Question on De Broglie

  • Thread starter redtree
  • Start date
  • #1
213
2

Main Question or Discussion Point

The wave-frequency relationship is as follows:

f = v / [tex]\lambda[/tex]

Therefore:
v = [tex]\lambda[/tex] * f

The de Broglie relations are as follows:

[tex]\lambda[/tex] = h / p

f = E / h

Using some basic algebra:

v = (h / p) * (E / h)
v = E / p
v = [tex]\gamma[/tex]mc[tex]^{}2[/tex] / [tex]\gamma[/tex]mv
v = c[tex]^{}2[/tex] / v

Now, assuming a natural unit system with c=1

v = 1 / v

That doesn't seem to make sense. Where is my error?
 
Last edited:

Answers and Replies

  • #2
2,257
7
are you sure it isnt v=v^2/v
 
  • #3
Doc Al
Mentor
44,867
1,114
The de Broglie relations are as follows:

[tex]\lambda[/tex] = h / p

f = E / h
The first gives the de Broglie wavelength. The second relates the frequency of a photon to its energy.
 
  • #4
213
2
Total energy is mc[tex]^{}2[/tex] not mv[tex]^{}2[/tex]
 
  • #5
213
2
The first gives the de Broglie wavelength. The second relates the frequency of a photon to its energy.
The relationship is not just for a photon but for any particle. Frequency and wavelength are related by v = f * [tex]\lambda[/tex]
 
  • #6
Doc Al
Mentor
44,867
1,114
The relationship is not just for a photon but for any particle.
What relationship? Not E = hf.
Frequency and wavelength are related by v = f * [tex]\lambda[/tex]
This is yet a third relationship, true for any wave.
 
  • #7
Redbelly98
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
12,098
129
Now, assuming a natural unit system with c=1

v = 1 / v

That doesn't seem to make sense. Where is my error?
It "makes sense" because you have chosen to have velocity be unitless. The solutions to your equation are

v = +1 or v = -1

I.e., the object moves at the speed of light.

Somewhere, you must have used a relation that holds only for objects of zero rest mass. I should be more up on this stuff than I am, and will defer to Doc Al for just where the "error" is.
 
  • #8
Gokul43201
Staff Emeritus
Science Advisor
Gold Member
7,051
16
That's right. The error is in the equation E = hf, which is the famous Einstein relation for the energy of a photon.

More generally, the error is in writing down a string of equations without defining what any of the quantities refer to.
 
  • #9
410
0
People, lets not get crazy and instead actually answer the fellow's question. What our friend has discovered is that the phase velocity of a deBroglie wave is indeed faster than the speed of light
[tex]v_p \frac{\omega}{k} = \frac{c^2}{v}[/tex]
Instead, he should consider particles as being represented by wave packets with some spread in momentum and energy (in order to be localizable). In this case, the relevant quantity is the group velocity
[tex]v_g = \frac{\partial\omega}{\partial k} = \frac{\partial E}{\partial p} = \frac{\partial (\sqrt{p^2 c^2 + m^2 c^4} )}{ p }= \frac{p}{\gamma m} = v[/tex]
I hope this clears things up.

Edit: [tex]E = h f[/tex] is true for particles of mass as well. It's a result of harmonic oscillators in general, of which quantum fields are one type (in the limit where "particle" makes sense).
 

Related Threads for: Question on De Broglie

Replies
1
Views
617
Replies
7
Views
2K
  • Last Post
Replies
14
Views
3K
  • Last Post
Replies
2
Views
3K
Replies
4
Views
2K
Replies
11
Views
811
Replies
25
Views
907
Top