Mathematical Description of the Photon Wave Function

In classical quantum mechanics, the wave function $\psi(\mathbf{r}, t)$ serves as the fundamental descriptor of a particle, where its squared modulus $|\psi|^2$ provides the probability density of finding a particle at a specific location. However, when we transition to the study of photons, this traditional framework encounters significant theoretical hurdles.

Unlike electrons or protons, photons are massless, relativistic particles that always travel at the speed of light. Because they lack a rest frame, the standard non-relativistic Schrödinger equation—$\hat{H}\psi = i\hbar \frac{\partial \psi}{\partial t}$—is fundamentally inadequate for describing their spatio-temporal evolution. Instead, the physics of photons is rooted in the quantization of the electromagnetic field, a process that leads us away from simple spatial wave functions and toward the more robust framework of Quantum Electrodynamics (QED) and Quantum Field Theory (QFT).

The Shift to Hilbert Space and State Vectors

In the quantum optical regime, we move away from describing a photon as a point-like particle moving through space. Instead, we describe the state of the electromagnetic field using state vectors within a complex Hilbert space. For a single mode of the field, the state is not defined by a position-based function, but by its occupation of specific modes.

The most intuitive way to bridge the gap between the discrete nature of photons and the continuous nature of classical waves is through the use of Coherent States.

Coherent States: The Bridge to Classicality

A coherent state, denoted as $|\alpha\rangle$, represents a state of the electromagnetic field that most closely resembles a classical monochromatic wave. Mathematically, a coherent state is defined as the eigenstate of the annihilation operator $\hat{a}$:

$$\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$$

where $\alpha$ is a complex number representing the amplitude and phase of the field. To understand the composition of such a state, we can expand it in terms of the Fock states (or photon number states) $|n\rangle$:

$$ |\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle $$

This expansion reveals several critical insights:

• Photon Number Distribution: A coherent state is not a state of definite particle number. Instead, it is a superposition of different number states, where the probability of finding $n$ photons follows a Poisson distribution.
• The Classical Limit: As the magnitude $|\alpha|$ becomes large, the relative fluctuation in photon number ($\Delta n / n \approx 1/\sqrt{n}$) becomes negligibly small. In this limit, the quantum uncertainties in amplitude and phase vanish, allowing the state to behave like a stable, classical electromagnetic wave—this is the mathematical foundation for the operation of lasers.

Statistical Descriptions: Thermal vs. Coherent Light

While coherent states describe highly ordered light, many physical systems involve "noisy" or disordered light. To account for these, we employ the density operator $\hat{\rho}$, which allows us to describe mixed states rather than just pure states.

A primary example is the thermal light field. Unlike the Poissonian statistics of a laser, thermal light follows Bose-Einstein statistics, characterized by much higher fluctuations (super-Poissonian statistics). The density operator for a single-mode thermal field is expressed as:

$$ \hat{\rho}{th} = \frac{1}{\bar{n}+1} \sum{n=0}^{\infty} \left( \frac{\bar{n}}{\bar{n}+1} \right)^n |n\rangle\langle n| $$

where $\bar{n}$ is the average photon number. This mathematical structure explains why thermal sources lack the stable phase required for high-precision interference, a stark contrast to the phase-coherent nature of the $|\alpha\rangle$ state.

Recovering the Wave: The Electric Field Operator

If the "wave function" of a photon is an abstract vector in Hilbert space, how do we recover the familiar oscillating waves of Maxwell's equations? The answer lies in the expectation values of field operators.

In quantum optics, we focus on the electric field operator $\hat{E}^{(+)}(\mathbf{r}, t)$. When we calculate the expectation value of this operator for a coherent state, we obtain:

$$ \langle \alpha | \hat{E}^{(+)}(\mathbf{r}, t) | \alpha \rangle \propto \text{Re}[\alpha e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}] $$

This result is profound: it demonstrates that the classical wave behavior—the sinusoidal oscillation of the electric field—emerges naturally from the statistical properties of a large collection of photons in a coherent state. The "wave" is not a physical medium, but the observable manifestation of the underlying quantum state.

The Normalization Dilemma and Wave Packets

A final technical challenge involves the normalization of the photon state. In standard quantum mechanics, we require $\int |\psi|^2 d^3r = 1$ to ensure total probability is conserved. However, for photons, the number of particles is not conserved in open systems, and the energy spectrum is continuous, making strict spatial normalization problematic.

To circumvent this, physicists often describe single photons using momentum-space wave packets. Instead of a position-based $\psi(\mathbf{r})$, we define a state as a superposition of momentum modes:

$$ |\psi\rangle = \int d^3k , f(\mathbf{k}) |1_{\mathbf{k}}\rangle $$

Here, $f(\mathbf{k})$ acts as the weight function in momentum space, describing the distribution of the photon's momentum. This approach allows for a mathematically consistent description of localized single photons, such as those used in quantum key distribution (QKD) and quantum computing.

Conclusion

The mathematical description of the photon wave function is a transition from the intuitive "particle-in-a-box" mechanics to the sophisticated language of operator algebra and Hilbert spaces. By treating the photon not as a localized point, but as an excitation of a field, we gain the ability to describe everything from the chaotic fluctuations of a lightbulb to the exquisite coherence of a laser. Understanding these mathematical structures is essential for anyone venturing into the frontiers of quantum communication, precision metrology, and the fundamental study of light-matter interaction.