Why does Bohr's derivation work?

[davidbenari]

Bohr assumed angular momentum was quantized as $L=n\hbar$. But really it is quantized as $L=\hbar \sqrt{l(l+1)}$.

What he does to derive,e.g., the Bohr radius is consider that the total energy of an electron orbiting a proton is

$E=\frac{L^2}{2mr^2}-\frac{k e^2}{r}$

and then he makes some clever substitutions. However, Bohr substituted the formula for $L_z$ ($n\hbar$) not the actual $L$ (which is $h\sqrt{l(l+1)}$). So why then does his procedure work?

Up until now I have considered this a mere accident but I've heard about people considering the so-called "Gravitational Bohr Radius" which is derived using the same procedure.

I don't understand why we asume its validity for the simple system of one particle orbiting another if we've got the wrong formula for angular momentum.

So then:

Why does his procedure work?

Why do we take on the similar (and wrong) derivation to the gravitational case?

 

[Dr. Courtney]

The hydrogen energy levels are degenerate with respect to angular momentum and only depend on principal quantum number, n.

It was a happy accident.

 

[davidbenari]

Hmm. I don't see how this explains why substituting the wrong formula for $L$ will give the correct energy levels. Maybe as you say it was a happy accident (which baffles me since this was so important for our history)...

But anyways, why do people like to talk about a Gravitational Bohr Radius ? Are they bringing Bohr's derivation into the gravitational case or are they actually solving the Schrödinger eqn for the gravitational case and calling the smallest radius "the Bohr Radius" ?

 

[davidbenari]

In case I'm not being clear this is the derivation I'm thinking about (for the G. Bohr Radius):

Suppose $L=n\hbar=rp$

Since $\frac{mv^2}{r}=\frac{GMm}{r^2}$

then $\frac{p^2}{m}=\frac{GMm}{r}$

Since $L=n\hbar$ then

$\frac{(n\hbar)^2}{r^2m}=\frac{GMm}{r}$

and therefore

$r=\frac{(n\hbar)^2}{GMm^2}$

for n=1

$r=\frac{\hbar^2}{GMm^2}$

and this what I've seen presented for "G Bohr Radius". My question is, did they solve the Schrödinger equation for the G. Potential case? Or did they derive this using a "Bohr Picture" ?

It seems to me there has to be some deep reason why this works for the smallest radius, other than it being a "brilliant blunder" by Bohr.

 

[jtbell]

why do people like to talk about a Gravitational Bohr Radius ?

Which people are these? I don't remember reading about a "gravitational Bohr radius".

 

[Heinera]

But anyways, why do people like to talk about a Gravitational Bohr Radius ?

Who are "people"? A quick Google search mostly turned up references to a particular R. Oldershaw. Crackpot if you ask me, and a subject not suited for this forum.

 

[davidbenari]

I'm taking a course on Coursera by Hitoshi Murayama where they use this formula and call it G. Bohr Radius. They use this formula to give a lower limit on the mass of dark matter particles (WIMPs) taking into consideration they have to be contained within a sphere whose radius is approximately the galactic radius.

Also (by a quick look on google) you can find articles like:

https://thespectrumofriemannium.wordpress.com/tag/gravitational-atom/

http://arxiv.org/pdf/0803.1197.pdf

Also I remember Griffiths QM consider a hypothetical "gravitational atom", but they only use the energy levels there not the radius.

Edit: I can't find exactly where Murayama refers to this formula by this name, but he does use it. (I had to get this name from somewhere and I'm pretty sure he used this term)

 

[davidbenari]

Perhaps this is crackpottery and I didn't know this :/ . Although Murayama is a respected guy who used this formula in his lectures.

 

[Vanadium 50]

Maybe as you say it was a happy accident (which baffles me since this was so important for our history)

Really? There are no other happy accidents in history?

The Bohr model was introduced in 1913, known to be wrong in 1913, and completely superseded in 1926.

The "happy accident" comes about, as Dr. Courtney says, from the k-l degeneracy in the hydrogen atom. That comes about because the Schroedinger Equation for a 1/r potential is a differential equation that can be solved by separation of variables two different ways. To me, that looks accidental. And happy.

 

[vanhees71]

It's not clear to me, why the Bohr-Sommerfeld model of the atom works (it was Sommerfeld who fully understood the mathematics of the Bohr model!). Perhaps there's an explanation from deriving it as an approximation from quantum theory, which should be somehow related to the WKB method at low orders. It's already amazing that you get the correct energy levels for the non-relativistic problem. What's even more amazing to me is that Sommerfeld got the correct fine structure. The modern way to understand it is to use QED, which boils down (in Coulomb gauge) to solve the time-independnet Dirac equation with a Coulomb potential. In the Bohr-Sommerfeld model there's nothing concerning spin 1/2, and naively, I'd expect to get rather some approximation of the energy levels of a "spinless electron", but that's given by the analogous calculation in scalar QED, and the corresponding "hydrogen-like" energy levels for a boson indeed do give a different fine structure for the hydrogen atom.

Another interesting detail is that Schrödinger started his investigation concerning the hydrogen atom indeed using the relativistic dispersion relation, which lead him, using the de Broglie-Einstein rule $\omega \rightarrow E/\hbar$ and $\vec{k} \rightarrow \vec{p}/\hbar$ to get the "wave equation" for "matter waves" to the Klein-Gordon equation, and of course he got the right spectrum for this problem, but it was the wrong fine structure. So he gave up the relativistic case for the time being and used the non-relativistic approximation, leading to the Schrödinger equation.

So still it's puzzling, why Sommerfeld got the correct fine structure using Bohr-Sommerfeld quantization. It's really an astonishing accident that the errors of a completely wrong model conspire in a way to give the correct hydrogen spectrum including fine structure!