Bohr model electron wavelengths

[copernicus1]

Hi

I know that in the Bohr model, electrons move between energy levels, but you don't hear much about the electron's wavelength at each particular level. If we assume the orbits contain an integer multiple of wavelengths, you get the usual $$2\pi r=n\lambda,$$ so, based on the expression for the Bohr radius, the wavelength at each level should be $$\lambda_n=\frac{2\pi r}{n}=\frac{2\pi n\hbar}{m_ec\alpha}.$$ Does anyone know if this a standard part of the theory? I've just never assumed the wavelength had to be fixed at each energy level, but that seems reasonable if each level has a fixed energy.

Thanks

 

[Simon Bridge]

Hi

I know that in the Bohr model, electrons move between energy levels, but you don't hear much about the electron's wavelength at each particular level.

That is because the deBroglie matter wave model is misleading. You don't need it for the Bohr model and for anything else we have better models.

 

[copernicus1]

Sure, I understand all that. I'm just curious about the theory more or less as an historical artifact.

Actually, what do you mean when you say the de Broglie matter wave model is misleading? I know you don't need it for the Bohr model since he got from the Balmer series, but what's misleading about it?

 

[Simon Bridge]

Actually, what do you mean when you say the de Broglie matter wave model is misleading?

It leads to misunderstandings along the lines of "wave-particle duality"... particles passing through many slits at the same time, interfering with themselves... that sort of thing. Mind you, it is difficult not to be misleading ...

Does anyone know if this a standard part of the theory?

FWIW: it is not part of the standard theory. There are theories it is a standard part of... but I don't know anyone who uses it outside of year I physics classes.

 

[jtbell]

It helps to know the timeline:

1913: Bohr publishes his original model with circular electron orbits which were determined by quantizing the orbital angular momentum. Sommerfeld later extends the theory to include elliptical orbits, with a second quantum number.

1924: de Broglie proposes that the electron is actually a wave, and that Bohr's original quantization condition comes from requiring an integer number of wavelengths around a circular orbit. I don't know if he ever generalized this to elliptical orbits.

1925-26: Schrödinger comes up with a differential wave equation for "electron waves." His model no longer has electron "orbits" in the sense of the Bohr-Sommerfeld model. The wave function is distributed around the nucleus in three dimensions. During the same period, Heisenberg comes up with his "matrix mechanics" approach.

After this, physicists abandoned the Bohr-Sommerfeld model very quickly. I don't think one can say that de Broglie's waves were ever a "standard part" of the Bohr-Sommerfeld model. Someone might have attempted to come up with a serious theory along those lines if Schrödinger and Heisenberg hadn't come up with their theories so quickly.