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What Are Bohmian Mechanics and the Pilot Wave Theory?

Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.

Bohmian Mechanics

Ask a group of scientists what quantum mechanics actually means and you are bound to find disagreement. So many different interpretations of the topic have so far eluded scientists in the ability to test and eliminate potential options. That is why after I heard about Bohmian mechanics, I was on board with that being my acceptable interpretation. So, how has it won me over?

Pilot Waves

Many people who disagreed with the complex and paradoxical results of quantum mechanics went on a hunt for hidden variables that when married to classical mechanics would account for all the weirdness of quantum physics. David Bohm’s search for hidden variables began in 1951 when he used thermodynamics to demonstrate that incomplete knowledge of a system leads one to seek hidden variables. By 1952, he had developed his theory fully and entitled it the “Quantum Theory in Terms of Hidden Variables,” later known as Bohmian Mechanics.

In it, particles do have wave functions that are not necessarily physical but are real in that they guide where our particle goes. It essentially imparts a quantum potential onto our particle, and so determinism returns because if we know our wave function then we can project forward, just like with classical mechanics. So, that is a big plus. Reality seems deterministic and so much science works based on such intuitions, so why should quantum mechanics be any different?

According to Bohm, it isn’t but seems to be because of hidden variables masking the particle’s true location, which is associated because of the uncertainty principle. And yet, it violates Newtonian physics…seemingly, but remember that such a field is reserved for the classical land. Without such violations, phenomena such as explaining light paths changing become nonsense.

Other things that the theory explains but traditional quantum mechanics cannot are uncertainty principles, probability functions, standing positions, and that pesky measurement problem that has dogged quantum mechanics for nearly 100 years (Ananthaswamy 149-150, 154; Smolin 98, 116-7).

Without realizing it, Bohm’s work-related back to de Broglie’s pilot wave idea, dating from 1927. In this, there is no particle/wave duality but instead a co-existence, with each being equally real at all times. The wave function now comes with a guidance function as well, showing the way for the particle to travel. Interestingly, the particle aims to be at the highest amplitude, known as the law of steepest ascent.

It’s complicated as to the why, but it relates back to the Born rule, where the probability of being found at a given location is proportional to the wave function squared. This causes low points on the wave to suddenly become positive upon being squared.

While Bohm’s work was more refined and completed, its relation to the prior now causes the theory to be known as the Bohm-de-Broglie theory. Bohm’s theory does have a few important differences like the guidance equation taking a different form and the law of steepest ascent being re-interpreted as a variation of Newton’s Law of Motion (where acceleration causes a force moving our particle to the highest point on wave function).

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Back in 1927, the theory was shut down for trying to restore a realist perspective to quantum mechanics, which wasn’t conceivable at the time. It made quantum mechanics a deterministic theory where the wave function never collapses. Besides, it upended the work of prominent quantum physicists at the time and so it was suppressed. Now, surely you may ask if any of these Bohmian mechanics is better to accept than the standard Copenhagen Interpretation. Is there any evidence for this work to even garner such a debate? (Ananthaswamy 155-6; Smolin 97-101, 109-110)

Proving One’s Worth

In 2006, an oil drop experiment was conducted by Yves Couder and Emmanuel Fort using silicone. The oil was vibrated but below the Faraday threshold, meaning no surface waves were generated. Now, a small drop about a millimeter in size is dropped onto this, and rather than being absorbed into the vat the drop seemingly floats along the surface as it deals with minute horizontal wave components that normally are below the surface. However, the drop exerts pressure that causes those components to now be a factor, and so the drop floats on.

The particle is guided by a wave, kind of like the pilot wave idea. But what we want to see is a quantum component to this, so enter the double slit experiment. A barrier with two slits was placed so it was just on the surface, and the drop was introduced into the system. Of course, the drop would only go down one path, but the waves on the other side interacted with each other, and upon recording all the places the drop ended up, an interference pattern arose! The particle had a wave result! However, follow-up experiments failed to replicate this result. It could be because the initial experiment was done over only 75 attempts, or perhaps some “environmental influence” unbeknownst to the experimenters was at play (Ananthaswamy 157-9).

Why should we then have confidence in Bohmian mechanics? Well, it offers some better solutions to common quantum problems. For example, it explains the uncertainty principle as a result of incomplete initial information leading to stacking uncertainties as the future progresses. The collapse of the wave function isn’t strictly literal so much as we receive only one function while the others are directed elsewhere. Therefore, measuring our state no longer causes causal headaches because it’s just us uncovering the state it already exists in. It is a collapse, just not a physical one but of the probability paths being eliminated that we were not privy to until we spotted the particle.

External influences clearly can influence that path because of wave interactions, and we can then backtrack its trajectory to find even more information. It is this level of ambiguity that truly explains the uncertainty principle, and the probability of where the particle is to be found is also accounted for. In fact, if we run enough particles under Bohmian mechanics, we can arrive at the Born rule, a hallmark of quantum mechanics, and this result is regardless of my initial set-up! Though this can we arrive at quantum equilibrium, where Bohmian mechanics matches traditional quantum mechanics (Ananthaswamy 164-5; Smolin 98, 117-120).

How about Schrodinger’s Cat, a favorite thought experiment demonstrating quantum weirdness at its finest? Via the pilot wave idea, we can have a given configuration of atoms correlated to the information we have at hand. Each atom has a spatial coordinate associated with it at any given moment, so we need x, y, and z information for each atom, for a cumulative total information of 3 times the number of atoms.

That will be a huge number, but pilot-wave allows for it because each atom does have a definite position at a given moment, unlike traditional quantum mechanics. But as each particle is at a given position, so is the wave it's riding on but in configuration space. This is not the same as 3D conditions but is instead a higher-dimensional region where a given value leads to a given 3D configuration. Once the interference of all those possible configurations plays out, we arrive at a specific value, meaning the uncertainty of the cat has now been eliminated from us engaging with it (Smolin 121-4).

Works Cited

Ananthaswamy, Anil. Through Two Doors at Once. Random House, New York. 2018. Print. 149-50, 154-9.

Smolin, Lee. Einstein’s’ Unfinished Revolution. Penguin Press, New York. 2019. Print. 97-101, 109-10, 116-24.

© 2021 Leonard Kelley

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