Exploring spontaneous emission through quantized electromagnetic fields
Following my previous post on absorption, which you can find here, I continue to explore the fascinating realm of quantum mechanics, specifically focusing on the phenomenon of spontaneous emission. This post aims to provide a detailed examination of how an electron in an excited state interacts with a quantized electromagnetic field, resulting in spontaneous emission, contrary to the behavior observed under semiclassical electromagnetic theories.
Introduction
In quantum mechanics, the behavior of particles is often counterintuitive, especially when transitioning from classical physics perspectives. One of the stark differences is observed in how electromagnetic fields are treated. In the semiclassical view, an electron in an excited state would theoretically remain in this state indefinitely because the electromagnetic field lacks the quantum fluctuations necessary to induce transitions between energy states. However, introducing a quantized electromagnetic field changes the scenario dramatically.
Quantum description of spontaneous emission
Consider an electron in an excited state denoted as state 2. In the absence of photons, the state of the system is represented as:
| \hat \psi \rangle = \mathbf b_2^\dag | \mathbf 0 \rangle
Here, E_q = E_b = E_2 represents the total energy of the system in this state. When a quantized field interacts with this electron, it introduces vacuum fluctuations and the possibility of spontaneous emission, leading to a transition to a lower energy state and the emission of a photon.
The perturbing Hamiltonian
The interaction of the electron with the quantized field can be described using the perturbing Hamiltonian, which in its action on the initial state yields:
\begin{aligned} \mathbf H_p | \hat \psi \rangle & = -\sum_{\lambda,j,k} \mathbf {\hat H}_{\lambda,jk} \delta_{k2} \mathbf b^\dag_j\mathbf a_\lambda^\dag | \mathbf 0 \rangle \end{aligned}
This equation highlights how the energy states interact under the influence of the quantized field, facilitating the transition by emitting a photon.
Transition probabilities and spontaneous emission
The transition probability for the electron to emit a photon and move to a lower energy state can be calculated using the Fermi Golden Rule:
w_r = \frac{2\pi}{\hbar}\left| \mathbf {\hat H}_{\lambda,j2} \right|^2 \delta\left(E_j - E_2 + \hbar\omega_\lambda\right)
This expression quantifies the likelihood of the electron’s transition from state 2 to state 1, accompanied by the emission of a photon with energy precisely matching the energy difference between these two states.
Conclusion
The concept of spontaneous emission in quantum mechanics illustrates the complex nature of particle interactions when exposed to a quantized electromagnetic field. Unlike in classical physics, where such processes are not readily explained, quantum mechanics provides a framework for understanding and predicting these events with remarkable precision. For more insights into this topic, you can find the details here.