Hamiltonian dynamics in electron-photon interactions
In my exploration of quantum mechanics, the interaction between light and matter presents one of the most fascinating studies. This blog post focuses on the Hamiltonian that describes electron-photon interactions, highlighting the sophisticated quantum mechanical processes that govern these fundamental interactions.
The quantum mechanical Hamiltonian
The classical view of the electric dipole given by the Hamiltonian:
\mathbf H = e \mathbf E \cdot \mathbf r
serves as the starting point in describing the interaction between an electric field and charged particles. In quantum electrodynamics (QED), this expression evolves into a more complex form once the quantum nature of the electric field is considered. By incorporating the electric field operator, the Hamiltonian for a set of electrons interacting with a quantized electromagnetic field is given by:
\mathbf H = \sum_{n=1}^N e i\sum_\lambda \left(\mathbf a_\lambda - \mathbf a_\lambda^\dag\right) \sqrt{\frac{\hbar \omega_\lambda}{2 \varepsilon_0}} \mathbf u_\lambda(\mathbf r_n) \cdot \mathbf r_n
This expression uses the creation and annihilation operators (\mathbf a_\lambda^\dag, \mathbf a_\lambda) that are fundamental in the second quantization formalism, which extends the concept to fields of quantum particles.
Transition to occupation number representation
To express the Hamiltonian in the occupation number representation, it is necessary to introduce the wavefunction operator \hat \psi and integrate over the fermionic field:
\mathbf {\hat H} = \sum_\lambda \left(\mathbf a_\lambda - \mathbf a_\lambda^\dag\right) \int \hat \psi^\dag \sum_{n=1}^N e i\sqrt{\frac{\hbar \omega_\lambda}{2 \varepsilon_0}} \mathbf u_\lambda(\mathbf r_n) \cdot \mathbf r_n \hat \psi \mathrm d^3 \mathbf r_1 \dots \mathrm d^3 \mathbf r_N
This formulation allows us to explore how individual photons interact with the fermionic quantum field, manifesting as observable phenomena such as absorption and emission of light.
Fundamental processes: absorption and emission
In QED, electron-photon interactions can be characterized by three fundamental processes: absorption, stimulated emission, and spontaneous emission. These are critical in various applications, from lasers to quantum computing.
Absorption
Absorption involves a photon being absorbed by an electron, which transitions to a higher energy state. Mathematically, this process can be represented as:
\mathbf{a}_\lambda^\dag |n\rangle \rightarrow |n+1\rangle
Stimulated emission
Conversely, stimulated emission occurs when an electron in a higher energy state emits a photon and transitions to a lower energy state, a process that can be induced by an incoming photon:
\mathbf{a}_\lambda |n\rangle \rightarrow |n-1\rangle
Spontaneous emission
Spontaneous emission describes an electron independently transitioning from a higher to a lower energy state, emitting a photon in the process. This is inherent to the quantum system and is described as:
|n\rangle \rightarrow \mathbf{a}_\lambda |n\rangle = \sqrt{n} |n-1\rangle
These processes are succinctly captured in the Hamiltonian framework, which can be expanded to include multiple particles and interaction terms.
Conclusion
The exploration of electron-photon interactions through the Hamiltonian provides profound insights into the quantum mechanical behavior of matter under electromagnetic influence. By dissecting the Hamiltonian and the quantum states involved, we can better understand the quantum dance between electrons and photons, which plays a crucial role in the field of quantum electrodynamics.
For more insights into this topic, you can find the details here.