Are the Maxwell equations a correct description of the wave character of photons?

Are the Maxwell equations a correct description of the wave character of photons?

Yes, Maxwell's equations are the wave equation for a photon, just as the Schrodinger equation is the wave equation for a nonrelativistic electron.

In other words: is the Schrödinger theory, in some way, equivalent to the Maxwell theory for the description of photons?

As others have pointed out, you can't apply the Schrodinger equation $i\hbar d\Psi/dt=-(\hbar^2/2m)\nabla^2\Psi$ to photons, because the Schrodinger equation is nonrelativistic, and a photon is never nonrelativistic.

Suppose history went a little differently: the corpuscular character would still be the generally accepted behaviour of light (as Newton preferred), and one discovered the wave character of photons and electrons simultaneously by doing interference experiments.

This is a very good question. There is a very nice discussion of this sort of thing in Peierls, Surprises in Theoretical Physics, section 1.3:

One of the most basic ideas of quantum mehcniacs is the analogy between light and matter [...] From this, it might appear an accident of history that physicists originally encountered only the wave aspects of light and only the corpuscular aspects of particles with mass, such as electrons. It therefore comes as a surprise to discover that this is no accident at all, and that the analogy between light and matter has very severe limitations.

I won't try to provide an analysis as complete as Peierls' in a physics.SE answer, but a crucial point is that because photons are bosons, you can have a coherent superposition of photons in which a large number of photons are packed into a volume equal to a cubic wavelength. Such a superposition can have a well-defined amplitude and phase that can be measured by classical measuring devices such as antennas.

But you can't do this with electrons, because they're fermions. This is why the electron wavefunction isn't a classical field that can be measured directly.

Would this equation give a exhaustive description of the electromagnetic (and vector like) behaviour of photons? In other words: is the Schrödinger theory, in some way, equivalent to the Maxwell theory for the description of photons? Or are the Maxwell equations some kind of limit for greater dimensions (like the Newton equations for mechanics)? What is the link between these two "wave character" descriptions of photons?

A photon is an excitation of a "mode", i.e. a solution of Maxwell's equations satisfying the appropriate boundary conditions. For example, a field constrained to be within a cavity has to satisfy the boundary conditions determined by the cavity. A field in free space might be a spherically symmetric solution, depending on the source properties etc. Once you have chosen your solution you can, in principle (in practice this may be very tricky - unless you're doing particle physics rather than quantum optics!), put a single excitation into it, creating a one-photon state.

Now, although the mode is a solution of Maxwell's equations, the state (at least in the Schroedinger picture), satisfies a Schroedinger equation. This is just equivalent to saying that it evolves unitarily in time.

This Schroedinger equation, however, isn't the "wave equation for the photon" in the same way that in single-particle quantum mechanics the Schroedinger equation is the wave equation for the particle. Rather, it's the time evolution equation for the state, which takes place in Hilbert space, not in spacetime. Trying to emulate the single particle QM description by constructing a wavefunction for the photon is difficult:

The wavefunction would be the inner product of the state with position eigenstates $|x\rangle$ $$ \Psi(x)=\langle x|\Psi\rangle$$ The difficulty comes about because there isn't a(n undisputed!) Lorentz invariant position operator $\bf{\hat{x}}$ for photons.

However, we can create single photon states. These, however, are not really "like" the classical field which corresponds to the mode you excited. See Lubos' answer here for discussion of the electric field in a single photon state for example. If you want something which looks like the classical field, you need to construct the corresponding coherent state. This does not have a definite number of photons.

Following the comment by Alfred Centauri, let me suppose that you discussed the Dirac theory of electron, and not the Schrödinger one. I will come back later on the possibility to describe matter-light interaction using the Schrödinger equation.

The Dirac theory describes the (special) relativistic behaviour of a particle. When complemented by the principle of gauge invariance (in particular the substitution of the normal derivative by the covariant one), it gives the basic playground for the simultaneous descriptions of the electromagnetic field (Faraday law and absence of magnetic monopole), the charge associated to the relativistic particle (the equation which replaces the Newton equation with the Lorentz force if you wish, but this has to be though with care) and their coupling (equations similar to the Maxwell-Ampère and Gauß, but there the current and charge densities have full quantum meaning, no more fluid interpretation as for the classical electromagnetism).

Obviously everything get more complicated when you try to quantise the electromagnetic field. The previous discussion didn't discussed the appearance of the photon.

I would say that the Wikipedia page related to the Dirac equation is not so helpful for understanding this point, but you could try to open the book by A. Messiah Quantum mechanics (volume II if not in an edition with the two volumes in one book), which contains all the pedagogical details you need, including the quantisation of the electromagnetic field in term of photon.

Schrödinger vs. Dirac description of matter

One can also describe the interaction between matter and light using the Schrödinger equation for the atom. This is the main study of the book

C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley (1992)

that I suggest you to read. In short, when the magnetic-like interaction is weak, the description using the Schrödinger equation is sufficient. You can understand this with the pictorial idea: when the (continuous laser) light field does not interact too much with the (gas) atoms, the effect can be describe by the first order term in interaction, which is already given by the Schrödinger prescription.

Historical perpectives

Now, regarding your historical perspective, it seems highly not probable that Dirac would have described the coupling between electrons and photons if he were unaware of the Maxwell's equations. This is once again because the gauge invariance is crucial in deriving the coupling. You may find more details about the history of gauge theory in the excellent collection of historical articles by

L. O'Raifertaigh The dawning of gauge theory, Princeton series in Physics (1997).

The same reasoning apply to the Schrödinger equation, because all these physicists were deeply influenced by the notion of field, that Maxwell really invented half a century before.

In short, the gauge invariance is the main ingredient of matter-field interaction, not the equation you're using to include it.

To be also noted:

  1. The particle behaviour of light was not the generally accepted behaviour of light (as you said) at the time of Maxwell's equations. Indeed, the Young two-slits experiment was already known by the end of the 18-th century.

  2. You do not really need to quantise the photon field to understand the photoelectric effect. This is discussed in a paper

    Lamb, W. E., & Scully, M. O. The photoelectric effect without photon, in Polarisation, matière et rayonnement (pp. 363–369). Presses Universitaires de France (1969).

where they calculate the photoelectric effect quantising only the electron / detector behaviour.