

(1) |ψ> = α|a> + β|b>,

with |α|2 + |β|2 = 1, and let |α| > |β|. The mass carried by each one of the two wave-packets is the same, i.e. the mass of the proton, approx. 1.6×10-24 g. No matter how weak is the intensity |β|2 of the wave-packet |b>, it carries the whole mass of the proton.

Do you want a proof? Please read the experiment with a proton in

Deleted research item The research item mentioned here has been deleted

I am indeed looking for a lab to perform it.

The article brings order between the intensity of a wave-packet and the physical properties of the type of particle it carries, by describing an experiment that proves the matters.

Thank you.

Dear Natalia,

In the most successful Standard Model of the Relativistic Quantum Fields Theory the charge is merely a coupling constant between fields. That is, the charge is associated with the interaction between the fields (rather than with the fields themselves, and hence with these fields` quanta).

Moreover, the climax of the "Standard Model" is an explanation for the emergence of "mass." One shows that "mass" results from interactions of abstract operators in Hilbert space (called fields operators) with each other. These fields by themselves are massless (Aitchison, 1989). In this way, the RQFT destroys the essence of the classical "massive matter" concept.

Ephraim Yeruham Levin

Dear Ephraim,

I believe that Natalia wanted to understand phenomenology, not formulas.

Anyway, your answer is wellcome, every piece of information is wellcome.

Now, maybe you can tell me more about the mass deviation from the m/e ratio that Natalia mentioned? I know nothing about this issue.

Dear Sofia,

Natalia did not write about "mass deviation from the m/e ratio". Rather, she wrote about "mass derivation from m/e ratio".

Ephraim Yeruham Levin

Dear Ephraim,

Thanks for the correction. I read now the description of Millikan's experiment.

Natalia S Duxbury

Dear Natalia,

I misunderstood your words "mass derivation" as "mass deviation". I know Millikan experiment and I re-read it now. But, I don't understand which connection do you see between this experiment and the wave-function. As far as I saw, all the equations Millikan used were of classical physics.

Best regards

Dear Sofia,

Never mind. Sometimes, it happened to me as well. Looking at things superficially (as well as mere phenomenologically) hinders the grasp of an evident truth.

Ephraim Yeruham Levin

Dear Ephraim,

Thanks for your kindness.

Deat Sofia, Dr. Millikan measured the electron charge ( the famous oil experiment, everyone from California Institute of Technology, including me,

knows that).

After the electron mass was derived from the ratio of those 2.

Your position is that electron is a wavepacket ALL THE TIME.

Does it mean that a wave has a mass and a charge?

Natalia S Duxbury

Dear Natalia, sometimes it's meaningless to say that the electron has a wave-packet. Assume that the electron is accelerated at a very high velocity, say, v = 62,800km/s. Let's calculate its wavelength and see if such an electron can be used for interference experiments. λ = 2πħ/m0v, where m0 is the electron mass, 10-27g. So, λ = 6.28×10-27/10-27×6.28×109 = 10-9cm. It's very hard to do interference experiments with such an electron, the wavelength is too much small, 1/10 from the Bohr radius. So, it's meaningless to speak of a wave-packet of such an electron.

However, if the electron is slow, say, v=628m/s, then λ = 1μm. If we create a wave-packet of coherence-length, say 50μm, split it by a beam-splitter into two copies and bring the copies to cross one another, one could get an interference pattern of a couple of fringes.

So, if the electron can be considered a particle, or a wave, depends on its energy.

I don't understand exactly which connection you see with Millikan's experiment. If the oil droplets were ionized with X rays, they they didn't carry suplementary electron, but they were loosing electrons. I mean the measured charges were positive. At least, this is the description that I read about this experiment.

Finally, the expression "the wave has mass and charge" is not rigorous. The wave has amplitude of probability, e.g. if the wave-function is a quantum superposition e.g. |ψ> = α|a> + β|b> + γ|c> + δ|d>, the wave-packet |a> has intensity |α|2, |b> has |β|2, etc. This is what matters at the encounter with a detector. The greater the intensity, the greater is the chance that the detector reacts to the wave-packet.

====> At the same time a wave-packet carries at every point in itself, all the physical properties of the type of particle: mass, charge, spin, isospin, etc. Split the wave-packet into 100 copies, each copy would be of very small intensity, but would carry all the physical properties.

Summary: a particle has a wave-packet at low energies. The wave-packet doesn't have mass and charge, it has intensity. But at each point and point in the wave-packet are carried all the physical properties of the type of particle.

Dear Sofia, thank you so much.