Question

Assume two energy levels of a gas laser are separated by \(1.4 \mathrm{eV},\) and assume that they are equally degenerate \(\left(g_{1}=g_{2}\right)\). The spontaneous emission Einstein coefficient for transitions between these energy levels is given by \(A_{12}=3 \cdot 10^{6} \mathrm{~s}^{-1}\). Find the other two Einstein coefficients, \(B_{12}\) and \(B_{21}\).

Short answer

\( B_{12} = B_{21} = 4.93 \times 10^{-12} \text{ m}^3 \text{s J}^{-1} \).

Step-by-step solution

Understanding Einstein coefficients

Einstein coefficients describe the probability of different types of transitions between energy levels in a quantum system. For a pair of energy levels, the coefficients are: \( A_{12} \) (spontaneous emission), \( B_{12} \) (stimulated absorption), and \( B_{21} \) (stimulated emission).

Using the relationship between Einstein coefficients

The relation between \( A_{12} \), \( B_{12} \), and \( B_{21} \) is given by the equations:1. \( \frac{A_{12}}{B_{12}} = \frac{8 \pi h u^3}{c^3} \)2. \( B_{12} = B_{21} \) (since \( g_1 = g_2 \)).Where \( h \) is Planck's constant, \( u \) is the frequency of the emitted photon, and \( c \) is the speed of light.

Calculating the frequency \( \nu \)

We first find the frequency \( u \) using the energy difference \( \Delta E \): \( \Delta E = h u = 1.4 \text{ eV} \).Convert the energy from eV to joules: \( 1.4 \text{ eV} = 1.4 \times 1.602 \times 10^{-19} \text{ J} \).Find \( u \):\( u = \frac{1.4 \times 1.602 \times 10^{-19}}{6.626 \times 10^{-34}} \).

Calculating \( B_{12} \) using the known \( A_{12} \)

Substitute the known \( A_{12} \) and calculated \( u \) into the relation:\( B_{12} = \frac{A_{12} c^3}{8 \pi h u^3} \).Using \( A_{12} = 3 \times 10^6 \text{ s}^{-1} \), \( h = 6.626 \times 10^{-34} \text{ J s} \), and \( c = 3 \times 10^8 \text{ m/s} \), find \( B_{12} \).

Obtaining \( B_{21} \) from \( B_{12} \)

Since \( B_{12} = B_{21} \), use the value of \( B_{12} \) obtained in the previous step for \( B_{21} \). This equality holds because the degeneracies \( g_1 \) and \( g_2 \) are equal.

Key concepts

Understanding Spontaneous Emission in Quantum Systems

Spontaneous emission is a fundamental concept in quantum physics. It plays a crucial role in how atoms and molecules transition between energy levels. When an atom is in a higher energy state, it can spontaneously emit a photon without any external influence. This photon release results from the inherent instability of an excited state. The Einstein coefficient for spontaneous emission, denoted as \( A_{12} \), quantifies this process. It represents the probability per unit time that an atom in a higher energy state will drop to a lower energy state, emitting a photon. Key points about spontaneous emission include:

• Occurs naturally without external cause.
• Chances of emission are determined by \( A_{12} \).
• Contributes to the natural linewidth of spectral lines.

Learning about spontaneous emission helps in understanding phenomena such as the luminescence of materials and the workings of devices like lasers and LEDs. Recognizing its role in quantum mechanics provides insight into the behavior of atoms and photons in various applications.

Exploring Stimulated Absorption and its Significance

Stimulated absorption is another vital mechanism of quantum transitions. Unlike spontaneous emission, this process requires external intervention. Atoms can move from a lower energy state to a higher energy state when they absorb photons whose energy matches the energy gap between these states. The Einstein coefficient \( B_{12} \) describes the likelihood of this transition. It relates the probability of absorption to the intensity of incoming radiation:

• Poppulation transitions solely occur with incoming photons.
• The rate of absorption is proportional to the radiation density.
• A higher intensity leads to more stimulated absorptions.

Stimulated absorption is crucial for developing technologies like lasers, where it is utilized to pump atoms to excited states. This process, combined with stimulated emission, sustains the laser beam's strength and coherence. Understanding \( B_{12} \) furthers our grasp of how energy is harnessed for technology innovation.

The Role of Quantum Transitions in Atomic Dynamics

Quantum transitions are the foundational movements between different energy states within an atom or molecule. These transitions can be of different types: spontaneous emission, stimulated absorption, and stimulated emission. All of these are described by Einstein coefficients, which mathematically frame how transitions occur. The concept of quantum transitions provides insights into:

• The nature of atomic excitations and relaxations.
• How photons interact with atoms to transfer energy.
• The principles behind practical applications like spectroscopy and laser technology.

Understanding quantum transitions is crucial because it aids in explaining observable phenomena like absorption and emission spectra. It helps in designing technologies that require precise energy control, enabling advancements in fields such as telecommunications and medicine. By exploring these transitions, we appreciate the interconnected nature of particles, energy, and light in a quantum world, laying the groundwork for progressing scientific inquiry and technological development.