Multimode radiation: observables and photoelectric effect
In my recent studies of multimode radiation, I have focused on the aspects of Hamiltonian eigenstates and their integral role as a complete basis for quantum mechanical systems. The eigenstates of a quantum system define the state space in which the system’s dynamics are described, a crucial concept for any quantum physicist.
Hamiltonian and eigenstates
The total Hamiltonian for a radiation field can be expressed as:
\mathbf{H}_R = \sum_\lambda \frac{\hbar \omega_\lambda}{2} \left(\mathbf{a}_\lambda^\dag \mathbf{a}_\lambda + \frac{1}{2}\right)
where \mathbf{a}_\lambda^\dagger and \mathbf{a}_\lambda are the creation and annihilation operators, respectively, satisfying the commutation relation:
[\mathbf{a}_i, \mathbf{a}_j^\dag] = i\delta_{ij}
These operators play a pivotal role in defining the eigenstates, known as Fock states, for each mode of the field.
Fock States and Renormalization
A Fock state | n_1, \dots, n_\lambda, \dots \rangle represents a state with a definite number of particles (photons) in each mode. The dimension of this state space grows exponentially, a challenge for computational and analytical treatments. To manage the infinities arising in energy calculations for these states, I apply renormalization techniques, which adjust the observed values to account for the vacuum energy:
E_V \equiv \sum_\lambda \frac{1}{2} \hbar \omega_\lambda | \mathbf{0} \rangle
Momentum and Field Observables
The linear momentum of the radiation field, expressed through the Poynting vector, also takes a quantized form:
\mathbf{P}_R = \sum_\lambda \hbar \mathbf{k}_\lambda \mathbf{a}_\lambda^\dagger \mathbf{a}_\lambda
The fields themselves, particularly the electric and magnetic fields, are expressed in terms of the creation and annihilation operators, making them measurable quantum observables. These field observables can exhibit quantum fluctuations, evident in phenomena such as the Lamb shift or the Casimir effect.
Photoelectric Effect and Detection
The photoelectric effect provides a direct method to measure the effect of radiation on matter, leading to electron emission by incident photons. In multimode radiation, the probability of detecting photons is not merely a product of single detections; it involves complex correlations between different modes and times:
w^{(1)}(\mathbf{r}, t) = \eta \frac{c}{L^3} \left|\sum_\lambda \mathbf{e}_\lambda \mathbf{a}_\lambda e^{i\mathbf{k}_\lambda \cdot \mathbf{r}} | \psi(t) \rangle \right|^2
Heisenberg Formalism
To manage the temporal dynamics of these measurements, I utilize the Heisenberg formalism, where operators evolve with time, keeping the state vector constant. This approach is vital for computing the outcomes of measurements at different times and under varying experimental conditions.
In conclusion, the study of multimode radiation through Hamiltonian eigenstates, Fock states, and field observables provides profound insights into the quantum nature of light and its interaction with matter.
For more insights into this topic, you can find the details here.