The wave function (Psi wave) of Schrodinger can be thought of as the de Broglie wave for a free particle with exactly defined momentum. But in general a psi wave is a superposition of de Broglie waves forming a wave packet.

These two things: de Broglie waves and Schrödinger wavefunctions are notions from two different theories. Although you can try to intuitively relate them, but sooner or later the analogy will break.

De Broglie postulate was that to each particle one can assign a plane wave and that particle "behaves" like such wave. And that's all.

On the other hand, wavefunction in the Schrödinger equation is a pure state of quantum mechanical system, i.e. a normalised vector from a infinite-dimensional separable Hilbert space. If you choose particular representation, say Hilbert space of L^2(R), you will see that the Schrödinger is basically a wave equations and solutions are superpositions of waves. That's where the naming come from. But this is much more general than very simple de Broglie hypothesis (position representation, where these two concepts seems to be related, is only one of many equivalent representations).

In my opinion, it is better to not to think to much about wave-matter duality. It was nice idea at some point, but mostly obsolete after development of quantum mechanics in 1930s. If you keep asking questions about nature of the matter you eventually hit the field theory, where the idea of wave-matter duality makes no sense at all.

This is related to the Einstein paradox: in 1D the wavefunction corresponding to a Hamiltonian eigenstate for a bound state is real, this means that S=0, therefore p=\partial_x S=0. This would imply that quantum mechanically the particle (e.g. a harmonic oscillator) is at rest and starts moving in the classical limit. The answer is that whether Re^{iS/\hbar} is a solution of the Schroediger equation, then also Re^{-iS/\hbar} is a solution. Therefore, the more general representation of the wave function is

\psi=R( A e^{iS/\hbar}+ B e^{-iS/\hbar})

If psi is real, then |A|=|B|, and you get a well defined nontrivial S with has the correct classical limit. These and other points can be found in my papers with Faraggi on the formulation of the QHJE. 

The de Broglie wave is a sub-quantum (hidden variables) notion,

describing reality on an ontological level different from the level

described by quantum mechanics. This can be compared with

descriptions of a billiard ball by means of an atomic theory or by

the classical theory of rigid bodies. Moreover, where a de Broglie

wave is assumed to be a description of a single particle

accompanied by a wave, is the Schrodinger wave function a

description of an ensemble of particles.

De Broglie's concepts never really were put in a clear theory, as far as I am aaware. A clear form of de Broglie's theory was given by Bohm. In Bohm's theory, the ``guiding wave'' is indeed the psi of quantum mechanics. However, this ``guiding wave'' has several properties which make it quite unlike any other wave. In particular, it does not propagate in physical space, but in some abstract space describing all configurations of the syste: thisi is  a relevant difference when the system has more than one particle. 

In the usual interpetation of quantum mechanics, however, psi is not a wave, but an abstract description of the state of the system. I fully agree with Tomasz, that particle-wave duality is not a helpful concept to understand quantum mechanics. Quantum systems are simply different from either classical particles or classical waves, and talking about a duality does not help understanding.

yes

Dear Leyvras,

The quantum mechanical wave function, or, more generally,  the density operator can be seen as a symbolic representation of a preparation procedure within the domain of application of quantum mechanics. In this sense it may be a helpful concept since in this way it is referring to physical reality.

Willem de Muynck

Dear Willem de Muynck:

I never meant to say that the wave function is not a helpful concept. Quite the opposite. I meant that thinking of the psi function, which I prefer to call the state function, as a physical wave, and further bringing this in opposition to a particle concept, is not really useful. That is, I do not think we should nowadays go on discussing quantum mechanics in terms of ``wave particle duality''

Dear Leyvraz,

My remark was not meant as a criticism, but rather as a way of implementing your remarks.

Willem de Muynck

It is a good question to ask in a classroom.  Historucally, of course, de Broglie's seminal idea came only a few years before Schrodinger's famous equation.  As I understand it, de Broglie's 'wave' is actually a wavepacket made of a linear superposition of a very large number of waves; it is so much 'localized' that it can be regarded as almost a 'particle'; hence wave-particle duality.  Psi in Schrodinger's wave equation for a one-particle system describes the time evolution of this wavepacket.  But, then, one can write down this equation for a more complicated system -- including even a many-body system, with interparticle interactions.  In principle, Psi incorporates all the physics of the system as well as the appropriate boundary conditions.  In practice, however, the complications involved  led physicists to invent other 'pictures' and approaches.  It is not only the emergence of these physical ideas, one after another, that is fascinating; but also the historic context and the protagonists who made it all possible.

Both models have the wave function, but the interpretation is different.

In Schrodinger the wave function itself has no physical meaning., or existance.

This is the mayority viewpoint, or copenhagen.

With de Broglie and other few later followers, there is both a real wave and a real

particle, the wave attracting the particle sort of within it, and this is the pilot model.

The particle rattles around in the wave.

In my view the de Broglie wave is a notion of a sub-quantum (hidden variables)  theory, underpinning quantum mechanics in an analogous sense classical statistical mechanics is underpinning thermodynamics. Taking the quantum mechanical wave function as an element of a sub-quantum theory is a methodological sin comparable with developing a sub-theory to the classical theory of rigid bodies by considering a model of a billiard ball as consisting of small rigid atoms which are glued together so as to constitute a macroscopic rigid body.

Basically The De Broglie wave is The offspring of The Higgs mecanism. The De Broglie wave is strictly connected to The dirac  spinors whose coupling mechanism is due to The Higgs field.

Well, this could be a recent development, but historically de Broglie is way before

Higgs

. de Broglie is also prior to Dirac, if Im not mistaken.

There is (or would be)an implicit underlying mechanism of very short scale,

Planck length noise which would make the electron rattle around. This seems to be

an implicit idea of Bohm and others...

Stefano, Do you mean a mixing of positive and negative

energy states coming from Dirac, due to a Higgs field as originating the wave?

In a recent paper I have shown that the Telegraph-like equation is more suitable to describe the particle than Schrodinger one. Here the duality is inherently embedded in the equation. The solution of the equation is a wavepacket. The phase velocity of this wave is complex reflecting that a quantum state is a mixture of wave and particle. How much particleness the state contains depends on the momentum of the packet.

The attached material explains what I say.

The Schrodinger Eqn (as also the Klein-Gordon and Dirac Eqns) were contrived as equations for the de Broglie wave.  See F. Bloch, Heisenberg and the early days of quantum mechanics. Physics Today, Dec. 1976, 

You may also be interested in a paper of mine at arXiv:1503.02534 - "The de Broglie Wave as Evidence of a deeper Wave Structure".

Regards

Dan Shanahan

Dear Dan,

Thank you for drawing my attention to your paper. it is so interesting and incorporated all points i was thinking of. I will send you later my latest manuscript to have a look.

Arbab

Interesting comments. The de Broglie hypothesis, although not formulated as a clear theory as it has been mentioned, predicts the wave behavior of matter, shown in interference experiments using electrons [Davisson-Germer, American Journal of Physics 42, 4 (1974)] and more massive objects like buckyballs [Nature 401, 680-682 (14 October 1999]. So, independently of any theoretical interpretation, it must have something true in it because it correspond to the way Nature behaves. However, as in the case of the double-slit experiment, there is no clear explanation of it using quantum mechanics (PNAS, June 20, 2017, vol. 114, no. 25, p. 6480).