(a) Use Eq. (9.37) (with the Bohr values L=2 ħand r=4 aB ) to show that the fine-structure separ

step 1 In this question we have to find that delta EFS is given by the follow expression from the equat

(a) Use Eq. (9.37) (with the Bohr values L=2 ħand r=4 aB )...

(a) Use Eq. (9.37) (with the Bohr values $L=2 \hbar$ and $r=4 a_{\mathrm{B}}$ ) to show that the fine-structure separation $\Delta E_{\mathrm{FS}}=2 \mu_{\mathrm{B}} B$ of the two $2 p$ levels of hydrogen can be written as $$ \Delta E_{\mathrm{FS}}=\frac{m_{\mathrm{e}}\left(k e^{2}\right)^{4}}{32 \hbar^{4} c^{2}} $$ $\left[\right.$ Hint: Since $\mu_{0} \varepsilon_{0}=1 / c^{2}$ and $k=1 / 4 \pi \varepsilon_{0}$, you can replace $\mu_{0}$ by $\left.\mu_{0}=4 \pi k / c^{2} .\right]$ (b) Show that you can rewrite $(9.38)$ as $$ \Delta E_{\mathrm{FS}}=\frac{\alpha^{2} E_{\mathrm{R}}}{16} $$ where $\alpha$ is the dimensionless fine-structure constant $$ \alpha=\frac{k e^{2}}{\hbar c} $$ (c) Show that $\alpha \approx 1 / 137$, which, together with (9.39), shows that fine structure is indeed a small effect.

(a) Use Eq. (9.37) (with the Bohr values $L=2 \hbar$ and $r=4 a_{\mathrm{B}}$ ) to show that the fine-structure separation $\Delta E_{\mathrm{FS}}=2 \mu_{\mathrm{B}} B$ of the two $2 p$ levels of hydrogen can be written as
$$
\Delta E_{\mathrm{FS}}=\frac{m_{\mathrm{e}}\left(k e^{2}\right)^{4}}{32 \hbar^{4} c^{2}}
$$
$\left[\right.$ Hint: Since $\mu_{0} \varepsilon_{0}=1 / c^{2}$ and $k=1 / 4 \pi \varepsilon_{0}$, you can replace $\mu_{0}$ by $\left.\mu_{0}=4 \pi k / c^{2} .\right]$
(b) Show that you can rewrite $(9.38)$ as
$$
\Delta E_{\mathrm{FS}}=\frac{\alpha^{2} E_{\mathrm{R}}}{16}
$$
where $\alpha$ is the dimensionless fine-structure constant
$$
\alpha=\frac{k e^{2}}{\hbar c}
$$
(c) Show that $\alpha \approx 1 / 137$, which, together with (9.39), shows that fine structure is indeed a small effect.

Key Concepts

Bohr Model Quantization

This concept explains how electrons in hydrogen-like atoms are restricted to discrete orbits with quantized angular momenta. It sets the basis for deriving energy levels and the size of electron orbits using angular momentum quantization, which is central to the early quantum model of the hydrogen atom.

Fine Structure Splitting

Fine structure refers to the small corrections to the energy levels of atoms, particularly hydrogen, arising from relativistic effects and the electron’s spin-orbit interaction. This splitting results in closely spaced sub-levels within a principal quantum number level and is essential for understanding high-resolution spectral lines.

Fine-Structure Constant

The fine-structure constant, denoted as ?, is a dimensionless parameter that quantifies the strength of the electromagnetic interaction. It plays a crucial role in determining the magnitude of fine structure effects and appears in many quantum electrodynamics equations, linking atomic energy scales with fundamental constants.

Electromagnetic Constants and Their Relationships

This concept encompasses constants like the Coulomb constant (k), the vacuum permittivity (??), and the vacuum permeability (??), along with their interrelationships. These constants are fundamental in linking classical electromagnetism to quantum phenomena, as seen in the derivation of quantities like the fine-structure splitting where ?? and ?? combine to relate magnetic and electric interactions via the speed of light (c).

Rydberg Energy

Rydberg energy (E_R) represents the energy scale characteristic of the hydrogen atom and is used to describe the binding energy of the electron. It provides a reference value that helps quantify small effects like fine structure by relating them to the larger scales of atomic energy levels.

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