What exactly is meant by "a wave is 'associated' with a particle" in De Broglie hypothesis?

Question

Im just referring to the basic De-Broglie hypothesis, which says that a wave of certain wavelength is 'associated' with the particle.

• Is the particle itself moving in a wavy motion?

OR

• Is there a wave (of something) moving around the particle?

OR

• Is there no particle, and just the wave i.e the wave is the particle?

I'm always attracted by this word 'associated' used in De Broglie hypothesis.

(I just want to know the physical meaning of wave proposed by De Broglie, and not the 'true' modern quantum nature of matter)

Answer 1

The way de Broglie himself understood his hypothesis is different from how it is understood in modern physics (wave-particle duality): de Broglie thought that there was a wave associated with every massive particle, but didn't really claim that they were one and the same thing. Thus, in terms of the OP, it is the second option: There is a wave (of something) moving around the particle. (In de Broglie's own terms the wave is carrying the particle=the particle is transported by a wave.)

French Wikipedia contains several quotes to this extent:

From the article on Louis de Broglie:

« L’idée fondamentale de [ma thèse de 1924] était la suivante : « Le fait que, depuis l’introduction par Einstein des photons dans l’onde lumineuse, l’on savait que la lumière contient des particules qui sont des concentrations d’énergie incorporée dans l’onde, suggère que toute particule, comme l’électron, doit être transportée par une onde dans laquelle elle est incorporée […] Mon idée essentielle était d’étendre à toutes les particules la coexistence des ondes et des corpuscules découverte par Einstein en 1905 dans le cas de la lumière et des photons. » « À toute particule matérielle de masse $m$ et de vitesse $v$ doit être associée une onde réelle »

Translation:
"The basic idea of [my 1924 thesis] was: "The fact that since Einstein introduced photons into the light wave, it was known that light contains particles which are concentrations of energy incorporated in the wave, suggests that any particle, like the electron, must be transported by a wave in which it is incorporated […] My essential idea was to extend to all particles the coexistence of waves and corpuscles discovered by Einstein in 1905 in the case of light and photons. "To every material particle of mass $m$ and speed $v$ must be associated a real wave"

(emphasis mine)

From article Hypothèse de de Broglie:

« On peut donc concevoir que par suite d’une grande loi de la Nature, à chaque morceau d’énergie de masse propre $m$, soit lié un phénomène périodique de fréquence $\nu_0$ telle que l’on ait : $h\nu_0 = mc^2$. $\nu_0$ étant mesurée, bien entendu, dans le système lié au morceau d’énergie. Cette hypothèse est la base de notre système : elle vaut, comme toutes les hypothèses, ce que valent les conséquences qu’on en peut déduire. »

Translation:
“We can therefore conceive that, as a result of a great law of Nature, to each piece of energy of proper mass $m$, is linked a periodic phenomenon of frequency $\nu_0$ such that we have: $h\nu_0 = mc^2$. $\nu_0$ being measured, of course, in the system linked to the piece of energy. This hypothesis is the basis of our system: it is worth, like all hypotheses, what the consequences that can be deduced from it are worth. »

(emphasis mine)

Update
It is worth pointing out that de Broglie died in 1987 - he thus lived through the development of the quantum theory and his views towards the end of his life were likely much closer to the modern ones, than in 1924, when he formulated his hypothesis. In his Nobel lecture given in 1929 (and freely available to read), he acknowledges the wave-particle dualism (although in careful terms) and admits that his work was superceded by Schrödinger's wave mechanics:

This new mechanics has since been developed, thanks mainly to the fine work done by Schrödinger. It is based on wave propagation equations and strictly defines the evolution in time of the wave associated with a corpuscle.

Thus to describe the properties of matter as well as those of light, waves and corpuscles have to be referred to at one and the same time. The electron can no longer be conceived as a single, small granule of electricity; it must be associated with a wave and this wave is no myth; its wavelength can be measured and its interferences predicted. It has thus been possible to predict a whole group of phenomena without their actually having been discovered. And it is on this concept of the duality of waves and corpuscles in Nature, expressed in a more or less abstract form, that the whole recent development of theoretical physics has been founded and that all future development of this science will apparently have to be founded.

(emphasis mine)

Answer 2

Welcome to the weird world of quantum mechanics. The question you are asking is the type of question one unavoidably asks as he tries to learn the counter-intuitive concepts of quantum theory. A book you might like, which is written in a simple style, is The Meaning of Quantum Theory by Jim Baggott.

In answer to your question, here are some points which may be found in the book. They approximate de Broglie's thinking. In special relativity, the relationship between E, p, $m_0$, and c is given by $$ E^2 = p^2 c^2 + m_0^2 c^4 $$

A photon moves at the speed of light. De Broglie thought that a photon had a very small mass, though we know today that it has zero mass. Nevertheless, if you assume that a photon has a vanishingly small mass, the equation reduces to $E=pc$. It was known at the time that the energy of a photon is given by $E=h\nu$. Now if you combine these last two simple equations you get $pc=h\nu$. Since the frequency $\nu$ is given by $\nu=c/ \lambda$, one gets $$\lambda=h/p$$

which is de Broglie's result. De Broglie suggested that this equation might work for matter as well as photons. This result was part of his PhD thesis, and his proposal influenced Schroedinger.

Clearly this result does not fit with a classical physical understanding of matter. Instead, it implies that the probability of finding a particle in a particular place will involve the superposition and interference of waves in the same way as light.

You can think of the wave nature of matter not as implying that matter moves in a wavy motion or that there are waves connected with matter, but rather that matter is not fixed in location and that its location (and momentum) is only describable in probabilities due to its wave nature. This is reflected in the uncertainty principle which applies to noncommutative properties such as position and momentum. The phrase "intrinsic to" might be substituted for “associated with.” Wave properties are intrinsic to matter, part of its nature, though that aspect of its nature is invisible in the macroscopic world. It becomes apparent in the sub-microscopic world of quantum mechanics.

The wave nature of matter is often viewed from our macroscopic, every-day vantage point as a property which is dual to the corporeal particle-like nature that we think of matter as possessing. I don’t especially care for that viewpoint since it puts one foot in the macroscopic world and one in the sub-microscopic world. I prefer to think of mass as having an intrinsic wave property. Because this property is normally only visible in the sub-microscopic world, we only perceive matter as possessing particle-like properties in everyday situations. Ascribing particle-like properties to matter is an excellent approximation in the macroscopic world in which we live and move. We don’t need quantum mechanics to play billiards but we do to probe the properties of atoms.

I hope this helps. I am still learning these concepts myself and precise language is needed to correctly describe the situation. Others more experienced than I may give a better (more accurate) explanation.