Determinism, Photons, and Bell Inequalities: Quantum Polarization and the Demise of Classical Predictability

This lecture traces how light, once treated as a pure wave, reveals its quantum nature through photons and polarization. It covers Malus’s law for transmission through a polarizer, the indivisible nature of photons, and the shift from deterministic classical physics to probabilistic quantum predictions. The talk then motivates hidden-variable ideas and introduces Bell inequalities, arguing that quantum mechanics cannot be completed by local hidden variables. It concludes with a primer on quantum states as vectors, the idea of superposition for photon polarization, and the emergence of probabilistic outcomes for single photons.

Introduction: From Classical Light to Photons

The discussion begins with the historical tension between light as a wave and light as quanta. Photons are described as packets of energy with energy E = h nu, propagating as indivisible quanta rather than classical point particles. A beam consists of billions of photons, each carrying the same energy quantum, defined by the light’s frequency and wavelength.

Polarization, Polarizers, and Malus's Law

The core experiment centers on a polarizer with a preferential direction. Light linearly polarized along the x axis passes completely, while light polarized along the y axis is absorbed. For a general angle alpha, the transmitted fraction of energy is cos^2(alpha). This classical picture implies that each photon either passes or is absorbed, with a fixed outcome for identical photons.

Quantum Perspective: One Photon at a Time

In quantum terms, a beam of identical photons can be described by a single-photon state. A linearly polarized state at angle alpha can be written as a superposition of x and y polarizations. The act of passing through a polarizer along x yields an amplitude proportional to cos(alpha), and the transmitted energy fraction is cos^2(alpha). The quantum description treats each photon as an indivisible unit that either goes through or does not, rather than a fraction of a photon.

Determinism vs Quantum Probabilities

The key conceptual leap is recognizing that identical photons, prepared in the same way, can yield different outcomes when measured. Classical physics would predict a fixed fraction of photons passing through, but the quantum picture assigns probabilities to individual photon outcomes. This introduces a fundamental randomness at the level of single events, challenging deterministic viewpoints.

Hidden Variables and the Bell Story

To salvage determinism, some proposed hidden-variable theories, positing that photons carry extra properties that determine their fate at the polarizer. John Bell showed that no local hidden-variable theory can reproduce all quantum predictions. Experiments testing Bell inequalities have shown results incompatible with local hidden variables, reinforcing the quantum mechanical view and signaling the impossibility of restoring determinism in this framework.

States, Vectors, and Superposition

Dirac notation is introduced to describe photon polarization states. A photon polarized along x is written as |X>, along y as |Y>, and a general alpha direction as cos(alpha)|X> + sin(alpha)|Y>. The linearity of quantum mechanics allows superposition of states, which is essential for understanding interference and measurement outcomes. After a polarizer, the photon state is projected onto the transmitted basis, illustrating how quantum states evolve under measurement.

Bell Inequalities and Entanglement (Preview)

The discussion foreshadows how entanglement and Bell inequalities further illuminate non-classical correlations that cannot be explained by hidden variables. While the polarizer example concerns single photons, the principles extend to more complex two-photon experiments that reveal deeper quantum connections.

Takeaway

The lecture emphasizes that determinism, as traditionally conceived in Newtonian physics, is not upheld in quantum measurements of photons. The framework of quantum states as vectors, superposition, and probabilistic predictions provides a consistent and experimentally supported description of light, polarizers, and measurement outcomes that surpass classical determinism.