Quantum Mechanics


May 27, 2020

Quantum mechanics is the field of physics that seeks to understand the movement and interaction of subatomic particles, such as quarks, hadrons, and gluons. In 1926, Erwin Schrödinger made a significant contribution to quantum mechanics when he successfully derived the psi-function. Later coined Schrödinger’s Equation, the psi-function describes the wave function of a quantum mechanical system. In describing the world with the psi-function, Schrödinger contributed one of the deepest philosophical problems of our time.

Gloss on Quantum Mechanics

In quantum mechanics, the world can be described in states. These states are mathematically described by Schrödinger’s Equation, which predicts the state of the world, or of a system, at some point in time. The crux of confusion in quantum mechanics arises from the strange reality that the mathematics predicts: a system can be in a superposition of states of the system, i.e. a superposition of possible properties of the system.

It’s unclear what superposition really means. On first glance, superposition seems to imply that our reality is in multiple states at once. However, that’s not the type of reality we live in, or at least the type of reality we think that we live in. Indeed, our experience is in deep contradiction with the very idea of superposition. We believe that there are determinate facts about the properties of objects that we observe, such as the color of a ball. We don’t see things as in superposition, nevertheless understand really what the idea means to begin with.

One poignant way to understand the problem of superposition is to think about the implications of superposition of a system involving a human observer: Schrödinger’s Equation leads to a state in which our beliefs can be in a superposition, and therefore there seems to be no fact about what the observer believes to be true. That’s in direct contradiction with one of the seemingly most reliable sources of information, the mind.

We reconcile the dynamical predictions of Schrödinger’s Equation by the collapse postulate. The collapse postulate claims that a state in a superposition probabilistically collapses to one of the constituent states with a probability dictated by the Born Rule. For example, the ball will collapse to either the state of being black or the state of being white with some specified probability calculated from that system’s wave function. Under collapse, then, we could be right about our experience of reality.

The Measurement Problem

The dynamics of quantum mechanics, stipulated by Schrödinger’s Equation, is fully deterministic, stating with mathematical certainty the state of a system at a particular point in time. However, the collapse postulate is probabilistic, painting a probabilistic picture of how the world will be at some point in time. The Measurement Problem asks how, then, the deterministic dynamics and the probabilistic collapse postulate can be consistent.

Bell's Dichotomy

Physicist John Bell thought there were two possible solutions to the Measurement Problem.
1. Schrödinger’s Equation is incorrect and does not contain all of the information needed to represent the state of the world. This view is embodied by the GRW Theory, in which an additional collapse postulate is added to Schrödinger’s Equation.
2. Schrödinger’s Equation is correct but needs an additional, separate postulate to ade

Ghirardi, Rimini, Weber (GRW) Theory

In 1986, Ghirardi, Rimini, and Weber (GRW) proposed a way to think about one of Bell’s possible solutions, modifying the wave function so that the wave function is a complete physical description of the world. Specifically, to fix the Measurement Problem, GRW proposed that the wave function itself needs an additional term which dictates that collapses occur with a fixed, small, law-like probability per unit time. This probability is proposed in such a way that the predictions given by the collapse follows our macroscopic observations of the world.

The GRW Theory is appealing because the probability proposed by GRW is consistent with empirical evidence, which is important because we have strong, repeated experimentation to support certain properties of quantum mechanics. On the macroscopic scale that humans interact on, it turns out that these predicted collapses occur very frequently, and thus we can be comfortable with viewing our world determinately on the human scale.

However, there are still some issues here. To follow the Uncertainty Principle, the GRW Theory follows a slightly different definition of collapse than we might think. When a collapse occurs, states actually have to still be in superpositions to make energy laws work, but the states simply become more similar (if you’re familiar with linear algebra, then think about the states as vectors, and the collapses make the vectors closer together). This implies that superposition persists, and it is unclear how there can be the factual certainty that we want.

Bohm's Theory

Bohm’s Theory takes an epistemic approach to the Measurement Problem. Theoretically, every particle has a determinate position, but as humans, we do not have access to all of the information necessary to determine the position of all particles. Therefore, probability in quantum mechanics exists as an epistemic idea because humans are ignorant about the exact state of the world.

Bohm’s Theory envisions the wave function given by Schrödinger’s Equation as correct, real, and physical. Under Bohm’s Theory, the wave function evolves with the dynamics and is theoretically able to be calculated with certainty at time t from some previous time t-1. The epistemic uncertainty shrouded as probability arises because we do not know the values of particles at time 0 (the beginning of the universe); if we did, then we would know with certainty the position of particles and wave function at any given time.

To fix the Measurement Problem, Bohm’s Theory makes a special additional postulate about what occurs after measurement when a system is in a superposition. After measurement, an effective collapse occurs. In an effective collapse, the other terms of the wave function (the other possible states of the world) equal to 0, but the wave function is still in a superposition. Therefore, all that we know about the state of a system is its effective wave function, and measurement simply narrows down the possibilities of position. In this way, Bohm’s Theory is appealing because it is in line with both our macroscopic and microscopic observations.

One important note is that Bohm’s Theory entails a foreign asymmetry to dynamics, in which the wave function dictates the motion of particles but the motion of particles does not dictate the wave function. There is something unintuitive about this notion, and many mention this worry as a possible objection to Bohm’s Theory.

Everett's Many Worlds Theory

Everett’s proposal doesn’t fall on either end of Bell’s possible solutions. Everett tries to stick with the predictions of Schrödinger’s Equation; unlike the GRW Theory or Bohm’s Theory, Schrödinger’s Equation is neither incorrect nor incomplete. Instead, he entertains a radically different explanation: superpositions are moments in which the world branches into multiple possibilities. When we make a measurement, we fall onto a particular branch of the world, unaware of the other branches.

One way to read Everett’s proposal is that Everett accepts that observers can be wrong about introspective reports, and that the Measurement Problem doesn’t really make sense in the first place.

A large epistemic problem arises with the notion of probability. It’s unclear what probability means in the context of Many Worlds, since each branch appears with certainty. Philosophical problems in statistics is a rich, adjacent field to philosophical problems in quantum mechanics.