Quantum physics: predicting one photon

Our question and scope

If we prepare the same light again and again, why can we predict the chances without knowing every result? We explore one property of a photon, a quantum of light: its polarization. This is an introduction to states, probabilities, and measurement through ideal linear polarizers. It does not cover atomic energy levels, entanglement, or every meaning of quantum uncertainty. Source: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4, The polarization states of the photon — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

A prediction has conditions

We assume individually prepared photons, ideal filters, and an ideal detector after the final filter. We ignore absorption unrelated to polarization, detector inefficiency, background counts, and any other changes between filters. The percentages in this guide are calculated model predictions, not experimental measurements. Real apparatus needs those imperfections accounted for.

What polarization means

For linearly polarized light, polarization specifies the orientation of the oscillating electric field, perpendicular to the direction of travel. For one photon, we use the corresponding polarization state; do not picture a little arrow wobbling inside a ball. Vertical here is a reference direction, not the direction the photon travels. Source: University of Waterloo, Institute for Quantum Computing, Light polarization — https://uwaterloo.ca/institute-for-quantum-computing/resources/teacher-resources/download/light-polarization

Why the starting point matters

The preparation filter may reject incoming photons. We count only those that have passed it, so the later probabilities do not include preparation losses. A transmitted photon has the preparation filter's polarization state. Source: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

State, amplitude, and superposition

A state is the mathematical description used to predict measurement results. Its amplitudes are numbers associated with possible outcomes; their absolute squares give probabilities, the Born rule. Relative to axes at 45° and 135°, a vertically prepared photon is a superposition: a linear combination of those two polarization states. Superposition depends on the basis used to describe the state. Source: IBM Quantum Learning, Quantum information, Quantum state vectors and Measuring quantum states — https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information

What the analyzer does

An analyzer is a second polarizing filter used as a test. In our ideal absorbing-filter model, a photon either passes or is absorbed. The downstream detector records transmitted photons. The labels describe the analyzer's transmission axis relative to the 0° preparation. Source: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

Read the axes and probabilities

This is a direct schematic of an ideal preparation–analyzer–detector experiment, not a scale drawing. The axis inside each filter symbol is its transmission direction; the horizontal arrows show light travel. The analyzer slider changes that angle and the calculated pass/absorb probabilities together. No dot motion, detector click, hidden photon path, or measured count is implied.

One checked rule for every condition

For an ideal linear polarizer, transmission probability is cos²(θ), with θ the angle difference between preparation and analyzer. At 0°, 45°, and 90°, this gives 1, 1/2, and 0: 100%, 50%, and 0%. These are theoretical predictions, not measured counts. Classical waves also give this relationship for transmitted intensity, so this comparison alone does not distinguish quantum theory from classical wave optics. Source: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

Probability belongs to possible outcomes

In this ideal polarization test, the photon is transmitted or absorbed. The filter does not transmit half of each photon. The quantum model assigns probabilities to those alternatives; it does not specify which result the next 45° trial will give. Source: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

Check the arithmetic and the assumption

For 100 independently prepared photons tested at 45°, the expected number transmitted is 100 × 0.5 = 50 photons. This is a calculated example, not experimental data. Expectation is an average over repeated batches; it does not require each batch to contain exactly 50 passes or the outcomes to alternate. Independence and unchanged apparatus are assumptions of this example.

Why repetition helps

Repeated preparations let us compare observed frequencies with the state's predicted probabilities. One result does not reveal the whole probability distribution. Source: IBM Quantum Learning, Quantum information, Measuring quantum states — https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information

Multiply the conditional probabilities

Both percentages start after the 0° preparation. Direct transmission is cos²(90°) = 0. With a middle filter, the first transmission probability is cos²(45°) = 1/2. A surviving photon is now polarized at 45°, so the next angle difference is also 45°. Final transmission is (1/2) × (1/2) = 1/4 = 25% of the original prepared photons. This is a derivation within the ideal model. If instead starting with unpolarized light before preparation, its extra factor of 1/2 makes the three-filter total 12.5%. Source for the filter rule and transmitted state: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

What changed, and what did not

The middle filter rejects some photons and prepares the survivors in its own polarization state. It does not add photons or energy to the transmitted light. Adding a filter changes the sequence of states being tested. Classical wave optics also predicts this intensity effect; it is not by itself proof that light lacks a classical description.

What observer means here

In this model, testing means interaction with a polarizer and recording transmission with a detector. A person can read the record later; looking at it is not an input to cos²(θ). This operational statement avoids claims about consciousness or a particular interpretation of quantum mechanics. Source for the apparatus-based polarization model: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

Do not turn state change into an absolute rule

A measurement need not make every later result uncertain. A photon transmitted at 45° is in a state that an immediate ideal 45° filter transmits with certainty, assuming no intervening change. Different measurement choices and different states can produce certainty or uncertainty. Source: IBM Quantum Learning, Quantum information, Measuring quantum states — https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information

Conditional on having passed the first filter, the photon is polarized at 45°. The next ideal filter has the same axis: cos²(0°) = 1.

Check the denominator

The question starts with a photon that has passed the first 45° filter. From an earlier 0°-prepared input, half would reach that point, and all of those survivors would pass the second 45° filter in the ideal model. The overall probability from that earlier input would therefore be 50%, while the conditional probability asked here is 100%. Source: Caltech, The Feynman Lectures on Physics, Vol. III, §11-4 — https://www.feynmanlectures.caltech.edu/III_11.html#Ch11-S4

A useful beginning, not the whole subject

The same photon preparation can yield certainty for one polarization test and uncertainty for another. That is a useful first mental model for quantum predictions. More general quantum states include relative phases, which matter when combining alternatives; a list of probabilities for just one test is not a complete state description. Source: IBM Quantum Learning, Quantum information — https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information

A next question

How do quantum amplitudes combine when there is more than one indistinguishable alternative, as in interference? That question goes beyond the filter probabilities explored here. Keep preparation, interaction, outcome, and repeated-trial evidence distinct as you learn further.