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Cobalt101

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In summary: Now if we discretise the waveform in this way it is easy to see that the momentum of a wave travels inversely with its wavelength.

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Cobalt101

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clarkvangilder

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Vanadium 50

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What do you mean by "underlying reality"?

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Cobalt101

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what physically is going on is what I mean by that.Vanadium 50 said:What do you mean by "underlying reality"?

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Nugatory

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There isn't any really satisfying answer to that sort of "why" question. Science is about discovering and describing the laws that govern the behavior of the universe we live in, and experiments show that we live in a universe in which gravity is an attractive force, like charges repel instead of attracting, ..., and momentum varies inversely with wavelength.Cobalt101 said:

Of course science does seem to offer answers to "why?" questions - everyone knows that Newton's law of gravitation explains why the planets move the way they do, for example. But that's just pushing the "why?" down one level. Why should the planets obey Newton's law of gravity and not some other law?

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Cobalt101

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Yes I get that there are often no answers to why questions, as you outline, but in this case I was not so much asking for why there is this relationship between momentum and wavelength, but rather what physical manifestation this might have ?Nugatory said:There isn't any really satisfying answer to that sort of "why" question. Science is about discovering and describing the laws that govern the behavior of the universe we live in, and experiments show that we live in a universe in which gravity is an attractive force, like charges repel instead of attracting, ..., and momentum varies inversely with wavelength.

Of course science does seem to offer answers to "why?" questions - everyone knows that Newton's law of gravitation explains why the planets move the way they do, for example. But that's just pushing the "why?" down one level. Why should the planets obey Newton's law of gravity and not some other law?

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muscaria

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Although there's always an infinite number of levels to why (As a parent knows only too well), here are some thoughts on the matter.Cobalt101 said:why wavelength varies (inversely) with momentum for all energy and matter

I think your question may be closely connected to action.

During the motion of a point particle in 1d say, the change in the action along the solution trajectory ##q(t)## is

$$dS=pdq-Hdt$$

where p is the momentum of the particle, q its position, H its energy, t is time. Notice here the pairing between momentum/space-displacement and energy/time-displacement in comparison to the de-Broglie relations. For a system of particles this generalises to a sum over degrees of freedom ##n##, the important thing being that each positional coordinate ##q_i## and momentum ##p_i## is dynamically paired in a special way. Also if you place ##t## on the same footing as the ##q_i## it turns out that the momentum paired with ##t## is in fact ##-H##.

##dq_i (i=1,2,..,n+1)## picks out a direction in extended configuration space and ##p_i## gives the flow along this direction. So this is what action differentials measure, the momentum which is carried over displacements.

Now back to 1 degree of freedom, so point particle on a line, and we imagine that time freezes but we can still investigate how ##S## would change if we virtually translated the particle by ##dq##, which is the displacement that was just about to take place during the motion i.e. ##dq=\dot{q}dt##. ##dS=pdq## is a geometric invariant so that if I deform my ##dq## in some way, ##p## will change also in a way to preserve ##dS##. So if I contract space such that the original coordinate differential ##dq## has now doubled with the new q-tics, we only need half the previous amount of flow for ##S## to tick off the same way as before and motion to be the same.

Now a wave is spread over space and not generally localised at a point. However, even without really knowing what property of the wave determines its momentum, if the wave travels with the same flow at every point along the line so that the wave just gets translated along homogeneously without saying anything about what shape the wave needs to be for this to happen. Now imagine we discretised the waveform and viewed it as being made up of infinitesimally evenly spaced points. For a given flow ##p##, there is an invariant ##dS## associated with ##dq##. Now if we contract the space in the same manner as before, the in-between spaces ##\textit{all}## grow evenly, and the flow ##p## thus decreases in order to preserve ##dS##.

A bit long winded but hope this helps..

The deBroglie equation is a fundamental equation in quantum mechanics that relates the wavelength of a particle to its momentum. It was first proposed by French physicist Louis de Broglie in 1924 and is expressed as λ = h/mv, where λ is the wavelength, h is Planck's constant, m is the mass of the particle, and v is its velocity.

The deBroglie equation is significant because it provides a connection between the wave-like and particle-like properties of matter. It implies that all particles, not just light, have a wave nature and can exhibit interference and diffraction effects. This has had a profound impact on our understanding of the micro world and has led to the development of quantum mechanics.

The underlying reality of the deBroglie equation is that all matter, at the subatomic level, has both particle and wave-like properties. This is a concept known as wave-particle duality, which suggests that matter can behave as both a wave and a particle, depending on the experimental conditions.

The deBroglie equation is related to the uncertainty principle through the concept of momentum. The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. The deBroglie equation shows that as the wavelength of a particle becomes smaller (i.e. its position becomes more precise), its momentum becomes larger, and vice versa.

The deBroglie equation has many practical applications, including in electron microscopy, where it is used to determine the wavelength of electrons and produce images of extremely small objects. It is also used in particle accelerators to study the properties of subatomic particles. In addition, the deBroglie wavelength is used in the development of quantum computers, which use the wave-like properties of particles to perform calculations.

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