Why does energy propagate outwards?

[Jilang]

I wasn't sure whether this question belonged in classical physics or here. It is something I never worried about before, but it is now bothering me. Given the reversibility of quantum processes why is one direction preferred over another? Sophiecentaur suggesting looking at Huygens Principle. This and later work (Kirchoff?) seems to suggest the backwards moving waves must suffer a self-cancelling effect, but that would imply that all wave propagation is fundamentally QM in nature.

 

[mfb]

What do you mean with "outwards"?

This and later work (Kirchoff?) seems to suggest the backwards moving waves must suffer a self-cancelling effect,

For propagating planar or spherical waves, this is true.

but that would imply that all wave propagation is fundamentally QM in nature.

Water waves show the same effect without quantum mechanics.

In general: entropy. There are more possible states with energy spread out "somewhere" than states with energy localized somewhere. You can still have states with converging waves, but then you need a very careful prepration of the setup.

 

[vanhees71]

I'm not sure whether I understand right what you mean.

I guess the point is that for a usual situation we use the retarded solution of the wave equation in electromagnetics to describe radiation of em. waves from some source (i.e., time-varying charge and current distributions). The answer is pretty simple: We use it, because it's the unique solution of classical electromagnetics (Maxwell theory) for the given physical situation, describing emission of electromagnetic waves from a source.

There is of course nothing "unphyscial" in principle with advanced solutions. However, it's much more difficult to prepare such a situation. For that you would have to create somehow an electromagnetic wave fields at a far distance of some charges and currents in such a way that it gets completely absorbed by making these charge/current distributions such that they completely transfer the field energy into motion of the charges. I think it is pretty impossible to control both the wave field and the charge-current distribution precisely in such a way.

All this has not so much to do with quantum theory but it's of course consistent with quantum theory either.

 

[Jilang]

What do you mean with "outwards"?

For propagating planar or spherical waves, this is true.
Water waves show the same effect without quantum mechanics.

In general: entropy. There are more possible states with energy spread out "somewhere" than states with energy localized somewhere. You can still have states with converging waves, but then you need a very careful prepration of the setup.

Thanks mfb, by outwards I meant outwards from the source of the disturbance. I got the impression though that the cancelling only worked for an odd number of dimensions, so strings yes, water waves no, spherical waves yes. From here and other places
http://mathpages.com/home/kmath242/kmath242.htm
Is that incorrect?

 

[mfb]

I got the impression though that the cancelling only worked for an odd number of dimensions, so strings yes, water waves no, spherical waves yes.

Where does that come from? You can prepare arbitrary 1- and 2-dimensional systems in 3 dimensions and arbitrary 1-dimensional systems in 2 dimensions. If something works in 1 dimension, it has to be able to work in 2 dimensions as well.

 

[Jilang]

All this has not so much to do with quantum theory but it's of course consistent with quantum theory either.

Schwartz appeared to regard the interpretation of Huygens Principle ( with each point being a new source of radiation) as nonsense. So perhaps classically there is no accounting for it? The propagation of light in QFT is consistent with it though as you say.

 

[Jilang]

Where does that come from? You can prepare arbitrary 1- and 2-dimensional systems in 3 dimensions and arbitrary 1-dimensional systems in 2 dimensions. If something works in 1 dimension, it has to be able to work in 2 dimensions as well.

From the link I gave it appears to be an outcome of the maths. It's intriguing and I hoped that someone could shed some light on it.

 

[Bill_K]

From the link I gave it appears to be an outcome of the maths. It's intriguing and I hoped that someone could shed some light on it.

It's true -- in an odd number of dimensions the solution of the wave equation (Green's function) is an impulse that propagates outward with the fundamental velocity. But in an even number of dimensions, although the Green's function has a sharp initial wavefront, it also has a "tail" that persists after the wavefront has passed.

This can be seen for example for the two dimensional wave equation by considering an equivalent problem -- the wave produced in three dimensions by an infinitely long line source extending along the z axis. At a given point the wave received will be zero until the wave from the nearest source point (z = 0) has arrived. From then on, contributions from more and more distant source points z ≠ 0 will keep coming, indefinitely.

[Also, one should not casually use water waves as an example, since they are TOTALLY different from the wave equation, being dispersive with an infinite propagation velocity!]

 

[Jilang]

Thanks Bill, I was starting to wonder if I was being lead up the garden path on that one! I am not familiar with Greens functions. Do they exist classically or just in the QM arena? The idea of new point sources seems to have more in common with QM than classical mechanics, but I am probably missing something fundamental.

 

[Bill_K]

For any partial differential equation such as the wave equation, a Green's function is the solution produced by a point source. The idea is that any source can be represented as a sum (or integral) over point sources, so the solution for an arbitrary source can be written as a sum or integral of Green's functions, and by finding the Green's function you have solved a wide variety of problems.

For the 3-D wave equation, for example, the Green's function is just an outgoing spherical impulse,

G(x, t) = (1/4πr) δ(r - ct)

 

[Jilang]

Thanks Bill, when you say the idea is that any source can be represented by the sum of sources is that a consequence of the mathematics of the classical differential equations?

 

[mfb]

Thanks Bill, when you say the idea is that any source can be represented by the sum of sources is that a consequence of the mathematics of the classical differential equations?

This is called superposition, and possible if the equations are linear.

 

[Jilang]

Thanks mfb, I'm going to read up about these Green functions.