Question on matter waves So,I just started this topic on modern physics in school that contained the concept of"de broglie waves" or "matter waves" and there are a few concepts that are unclear to me.
Firstly, does a single electron constitute a matter wave?Or does there have to be a continuous stream of particles for us to label it as a matter wave?
And secondly if a single electron does indeed constitute a matter wave,what happens when a wave due to a single electron is incident on a boundary and gets split into two parts, the refracted part and the reflected part.. Which of these two parts contain the original electron now?
 A: The de Broglie approach is somewhat archaic and confusing, but yes, it is the case that what is going on applies to a single particle.
The easiest way to see this is to think about the (very famous) double-slit experiment.  Here you have a stream of particles being fired at a barrier with two slits in, and a particle detector (or an array of them, or a strip of sensitive material like film) somewhere beyond the barrier.  Then what you see is that, although the particle detector only detects particles (ie, a particle either arrives or does not arrive at the detector, you never see half a particle), the particles are more likely to arrive in some places than others, and over time this builds up an interference pattern.  And this pattern goes away if you close one of the slits.
So far so easy.  Now the trick is that you can turn down the intensity of whatever is producing the particles.  And you can turn it down so far that there is only ever one particle in the system at a time (for instance, turn it down until it produces one particle per hour or per day or something).  Now it will take a long time to get reasonable statistics, but if you wait long enough you still get the pattern of interference fringes.
So it is clear from this that whatever is happening is not dependent on there being many particles, but is a property of each particle.
A: 
Firstly, does a single electron constitute a matter wave?Or does there have to be a continuous stream of particles for us to label it as a matter wave?

A single electron is a matter wave. The integral of the square of the amplitude of the wave over a region gives the probability of finding the electron in that region if you do a measurement.

And secondly if a single electron does indeed constitute a matter wave,what happens when a wave due to a single electron is incident on a boundary and gets split into two parts, the refracted part and the reflected part. Which of these two parts contain the original electron now?

The electron is not located at a single place in the wave. Rather, the wave can be decomposed into two components after the reflection. Let's call the forward propagating part of the wave $|F\rangle$ and the reflected part of the wave $|R\rangle$. The total wave function is then $|F\rangle + |R\rangle$. And in general $|F\rangle$ and $|R\rangle$ can destructively interfere. Rather, there are multiple instances of the electron, but in any given measurement of a single electron wave you will only detect one electron.
If you want to understand the issues involved here in more depth, I suggest starting with "The Fabric of Reality" by David Deutsch, chapters 2 and 9.
