Some light quantum mechanics (with minutephysics)

Quantum mechanics’ most counterintuitive feature—probabilities replacing classical “splits” of energy—can be built from the ordinary wave physics of light. Starting with Maxwell’s electromagnetic waves, the discussion shows how superposition, amplitudes, phases, and choice of basis already appear in classical polarization. The key shift comes when experiments reveal that light energy is quantized in indivisible packets: a photon carries a fixed amount of energy hf, so it cannot divide its energy between polarization components the way a classical wave would. That mismatch forces the squared amplitudes of a superposition to be interpreted not as energy fractions, but as probabilities for where the photon’s entire energy will be found after measurement.

The foundation begins in the late-1800s picture of light as coupled electric and magnetic vector fields. Maxwell’s equations imply that changing electric fields generate changing magnetic fields and vice versa, producing propagating electromagnetic waves whose electric and magnetic components oscillate perpendicular to each other and to the direction of travel. With that classical wave machinery in place, polarization becomes a clean laboratory for quantum-style math: a horizontally polarized electric field can be written as a cosine oscillation with frequency f, amplitude a, and phase shift ϕ, while vertically polarized light uses the orthogonal component. Because Maxwell’s equations are linear in a vacuum, two solutions can be added to form a superposition—exactly the same mathematical operation that later becomes central to quantum state descriptions.

Superposition in polarization naturally introduces phase-sensitive behavior. When horizontal and vertical components are in phase, the resulting polarization can be described in a rotated basis (for example, diagonal versus anti-diagonal). When the phase difference is 90 degrees and amplitudes match, the wave becomes circularly polarized. The choice of basis matters because different measurement devices “project” onto different directions: a polarizing filter absorbs energy from one polarization component, so it’s convenient to describe the incoming state in the filter’s aligned basis.

The quantum turn arrives with energy quantization. Classical wave energy scales like the square of amplitude, so a superposition with horizontal amplitude Ax and vertical amplitude Ay would suggest energy proportional to Ax² + Ay², and—crucially—energy could be continuously redistributed between components. Experiments instead indicate that electromagnetic energy comes in discrete amounts: for frequency f, the smallest nonzero energy is hf (Planck’s constant h times f). A photon therefore cannot exist as “half a photon” in one polarization and “half a photon” in another. When a diagonally polarized photon hits a vertically oriented polarizer, the classical expectation of partial absorption gives way to an all-or-nothing outcome: roughly half the photons pass and half are absorbed, with the passing photons emerging polarized along the filter.

That same logic extends to intermediate angles. For a photon whose polarization is 22.5 degrees off a filter, classical amplitude-squared reasoning predicts about 15% of the energy would align with the blocked component (sin² 22.5° ≈ 0.15). The quantum observation instead is probabilistic: about 15% of photons are fully blocked and about 85% pass entirely. The wave equations remain intact—superposition with amplitudes and phases still describes the state—but the interpretation changes. Squared amplitudes now encode probabilities for measurement outcomes, not divisible energy shares. In this way, the familiar wave math of polarization becomes a stepping stone to the broader quantum rule: quantum states are superpositions in a chosen basis, and measurement yields outcomes with probabilities proportional to the squared magnitudes of the corresponding amplitudes.