Classical and Quantum

The Double-Slit Experiment

The motivation to resort to quantum theory is best understood by the double-slit experiment. As a reminder of the setup, there's a source of photons directed at two walls, and the first wall has two small slits that a photon can go through.

Consider a small region R on the second wall. Let P₁ be the probability a photon lands in R when only Slit 1 is open. Let P₂ be the probability a photon lands in R when only Slit 2 is open. When both slits are open, you’d think the total probability of hitting region R is P = P₁ + P₂. However, we observe that our expectation fails. Most bizarrely, P can decrease such that fewer photons land in R compared to when only one slit is open. What's more amazing is that we observe the same interference pattern even when we slow the photon source and send photons one-by-one!

Then to account for this phenomenon, instead of assuming probability preserves the 1-norm (total P = P₁ + P₂), what if we assume probability preserve the 2-norm? Let’s assume photons have complex-value amplitudes which add and subtract linearly (total A = A₁ + A₂), but the probability of observing some event is based on the square of the associated amplitudes (total P = |A|² = |A₁ + A₂|² = |A₁|² + |A₂|² + A₁A₂* + A₁*A₂). A₁A₂* + A₁*A₂ are cross terms that produce the unexpected "quantum" effect. The result of this inquiry is what's known as the Born Rule, a fundamental law of quantum mechanics.

The Born Rule is a postulate! You could just as easily imagine preserving the 3-norm or 4-norm, but the Born Rule fits the empirical observations. The justification for postulates can only be empirical, since they have no prior reason to fall back upon.

The Measurement Problem

Alright then, what if we watch the photons carefully? If we open up both slits, and keep track of which photons passes which slit, we can then plot the distribution for a single slit. Surely, we should expect to see a different outcome compared to when only one slit is open. That's not the case.

When we measure which slit photons goes through, the photons behave exactly like one slit is open. The final distribution of photons on the second wall looks like the sum of the distributions we get from either slit individually, in line with standard probability theory. Why does measuring the photons affect their behavior?

At the moment, we can only explain this as an update to the amplitudes of the possible events. It doesn't matter whether a sentient being measure the photon, all that matters is that the photon intereacts with the environment during what we call a "measurement." Measuring the photon going through Slit 1 "updates" the amplitudes so that A₂ = 0. Consequently, all quantum effects from the cross terms are eliminated.

Key Takeaway

After mathematical experimention, we now understand that we must take the Born Rule onboard as posulate. We apriori know that the 2-norm must be preserved through all quantum transformations. Unsatisfied? So am I, but it's currently the best we can do. However, it does allows us to make awesome inferences, some of which I'll present later!