EM Spectrum of Hydrogen-like Atoms

Hydrogen-like atoms, also known as hydrogenic atoms, are atoms with only one electron. This simplicity allows for a relatively straightforward analysis of their electromagnetic (EM) spectrum. The EM spectrum of hydrogen-like atoms is crucial for understanding atomic structure, quantum mechanics, and spectroscopy.

The Bohr Model

The Bohr model of the hydrogen atom was an early theory that explained the discrete energy levels of electrons in hydrogen-like atoms. According to the Bohr model, the energy levels are quantized and are given by the formula:

$$ E_n = -\frac{Z^2 R_H}{n^2} $$

where:

• $E_n$ is the energy of the electron in the nth orbit,
• $Z$ is the atomic number (for hydrogen $Z = 1$),
• $R_H$ is the Rydberg constant for hydrogen ($R_H \approx 2.18 \times 10^{-18}$ J),
• $n$ is the principal quantum number ($n = 1, 2, 3, \ldots$).

Quantum Mechanical Model

The quantum mechanical model, which includes the Schrödinger equation, provides a more accurate description of hydrogen-like atoms. The energy levels in this model are given by:

$$ E_n = -\frac{Z^2 \mu e^4}{8 \epsilon_0^2 h^2 n^2} $$

where:

• $\mu$ is the reduced mass of the electron-proton system,
• $e$ is the elementary charge,
• $\epsilon_0$ is the vacuum permittivity,
• $h$ is Planck's constant.

The EM Spectrum

When an electron transitions between energy levels, it either absorbs or emits a photon with energy equal to the difference between the two levels. The frequency $\nu$ of the emitted or absorbed radiation is related to the energy difference $\Delta E$ by:

$$ \Delta E = h \nu $$

For a transition from a higher energy level $n_i$ to a lower energy level $n_f$, the wavelength $\lambda$ of the emitted photon can be calculated using the Rydberg formula:

$$ \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) $$

Series in Hydrogen Spectrum

The hydrogen spectrum consists of several series of lines named after their discoverers. Here are the most important ones:

Series Name	Transition From	To (n_f)	Wavelength Range
Lyman	$n_i \geq 2$	1	Ultraviolet
Balmer	$n_i \geq 3$	2	Visible
Paschen	$n_i \geq 4$	3	Infrared
Brackett	$n_i \geq 5$	4	Infrared
Pfund	$n_i \geq 6$	5	Infrared

Examples

Example 1: Wavelength of a Photon

Calculate the wavelength of the photon emitted when an electron in a hydrogen atom transitions from the $n = 3$ level to the $n = 2$ level.

Using the Rydberg formula:

$$ \frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{3^2} \right) $$

$$ \frac{1}{\lambda} = R_H \left( \frac{1}{4} - \frac{1}{9} \right) $$

$$ \frac{1}{\lambda} = R_H \left( \frac{5}{36} \right) $$

$$ \lambda = \frac{36}{5 R_H} $$

Plugging in the value of $R_H$:

$$ \lambda \approx \frac{36}{5 \times 1.097 \times 10^7 \text{ m}^{-1}} $$

$$ \lambda \approx 6.56 \times 10^{-7} \text{ m} $$

Example 2: Energy of a Photon

Calculate the energy of a photon emitted during a transition from $n_i = 4$ to $n_f = 2$ in a hydrogen-like atom with $Z = 2$ (Helium ion, He+).

Using the energy level formula:

$$ \Delta E = -Z^2 R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) $$

$$ \Delta E = -2^2 R_H \left( \frac{1}{2^2} - \frac{1}{4^2} \right) $$

$$ \Delta E = -4 R_H \left( \frac{1}{4} - \frac{1}{16} \right) $$

$$ \Delta E = -4 R_H \left( \frac{3}{16} \right) $$

$$ \Delta E = -3 R_H $$

Plugging in the value of $R_H$:

$$ \Delta E \approx -3 \times 2.18 \times 10^{-18} \text{ J} $$

$$ \Delta E \approx -6.54 \times 10^{-18} \text{ J} $$

The negative sign indicates that energy is released (photon emitted) during the transition.

Conclusion

The EM spectrum of hydrogen-like atoms is a fundamental topic in modern physics, providing insights into quantum mechanics and atomic structure. The discrete energy levels and spectral lines can be calculated using the Bohr model or quantum mechanics, and they are observable in the form of spectral series in the EM spectrum. Understanding these concepts is essential for students preparing for exams in physics and related fields.