Reasons behind de Broglie hypothesis What reasons does de Broglie had when he proposed the idea that electrons behave like waves?
 A: The answer to this question can be found in de Broglie's Nobel Lecture. The pdf file is available here. I quote the relevant part.

The necessity of assuming for light two contradictory theories -that of
  waves and that of corpuscles- and the inability to understand why, among
  the infinity of motions which an electron ought to be able to have in the
  atom according to classical concepts, only certain ones were possible: such
  were the enigmas confronting physicists at the time I resumed my studies of
  theoretical physics.
When I started to ponder these difficulties two things struck me in the main.
  Firstly the light-quantum theory cannot be regarded as satisfactory since it
  defines the energy of a light corpuscle by the relation $W=h\nu$ which contains
  a frequency $\nu$. Now a purely corpuscular theory does not contain any
  element permitting the definition of a frequency. This reason alone renders
  it necessary in the case of light to introduce simultaneously the corpuscle
  concept and the concept of periodicity.
On the other hand the determination of the stable motions of the electrons
  in the atom involves whole numbers, and so far the only phenomena in
  which whole numbers were involved in physics were those of interference
  and of eigenvibrations. That suggested the idea to me that electrons themselves
  could not be represented as simple corpuscles either, but that a periodicity
  had also to be assigned to them too.
I thus arrived at the following overall concept which guided my studies:
  for both matter and radiations, light in particular, it is necessary to introduce
  the corpuscle concept and the wave concept at the same time. In other words
  the existence of corpuscles accompanied by waves has to be assumed in all
  cases. However, since corpuscles and waves cannot be independent because,
  according to Bohr’s expression, they constitute two complementary forces
  of reality, it must be possible to establish a certain parallelism between the
  motion of a corpuscle and the propagation of the associated wave. The first
  objective to achieve had, therefore, to be to establish this correspondence.

