# I Underlying reality re deBroglie equation

1. Jan 2, 2017

### Cobalt101

While the mathematics and the experimental results of de Broglie's equation are clear, what is the best understood explanation as to what is happening in underlying reality of why wavelength varies (inversely) with momentum for all energy and matter ?

2. Jan 2, 2017

### clarkvangilder

It seems as though the wave function amplitude would be larger where a particle is (classically) slower, and therefore at a lower kinetic energy. The slower the motion, the farther apart the nodes, and thus a larger wavelength?

3. Jan 2, 2017

Staff Emeritus
What do you mean by "underlying reality"?

4. Jan 2, 2017

### Cobalt101

what physically is going on is what I mean by that.

Last edited: Jan 2, 2017
5. Jan 2, 2017

### Staff: Mentor

There isn't any really satisfying answer to that sort of "why" question. Science is about discovering and describing the laws that govern the behavior of the universe we live in, and experiments show that we live in a universe in which gravity is an attractive force, like charges repel instead of attracting, ...., and momentum varies inversely with wavelength.

Of course science does seem to offer answers to "why?" questions - everyone knows that Newton's law of gravitation explains why the planets move the way they do, for example. But that's just pushing the "why?" down one level. Why should the planets obey Newton's law of gravity and not some other law?

6. Jan 2, 2017

### Cobalt101

Yes I get that there are often no answers to why questions, as you outline, but in this case I was not so much asking for why there is this relationship between momentum and wavelength, but rather what physical manifestation this might have ?

7. Jan 11, 2017 at 6:04 AM

### muscaria

Although there's always an infinite number of levels to why (As a parent knows only too well), here are some thoughts on the matter.
I think your question may be closely connected to action.
During the motion of a point particle in 1d say, the change in the action along the solution trajectory $q(t)$ is
$$dS=pdq-Hdt$$
where p is the momentum of the particle, q its position, H its energy, t is time. Notice here the pairing between momentum/space-displacement and energy/time-displacement in comparison to the de-Broglie relations. For a system of particles this generalises to a sum over degrees of freedom $n$, the important thing being that each positional coordinate $q_i$ and momentum $p_i$ is dynamically paired in a special way. Also if you place $t$ on the same footing as the $q_i$ it turns out that the momentum paired with $t$ is in fact $-H$.
$dq_i (i=1,2,..,n+1)$ picks out a direction in extended configuration space and $p_i$ gives the flow along this direction. So this is what action differentials measure, the momentum which is carried over displacements.
Now back to 1 degree of freedom, so point particle on a line, and we imagine that time freezes but we can still investigate how $S$ would change if we virtually translated the particle by $dq$, which is the displacement that was just about to take place during the motion i.e. $dq=\dot{q}dt$. $dS=pdq$ is a geometric invariant so that if I deform my $dq$ in some way, $p$ will change also in a way to preserve $dS$. So if I contract space such that the original coordinate differential $dq$ has now doubled with the new q-tics, we only need half the previous amount of flow for $S$ to tick off the same way as before and motion to be the same.
Now a wave is spread over space and not generally localised at a point. However, even without really knowing what property of the wave determines its momentum, if the wave travels with the same flow at every point along the line so that the wave just gets translated along homogeneously without saying anything about what shape the wave needs to be for this to happen. Now imagine we discretised the waveform and viewed it as being made up of infinitesimally evenly spaced points. For a given flow $p$, there is an invariant $dS$ associated with $dq$. Now if we contract the space in the same manner as before, the in-between spaces $\textit{all}$ grow evenly, and the flow $p$ thus decreases in order to preserve $dS$.
A bit long winded but hope this helps..