21

Electron orbit circumferences have to be in multiples of its de Broglie wavelength, but what do those 2 have in common?

Qmechanic
  • 201,751
Yusuf Abukar
  • 531

5 Answers5

21

I will assume you are familiar with the properties of waves such as interference and diffraction.

Consider an electron orbiting the nucleus. By de Broglie's hypothesis, we would consider it to be a wave orbiting around the nucleus. Now, once the electron wave orbits once, the second time it would interfere with the first wave. For the system to be stable, the electron shouldn't cancel itself out, so the wave must constructively interfere. That means, the circumference must be an integral multiple of the electron's wavelength.

Here's a picture that shows how the electron wave is expected to exist in the atom according to the Bohr's theory.

source: www.astronomyclub.xyz

Greg
  • 143
Pritt Balagopal
  • 3,187
  • 1
    @ Pritt Balagopal Does that mean that the electron instantaneously cancels itself out? If it it travelling in a given time to return to its starting position, how can it cancel itself out? – Yusuf Abukar May 07 '17 at 19:15
  • Electron shouldn't cancel itself out. Thats why circumference is a multiple of wavelength. – Pritt Balagopal May 08 '17 at 03:00
  • 1
    @PrittBalagopal : would be better to call the electron a standing wave, which is the original de-Broglie hypothesis. – vs_292 Sep 09 '17 at 14:18
  • That's right, the electron is in a standing wave. – Pritt Balagopal Sep 09 '17 at 15:11
  • Why does it constructively interfere when the circumference is an integral multiple of the wavelength? – Kaushik Sep 03 '19 at 19:12
  • Whay do we say that a destructive interference result in the electron canceling out? How would it cancel out? – Marco Disce May 22 '21 at 12:42
  • Doesn't that imply an absolute frame of reference in which the waves occur? Or does the nucleus define a frame of reference (in which its angular momentum is zero) for the electron? – Jus12 Aug 12 '22 at 11:49
  • Also Mercury's orbit does not overlap, but we don't see Mercury cancelling itself out. – Jus12 Aug 12 '22 at 13:09
17

Nothing actually. It was quite a wild guess by Bohr and supplied him with the spectrum of hydrogen. Pretty good guess indeed.

ZeroTheHero
  • 45,515
  • 2
    Most of the advancements in quantum science has been done by equation guessing and then trying to figure out the "whudunnit" of it later. – Stian Mar 15 '17 at 09:35
16

De Broglie suggested the existence of matter waves and gave a relation between their wavelength and momentum.

$\lambda=\frac{h}{p}$ ,

He said that this relation is completely general. It can be applied to matter particles and even photons.

enter image description here

Bohr, in his atomic model, considered an electron to be in form of a standing electron wave and if this wave is to be continuous over the circumference of the stationary orbit that the electron lie in, the circumference must be a integral multiple of its wavelength $(n\lambda)$.

$2\pi r=n\lambda$

$2\pi r=\frac{nh}{p}$

$pr=\frac{nh}{2\pi}$

And finally,

$mvr=\frac{nh}{2\pi}$

Mitchell
  • 4,907
  • 1
    your answer has really nice start, clarifying that de Broglie's idea was applicable in general and not just sub atomic particles. However your answer and especially the equations do not answer the question asked. It looks like what you write in your school's exam - rote memorization. Please see the accepted answer and thrive to understand physics – RinkyPinku Apr 25 '17 at 06:25
7

This comes from the Bohr-Sommerfeld quantization formula, which can be derived from the semiclassical WKB approximation of quantum mechanics, cf. e.g. this Phys.SE post. The quantization condition follows from requiring single-valuedness of the wavefunction.

Community
  • 1
Qmechanic
  • 201,751
4

Quite new here. Anyways, it was not a guess by Bohr in the first place. And it is so as to save energy else once it'll be out of phase it will start losing energy and Niels quantization will not be valid. That's the logical explanation though still incomplete considering that there's a lot going on and the theory we are referring to is outdated to a LOT of extent. Happy learning!

ZeroTheHero
  • 45,515
Rishabh Jain
  • 49