Quantum Formalism

What is the quantum formalism, precisely, and how does it follow from Bohmian mechanics?

Quantum theory is a very successful theory. But it seems to involve collapse of the wave function when a measurement is performed, eigenvalues of operators appearing as the results of measurements, and a probability rule governing the distribution of the results. How do these fit in with Bohmian mechanics?

The textbook quantum mechanics is what we call the quantum formalism. The formalism is a set of rules which give us the experimental predictions. No one suggests solving Bohmian equations of motion to compute out experimental predictions except, perhaps, in special circumstances.

Instead, we want to derive the rules from the equations of motion of Bohmian mechanics. That’s right. Given Schrödinger’s equation and Bohm’s equation, we can derive the effective collapse of system wave functions, clarify the role of operators in measurements, and explain the role of probability in this deterministic theory.

The key to all of this is to understand the difference between a system under consideration (some experimental setup) and the system encompassing everything. Each experiment only has an effective wave function; generally subsystems do not have effective wave functions. They do precisely in the situations under consideration in textbook quantum mechanics. So that effective wave function, being a mathematical fiction, albeit a useful one, can behave in odd ways if the larger dynamics that it is embedded in deems it so.

Equivariance, which is the preservation of $|\psi|^2$ probability under the particle evolution, is the central mathematical fact governing the results.

Quantum Equilibrium Hypothesis

Why do we demand that the particles are distributed according to $|\psi|^2$? What does that mean for a deterministic theory?

Operators as Observables

A quantum measurement is nicely summarized by operators acting on the space of wave functions. Why? Why do eigenvalues appear as results? What does that even mean?

Collapse of the Wave Function

Schrodinger’s equation never collapses the wave function, but yet we see what looks like a collapse of the wave function into eigenstates of the operators. How can this arise from Bohmian mechanics? Why is the probability distribution of the results $|\psi|^2$?

Examples of Quantum Measurements

What do momentum “measurements” measure in Bohmian mechanics? What about spin measurements? How do the eigenvalues of the Hamiltonian get revealed by the dynamics of the particles? What happens in the ground state?

The Uncertainty Principle

In a perfectly deterministic theory, what role does uncertainty have? How can there be an absolute limit to our knowledge when the model gives us a precise evolution?