I'm having difficulty understanding de Broglie matter wave hypothesis. It is a mass or a particle hypothesis? According to de Broglie a particle with mass $m$ moving at a constant speed has an associated matter wave with a frequency
\begin{equation*}
\nu\:=\:E/h
\end{equation*}
where $E$ is particle energy. Suppose this is just a mass relationship. Then, we can conceptually imagine the particle composed of two halves traveling at the same speed. Since each part has now half of the total energy they have an associated frequency that is half of the original
\begin{equation*}
\nu_{1/2}\:=\:\frac{1}{2}\:\nu
\end{equation*}
and so in general by splitting the particle in fractions of any proportions we can get all sorts of matter frequencies associated with the particle parts. In a sense this is the situation with a molecule where each atom that composite it has an associated frequency different from the whole (without considering the waves associated with the individual particles that compose the atoms themselves).
So, is this interpretation correct or I'm missing something?
@Andrew: I read about bi-photons a while ago and was searching for a physical interpretation in the same lines. If I understood correctly, each photon has its own frequency but when they get entangled they behave very much as a single object with a frequency proportional to the total energy. I guess there are other requirements for a combination of two particles to be treated as a composite beside that both particles travel at the same speed. In any case I guess we can write a wave function for the composite traveling at a constant speed as $\Psi=\Psi_1(x_1,t)\Psi_2(x_2,t)$ where $\Psi_1=e^{i(k_1 x_1-\omega_1 t)}$ and $\Psi_2=e^{i(k_2x_2-\omega_2t)}$. Then assuming that $x_1= x_2\equiv x$ and $v_1=v_2\equiv v$ we get $\Psi(x_,t)=e^{i((k_1+k_2)x-(\omega_1+\omega_2)t)}$ which has a frequency that is the sum of the individual frequencies. I suppose this is equivalent to the center of mass approach that you suggest. Nevertheless, I just found out a similar question posed in this forum (Validity of naively computing the de Broglie wavelength of a macroscopic object) that treats the subject in some detail.
Best Answer
In it's simplest form, de Broglie's hypothesis is meant to be applied to fundamental, indivisible particles, like an electron (an electron is fundamental and indivisible to within our current experimental precision at least). In that case it doesn't make semse to talk about half an electron, or to divide the mass of the electron among its parts. There is a single well defined frequency/energy for an electron at rest (but not a position :) ).
For composite particles like molecules things are more complicated. De Broglie's hypothesis only applies to free particles, so you shouldn't apply it naively to the bound degrees of freedom within a composite particle. You can however think of a de Broglie wavelength for the center of mass degree of freedom, which is particularly useful when you do experiments that don't probe the internal structure of the object.