Transition of Electrons in Hydrogen-like Atoms

In the study of atomic physics, one of the fundamental concepts is the transition of electrons in hydrogen-like atoms. Hydrogen-like atoms are atoms that have only one electron, similar to the hydrogen atom. The transition of electrons refers to the movement of an electron from one energy level to another within the atom.

Energy Levels in Hydrogen-like Atoms

In hydrogen-like atoms, the electron is bound to the nucleus by the electrostatic force of attraction. The electron can exist only in certain discrete energy levels, which are quantized. These energy levels are represented by the principal quantum number, denoted by n. The energy of an electron in a particular energy level is given by the formula:

$$E_n = -\frac{{13.6 \, \text{{eV}}}}{{n^2}}$$

where E_n is the energy of the electron in the nth energy level.

The energy levels in a hydrogen-like atom are arranged in a series, with the lowest energy level being n=1. As the value of n increases, the energy of the electron becomes less negative, indicating that the electron is further away from the nucleus.

Transition of Electrons

When an electron in a hydrogen-like atom undergoes a transition from one energy level to another, it absorbs or emits energy in the form of electromagnetic radiation. This transition is governed by the conservation of energy, which states that the total energy before and after the transition must be the same.

The energy of the photon emitted or absorbed during the transition can be calculated using the formula:

$$E = hf$$

where E is the energy of the photon, h is Planck's constant (6.626 x 10^-34 J·s), and f is the frequency of the radiation.

The frequency of the radiation can be related to the energy difference between the initial and final energy levels using the equation:

$$\Delta E = hf = E_i - E_f$$

where ΔE is the energy difference between the initial (E_i) and final (E_f) energy levels.

Types of Transitions

There are several types of transitions that can occur in hydrogen-like atoms, depending on the initial and final energy levels of the electron. The most common types of transitions are:

1. Emission: When an electron transitions from a higher energy level to a lower energy level, it emits a photon of energy equal to the energy difference between the two levels. This is known as emission. The emitted photon can have different wavelengths, corresponding to different regions of the electromagnetic spectrum.

Example: An electron in the n=3 energy level of a hydrogen-like atom transitions to the n=2 energy level. The energy difference between these levels is given by ΔE = E_3 - E_2. The emitted photon will have an energy equal to ΔE and a corresponding wavelength.

1. Absorption: When an electron absorbs a photon of energy equal to the energy difference between two energy levels, it transitions from a lower energy level to a higher energy level. This is known as absorption.

Example: An electron in the n=2 energy level of a hydrogen-like atom absorbs a photon of energy equal to the energy difference between the n=2 and n=3 energy levels. The electron transitions to the n=3 energy level.

1. Ionization: Ionization occurs when an electron in a hydrogen-like atom transitions from the highest energy level (n>1) to a level outside the atom. In this case, the electron gains enough energy to overcome the attractive force of the nucleus and becomes a free electron.

Example: An electron in the n=3 energy level of a hydrogen-like atom absorbs a photon of energy equal to the energy difference between the n=3 energy level and the ionization energy. The electron is then ionized and becomes a free electron.

Selection Rules

The transition of electrons in hydrogen-like atoms is subject to certain selection rules, which determine the allowed transitions between energy levels. These selection rules are based on the conservation of angular momentum and the conservation of parity.

The selection rules for transitions in hydrogen-like atoms are as follows:

1. Δl = ±1: The change in the orbital angular momentum quantum number (l) between the initial and final energy levels must be ±1. This means that the electron can transition from an orbital with angular momentum l to an orbital with angular momentum l+1 or l-1.

2. Δm = 0, ±1: The change in the magnetic quantum number (m) between the initial and final energy levels must be 0, ±1. This means that the electron can transition to an orbital with the same magnetic quantum number or with a magnetic quantum number that differs by ±1.

3. Δn = ±1: The change in the principal quantum number (n) between the initial and final energy levels must be ±1. This means that the electron can transition to an energy level that is one level higher or one level lower.

These selection rules determine which transitions are allowed and which are forbidden in hydrogen-like atoms. For example, a transition from the n=2 energy level to the n=1 energy level is allowed, while a transition from the n=2 energy level to the n=3 energy level is forbidden.

Summary

The transition of electrons in hydrogen-like atoms is a fundamental concept in atomic physics. The electron can exist only in certain discrete energy levels, and transitions between these levels result in the emission or absorption of photons. The energy of the photon emitted or absorbed during a transition can be calculated using the energy difference between the initial and final energy levels. There are different types of transitions, including emission, absorption, and ionization. The selection rules for transitions in hydrogen-like atoms determine the allowed transitions based on the conservation of angular momentum and parity.

Type of Transition	Description
Emission	Electron transitions from higher to lower energy level, emitting a photon
Absorption	Electron absorbs a photon and transitions from lower to higher energy level
Ionization	Electron transitions from highest energy level to a level outside the atom, becoming a free electron

Example: An electron in a hydrogen-like atom is initially in the n=3 energy level. It undergoes an emission transition to the n=2 energy level. The energy difference between these levels is given by ΔE = E_3 - E_2. The emitted photon will have an energy equal to ΔE and a corresponding wavelength.

In conclusion, understanding the transition of electrons in hydrogen-like atoms is crucial for studying atomic physics. It involves the quantized energy levels of the electron, the emission and absorption of photons, and the selection rules that govern the allowed transitions.