Unraveling photon absorption and electron transition
In my recent work, I have explored the fascinating quantum mechanics of photon absorption, a fundamental process that occurs when an electron absorbs a photon and transitions to a higher energy state. This analysis is rooted in the principles of perturbation theory, which helps to understand the microscopic interactions underpinning quantum phenomena.
Initial state and perturbation theory
The scenario begins with an electron in state 1 and a photon in mode 1. The initial state can be represented as:
\hat \psi = \mathbf b_1^\dag \mathbf a_1^\dag | \mathbf 0 \rangle
where \mathbf b_1^\dag and \mathbf a_1^\dag are the creation operators for the electron and photon, respectively. This setup allows us to calculate the total energy of the system:
E_q = E_b + E_f = E_1 + \hbar\omega_1
where E_1 is the energy of the electron in state 1, and \hbar\omega_1 is the energy of the photon.
To examine the interaction, I apply a perturbing Hamiltonian, expressed as:
\mathbf H_p = \sum_{\lambda,j,k} \mathbf {\hat H}_{\lambda,jk} \mathbf b^\dag_j \mathbf b_k\left(\mathbf a_\lambda - \mathbf a_\lambda^\dag\right)
This Hamiltonian perturbs the initial state, leading to potential transitions driven by the absorption of the photon.
Quantum transition analysis
Through careful manipulation and applying commutation relations, the perturbed state becomes:
\mathbf H_p \hat \psi = \sum_{\lambda,j,k} \mathbf {\hat H}_{\lambda,jk} \left(\delta_{k1} \delta_{\lambda1} \mathbf b^\dag_j| \mathbf 0 \rangle - \delta_{k1} \mathbf b^\dag_j\mathbf a_\lambda^\dag\mathbf a_1^\dag | \mathbf 0 \rangle\right)
This result highlights the primary transitions, where the electron may end up in a different state and the photon is either absorbed or leads to the creation of additional photons.
Calculating transition rates
Applying Fermi’s Golden Rule, the transition rate for an electron absorbing a photon and transitioning from state 1 to state 2 can be derived:
w_r = \frac{2\pi}{\hbar}\left| \mathbf {\hat H}_{1,j1} \right|^2 \delta\left(E_2 - E_1 - \hbar\omega_1\right)
This formula provides a quantitative measure of the likelihood of the electron’s transition, dependent on the energy conservation between the initial and final states.
Conclusion and further applications
The insights from this analysis not only deepen our understanding of quantum transitions but also provide a foundation for further research into quantum dynamics and photon-electron interactions. The precise conditions under which these transitions occur help in designing experiments and technologies based on quantum mechanics.
For more insights into this topic, you can find the details here.