# Bohr-Sommerfeld Quantization Rule for 3D Systems

• paweld
In summary, the "old" quantum mechanics involved the famous Bohr-Sommerfeld quantization rule, which could be applied to three-dimensional systems such as the hydrogen atom. This involved dividing the electron's motion into two orthogonal directions and using the de Broglie wavelength to represent the number of waves contained in each segment of motion. However, this model was later updated with the understanding that the electron's motion is actually in one direction, and the number of de Broglie waves is determined by the sum of the number of waves in each direction.
paweld
"Old" quantum mechanics

There is famous Bohr Sommerfeld quantization rule which says that $$\oint p dx = n h$$. Is it possible to apply this rule to three dimensonal system e.g. hydrogen atom.
The integral will probably look as follow:
$$\int \sqrt{2E - \frac{l^2}{r^2}+\frac{\alpha c \hbar}{r}} = n h$$
But I don't know which the contour I should chose.

paweld said:
There is famous Bohr Sommerfeld quantization rule which says that $$\oint p dx = n h$$. Is it possible to apply this rule to three dimensonal system e.g. hydrogen atom.
The integral will probably look as follow:
$$\int \sqrt{2E - \frac{l^2}{r^2}+\frac{\alpha c \hbar}{r}} = n h$$
But I don't know which the contour I should chose.

In the hydrogen of the Bohr-Sommerfeld model, the orbital becomes elliptical or circular.
So simply I use the two-dimensonal system.
The orbital length becomes a integer times the de Broglie's wavelength ($$\lambda =h/mv =h/p$$).

For example, when the orbital is elliptical, we divide the electron's movement into the two directions at short time intervals.

The $$\perp$$ is the direction of the angular momentum(tangential), and the $$r$$ is the radial direction.
The two directions are rectangular at each point.

$$\oint p_{\perp} dq_{\perp} = n_{1}h, \qquad \oint p_{r} dq_{r} = n_{2}h$$

At each point, the de Broglie's wavelengths are $$\lambda_{\perp}= h/p_{\perp}, \lambda_{r}= h/p_{r}$$
The wavelengths are changing at each point. So, the above equations are equal to,

$$\oint dq_{\perp}/\lambda_{\perp} = n_{1}, \qquad \oint dq_{r}/\lambda_{r}= n_{2}$$

This means that in the Bohr-Sommerfeld model, the sum of the number of the de Broglie's waves contained in each short segment becomes a interger.

In the case of the three-dimensional system or more complex system, we had better use the computer or something, I think.

Last edited:

Sorry. I should have added one more thing in the above statement(#2).

I divided the electron's motion into the two directions which are rectangular at each point on the elliptical orbital.

But actually the electron is moving in one direction.
And its momentum $$p$$ satisfies $$p^2 = p_{\perp}^2 + p_{r}^2$$ at each point.

So in a short time($$dt$$), the electron moves $$dq=\frac{p}{m} dt$$.
Of course this satisfies,

$$dq^2 = dq_{\perp}^2 + dq_{r}^2$$

The number of de Broglie's waves ($$\lambda$$ in length) contained in the short segment($$dq$$) is,

$$dq/\lambda=dq/(\frac{h}{p})=p dq /h = p^2dt /hm$$

As I said above, due to $$p^2 = p_{\perp}^2 + p_{r}^2$$, we arrive at the following relation,

$$dq /\lambda = dq_{\perp}/\lambda_{\perp} + dq_{r}/\lambda_{r}$$

As a result, the number of de Broglie's waves contained in one-round of the orbital satisfies,

$$n = n_{1} + n_{2}$$

For example, in the energy level (n=2) of the hydrogen(the Bohr model), there are two patterns as follows,
n1=1, n2=1 ----- elliptical (angular momentum=1)
n1=2, n2=0 ----- circular (angular momentum =2)

## 1. What is the Bohr-Sommerfeld Quantization Rule for 3D Systems?

The Bohr-Sommerfeld Quantization Rule for 3D Systems is a principle developed by Danish physicist Niels Bohr and German physicist Arnold Sommerfeld in the early 20th century. It states that in a three-dimensional system, the total energy of an electron in an atom or molecule can only take on certain discrete values, known as quantized energy levels.

## 2. How does the Bohr-Sommerfeld Quantization Rule differ from the Bohr Model of the atom?

The Bohr Model of the atom proposed by Niels Bohr in 1913 only applied to hydrogen atoms and did not take into account the electrons' wave-like nature. The Bohr-Sommerfeld Quantization Rule, on the other hand, applies to all atoms and molecules and takes into account the wave-like behavior of electrons, providing a more accurate description of their energy levels.

## 3. What is the significance of the Bohr-Sommerfeld Quantization Rule for 3D Systems?

The Bohr-Sommerfeld Quantization Rule was a crucial development in the understanding of atomic and molecular structure. It provided a more accurate and comprehensive model of electron energy levels and helped explain the stability of atoms and molecules. It also laid the foundation for further developments in quantum mechanics.

## 4. How is the Bohr-Sommerfeld Quantization Rule applied in modern physics?

The Bohr-Sommerfeld Quantization Rule is still used in modern physics, particularly in the study of atomic and molecular structures. It is also an important concept in the field of spectroscopy, which is used to study the interactions between matter and electromagnetic radiation.

## 5. Are there any limitations to the Bohr-Sommerfeld Quantization Rule for 3D Systems?

While the Bohr-Sommerfeld Quantization Rule is a significant advancement in our understanding of atomic and molecular structures, it is not a complete description. It does not account for the effects of relativity and quantum mechanics, which are necessary for a more accurate understanding of subatomic particles and their behavior.

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