A deBroglie wave has two interpretations, which are generalizations in different domains, and which are conflated for a single particle. This leads to a lot of confusion.
- A classical field describing the motion of a single particle or many coherent bosons in a Bose-Einstein condensed state.
- A wavefunction, a probability wave over configurations of particles.
historically, Schrodinger interpreted the deBroglie wave as the first thing initially, as a physical scalar wave. This is the wrong interpretation, as it is not equivalent to matrix mechanics, and it is experimentally untenable since a physical wave doesn't allow for entanglement. The battle over this was settled by Schrodinger (and Einstein and deBroglie too, who understood the deBroglie wave was like the solution to the Hamilton Jacobi equation, something that lives in configuration space), who demonstrated that the wave was in configuration space in 1926, and proved that with this interpretation, the Heisenberg formalism was a consequence of the wave formalism.
To quickly answer the questions
- It looks like a solution to the Schrodinger equation--- a wave peaked at where the particle is most likely to be (or where there are the most particles, in the field interpretation, see below), whose complex phase is twisting in the direction of motion at a rate which is proprotional to the local velocity of the particle (or the local velocity of the superfluid flow in the BEC in the field interpretation).
- Neither--- they aren't deformations in a material, so the idea is nonsensical. If you have a sound wave in a solid, you can ask if it is transverse or longitudinal, because it is motion of atoms. The deBroglie waves are wave of possibilities (except, you can ask this question in the field intepretation, see below).
- If the deBroglie wave is for a spinless particle, it has no analog of polarization. There is only one component. If you have a deBroglie wave for a particle with spin, it has several components. For the spinning electron, there are two components, for the two different spins, so that there are two deBroglie waves. The polarization for electron waves is spin-1/2, so it isn't like a photon polarization which is spin 1.
- The ground state of a He atom is highly entangled--- the configurations where one electron is on one side of the atom, the other electron tends to be on the other side, due to the repulsion. The entanglement is highest in the case of He (actually, highest of all in the case of the H negative ion, but this ion is marginally unstable, since it is only the entanglement which keeps it bound at all), because as the nucleus becomes more highly charged, the innermost electrons mutual repulsions are relatively weaker compared to their attraction to the nucleus. The precise description was worked out in the 1930s using the variational approximation, and it is essentially as exact as you like, because the variational ansatz, after you take into account rotational invariance and the spin being all locked up between the two electrons, is parametrized by one function of 3 dimensions which you can approximate very very precisely as an exponential of a polynomial with appropriate asymptotics.
- The experimental evidence for the new quantum mechanics, with its entanglement, in the 1920s-1930s consisted of the following: The precise spectrum of the H-ion and He atom, which could be worked out variationally. The approximate spectrum and specific heat of metals, where the electrons form a quantum Fermi gas, the spectroscopic entanglement of radiation with atoms which followed from the Heisenberg Jordan Dirac treatement of electrodynamics, and which resolved the paradoxes of photon absorption and emission in the older, entanglement free, Kramers Bohr Slater theory. In the 1940s, you get more precise evidence in the Lamb shift and countless condensed matter systems, and by the 1960s, you have Bells theorem and superconductivity. Essentially the only thing we haven't verified experimentally is quantum computation.
The points above require a little more discussion, regarding the field and particle intepretation.
When deBroglie understood the matter waves, it wasn't clear if these are physical waves in space, like an electromagnetic wave, or if they are something more abstract, like the solution to the Hamilton Jacobi equation. The Hamilton Jacobi solution is over all classical configurations, and it tells you what the integrable motion frequencies are. Einstein established the character of the deBroglie waves in 1924, by showing that the semiclassical limit description, they are the solution to the Hamilton Jacobi equation. When Schrodinger found the right equation, Einstein and Schrodinger discussed the interpretation, and it became clear that the Schrodinger equation too was to be thought of as a wave over configurations.
What this means is that the wave for 2 electrons is in 6 dimensions, for 3 electrons in 9 dimensions, describing all possible mutual positions of these. This led Einstein to ask how physical these waves are, considering that if you have a powderkeg in quantum mechanics, you can set up a situation where its wave is superposed between exploded and unexploded. This observation of Einstein's is the origin of Schrodinger's cat, and it is the reason Einstein could never be convinced to take the quantum formalism seriously as a description of physical reality--- it was just too enormous to be physical. It looked like a statistical description of something else. This has not been a common interpretation, because if it is a statistical description of something else underneath, we don't know exactly what that other thing could be.
But before chatting with Einstein, Schrodinger believed his equation described ordinary scalar waves in space. This interpretation made the amplitude $|\psi|^2$ a charge density, and the Schrodinger current an actual electromagnetic current.
While this interpretation is incorrect for the fundamental quantum deBroglie wave, it is correct for a Bose Einstein condensate. If you have many Bosons in a superposition state where they all share the same quantum state, their wavefunction becomes a classical field which obeys the Schrodinger equation, a Schrodinger field. The Schrodinger field description does not require linearity, it is just a scalar field (or a vector/tensor field for bosons with spin) which describes the density and matter-current in a Bose Einstein condensate. In this context, it is called a Gross-Pitaevsky equation, or in other contexts, a Bogoliubov-deGennes equation, or something else, but this field interpretation is very important, because it is the only limit in which Schrodinger waves turn into waves in space.
In this context, the deBroglie wave shared by the Bose particles turns into a classical scalar field, and it has an intepretation which is identical to the one proposed by Schrodinger. But such a description cannot describe entanglements in nature, and the simplest case where entanglement is seen to be necessary is in the ground state of the Helium atom.
Best Answer
Different possible polarizations of a "matter particle wave" corresponds to the different possible degrees of freedom of the quantum field describing the "particle".
For a photon, we have 2 possible polarizations (for instance : vertical polarization, horizontal polarization). For a electron, we have also 2 possible polarizations (for instance : left handed, right handed). For the positron, we have also the same 2 possible polarizations , and the whole electron/positron quantum Dirac field describes 4 possible polarizations.
However, transversality has to do with a precise space-time condition, and this notion is only available for some Lorentz representations. A transverse relation will be written : $\vec k.\vec \epsilon_\lambda (k) = 0$. However, it suppose that the Lorentz representation of the field is a "vector", which is (roughly) true for the photon field, but false for the electron/positron Dirac field. In the latter case, the representation is a bi-spinor, so you cannot get a transversality relation directly between the momentum $\vec k$ and a bi-spinor like $u(\lambda, \vec k), v(\lambda, \vec k)$ (you will have to involve bilinear (quadratic) quantities based on bi-spinors to get "vectors").
In the same way, the notion of longitudinal wave $\vec k$ parrallel to $\vec \epsilon_\lambda (k)$, is a nonsense in the case of the Dirac field.