Hidden Variables

My assigned prompt in G-Chem 2 this past week has students scratching their heads. I tell them that the equations we’ve been using in thermodynamics and kinetics were elucidated in the nineteenth century, before the widespread belief in tiny invisible entities known as atoms and molecules. But everything we discuss in class on these topics has been suffused with molecular language. Is it possible to imagine developing and using these equations without knowing about the atomic world?

 

I’ve had some thoughtful responses so far. Students have recognized that to some extent it is easier to connect something you observe with potentially vague yet macroscopic quantities such as pressure and temperature. They’ve also started to muse on how one grafts new ‘theories’ on to existing equations that work. Essentially, previously hidden variables are being brought to light!

 

A similar narrative arc is how Manjit Kumar ends his book, Quantum, that focuses on the debate between Bohr and Einstein (and their supporters and detractors) on the nature of reality in the physical world. This, I think, distinguishes Kumar’s telling of the tale from the many other histories I’ve read about the development of quantum mechanics. The reader gets the sense of why the Copenhagen interpretation reigned supreme for many years, but also where it has proved less than satisfying. Kumar argues effectively that Einstein was not so much against the probabilistic nature of quantum mechanics (associated with his famous “God does not play dice” statement), but instead was philosophically wedded to the idea that nature has an independent reality not necessarily dependent on the observer.

 

Einstein thought quantum mechanics was incomplete, and found anathema Bohr’s view of reality being undefinable until the act of observation causes the superposition of states in the wavefunction to ‘collapse’. In essence, Einstein would have supported the idea that there are hidden variables to be discovered, that would underpin quantum mechanics. He didn’t propose hidden variables explicitly as such, but others after him have pursued the idea – Bell’s theorem being the most famous of these ‘tests’ that eventually led to other strange ideas such as quantum entanglement and nonlocality.

 

In one of my G-Chem 1 prompts last semester (Week 5), I ask the students what they think about indeterminacy as illustrated by Heisenberg’s Uncertainty Principle. The students have trouble with the thought that there is a fundamental limit to measuring the two variables (momentum and position) simultaneously. Many of them argued that we’ve seen how technology has allowed us to access what seemed in accessible, including revealing the invisible entities we now call atoms. The idea of atoms far preceded observing them with the scanning tunneling microscope, but eventually scientists and technologists succeeded. Shouldn’t we expect that behind indeterminacy there are hidden variables which we’ve yet to uncover?

 

In P-Chem 2 this semester, I emphasize the statistical underpinnings of thermodynamics (from a Boltzmann view) and we work on understanding how the mathematics of the microscopic world lead to the macroscopic variables of thermodynamics. I hadn’t thought about couching this discussion in terms of hidden variables until reading Manjit’s book, but essentially that’s what we’re doing. Early in the semester when I introduce the full virial equation to describe non-ideal gases and contrast it to the two-parameter van der Waals equation, we talk about the introduction of variables to ‘model’ an invisible picture underlying reality. I’ve been emphasizing the use of models and their limitations, but I hadn’t explicitly addressed the general idea behind hidden variables (even when we use them). Something for me to consider doing!