Why Does This Seemingly Nonsensical Argument Hold?

  • Thread starter Thread starter aaaa202
  • Start date Start date
  • Tags Tags
    Argument
AI Thread Summary
The argument about the number of vectors with a given velocity being proportional to the area of a sphere, 4πv², raises questions about its mathematical validity, particularly concerning the concept of infinity. While it seems nonsensical, the argument holds because the shell volume, 4πv² dv, increases with velocity, leading to more vectors in that shell. This discussion connects to quantum mechanics, where the distribution of vectors can be understood through a lattice of points in velocity space. The relationship between particle velocity and wavelength, as described by de Broglie's relation, supports this lattice concept. Ultimately, the exploration of these ideas reveals deeper insights into the nature of velocity and quantum mechanics.
aaaa202
Messages
1,144
Reaction score
2
I've sometimes seen this argument being used:

The amount of vectors with a given velocity is propotional to the area of the sphere given by:
4πv2, because there are more vectors corresponding to bigger speeds.

But mathematically this is nonsense to me, pretty much like comparing infinities. There are an infinite amount of vectors corresponding to any speed apart from zero speaking strictly mathematical.

So why is that on a deeper level makes this argument of "nonsense" hold?
 
Physics news on Phys.org
It's the shell volume 4\pi v^2 dv which is larger. If we imagine different vectors v, distributed as a uniform fine lattice of points in velocity-space, then the number of points in the shell will be proportional to the shell volume.

I realize this 'answer' raises other issues, but I hope it is of some help.
 
Yes exactly, and it is probably these other "questions" that I think about. Is it something quantum mechanical?
 
Yes. Boltzmann (working before quantum theory) did effectively use a lattice of points, but it was arbitrary. How brilliant! Now we can justify the lattice quantum mechanically. In a crude treatment the molecules are matter waves of wavelength related to particle velocity by de Broglie's relation,
mv=\frac{h}{\lambda}. The wavelengths, \lambda, are fixed by boundary conditions for standing waves in a box. The lattice of points in velocity space emerges very simply from this.
 
I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
Back
Top