The standard interpretation at the time was that pioneered by the Copenhagen group – notably Bohr, Heisenberg, and Schrödinger – and is today usually referred to as the Copenhagen Interpretation. It works as follows. In quantum mechanics, everything is described by a wave-function, usually denoted Psi. Psi is a function of time. One can calculate how it changes in time with a differential equation known as the Schrödinger equation. When one makes a measurement, one calculates probabilities for the measurement outcomes from the wave-function. The equation by help of which one calculates these probabilities is known as Born’s Rule. I explained in an earlier video how this works.
The peculiar thing about the Copenhagen Interpretation is now that it does not tell you what happens before you make a measurement. If you have a particle described by a wave-function that says the particle is in two places at once, then the Copenhagen Interpretation merely says, at the moment you measure the particle it’s either here or there, with a certain probability that follows from the wave-function. But how the particle transitioned from being in two places at once to suddenly being in only one place, the Copenhagen Interpretation does not tell you. Those who advocate this interpretation would say that’s a question you are not supposed to ask because, by definition, what happens before the measurement is not measureable.
Bohm was not the only one dismayed that the Copenhagen people would answer a question by saying you’re not supposed to ask it. Albert Einstein didn’t like it either. If you remember, Einstein famously said “God does not throw dice”, by which he meant he does not believe that the probabilistic nature of quantum mechanics is fundamental. In contrast to what is often claimed, Einstein did not think quantum mechanics was wrong. He just thought it is probabilistic the same way classical physics is probabilistic, namely, that our inability to predict the outcome of a measurement in quantum mechanics comes from our lack of information. Einstein thought, in a nutshell, there must be some more information, some information that is missing in quantum mechanics, which is why it appears random.
This missing information in quantum mechanics is usually called “hidden variables”. If you knew the hidden variables, you could predict the outcome of a measurement. But the variables are “hidden”, so you can only calculate the probability of getting a particular outcome.
Back to Bohm. In 1952, he published two papers in which he laid out his idea for how to make sense of quantum mechanics. According to Bohm, the wave-function in quantum mechanics is not what we actually observe. Instead, what we observe are particles, which are guided by the wave-function. One can arrive at this interpretation in a few lines of calculation. I will not go through this in detail because it’s probably not so interesting for most of you. Let me just say you take the wave-function apart into an absolute value and a phase, insert it into the Schrödinger equation, and then separate the resulting equation into its real and imaginary part. That’s pretty much it.
The result is that in Bohmian mechanics the Schrödinger equation falls apart into two equations. One describes the conservation of probability and determines what the guiding field does. The other determines the position of the particle, and it depends on the guiding field. This second equation is usually called the “guiding equation.” So this is how Bohmian mechanics works. You have particles, and they are guided by a field which in return depends on the particle.
To use Bohm’s theory, you then need one further assumption, one that tells what the probability is for the particle to be at a certain place in the guiding field. This adds another equation, usually called the “quantum equilibrium hypothesis”. It is basically equivalent to Born’s rule and says that the probability for finding the particle in a particular place in the guiding field is given by the absolute square of the wave-function at that place. Taken together, these equations – the conservation of probability, the guiding equation, and the quantum equilibrium hypothesis – give the exact same predictions as quantum mechanics. The important difference is that in Bohmian mechanics, the particle is really always in only one place, which is not the case in quantum mechanics.
As they say, a picture speaks a thousand words, so let me just show you how this looks like for the double slit experiment. These thin black curves you see here are the possible ways that the particle could go from the double slit to the screen where it is measured by following the guiding field. Just which way the particle goes is determined by the place it started from. The randomness in the observed outcome is simply due to not knowing exactly where the particle came from.
What is it good for? The great thing about Bohmian mechanics is that it explains what happens in a quantum measurement. Bohmian mechanics says that the reason we can only make probabilistic predictions in quantum mechanics is just that we did not exactly know where the particle initially was. If we measure it, we find out where it is. Nothing mysterious about this. Bohm’s theory, therefore, says that probabilities in quantum mechanics are of the same type as in classical mechanics. The reason we can only predict probabilities for outcomes is because we are missing information. Bohmian mechanics is a hidden variables theory, and the hidden variables are the positions of those particles.
So, that’s the big benefit of Bohmian mechanics. I should add that while Bohm was working on his papers, it was brought to his attention that a very similar idea had previously been put forward in 1927 by De Broglie. This is why, in the literature, the theory is often more accurately referred to as “De Broglie Bohm”. But de Broglie’s proposal did, at the time, not attract much attention. So how did physicists react to Bohm’s proposal in fifty-two. Not very kindly. Niels Bohr called it “very foolish”. Leon Rosenfeld called it “very ingenious, but basically wrong”. Oppenheimer put it down as “juvenile deviationism”. And Einstein, too, was not convinced. He called it “a physical fairy-tale for children” and “not very hopeful.”
Why the criticism? One of the big disadvantages of Bohmian mechanics, that Einstein in particular disliked, is that it is even more non-local than quantum mechanics already is. That’s because the guiding field depends on all the particles you want to measure. This means, if you have a system of entangled particles, then the guiding equation says the velocity of one particle depends on the velocity of the other particles, regardless of how far away they are from each other.
That’s a problem because we know that quantum mechanics is strictly speaking only an approximation. The correct theory is really a more complicated version of quantum mechanics, known as quantum field theory. Quantum field theory is the type of theory that we use for the standard model of particle physics. It’s what people at CERN use to make predictions for their experiments. And in quantum field theory, locality and the speed of light limit, are super-important. They are built very deeply into the math.
The problem is now that since Bohmian mechanics is not local, it has turned out to be very difficult to make a quantum field theory out of it. Some have made attempts, but currently there is simply no Pilot Wave alternative for the Standard Model of Particle Physics. And for many physicists, me included, this is a game stopper. It means the Bohmian approach cannot reproduce the achievements of the Copenhagen Interpretation.
Bohmian mechanics has another odd feature that seems to have perplexed Albert Einstein and John Bell in particular. It’s that, depending on the exact initial position of the particle, the guiding field tells the particle to go either one way or another. But the guiding field has a lot of valleys where particles could be going. So what happens with the empty valleys if you make a measurement? In principle, these empty valleys continue to exist. David Deutsch has claimed this means “pilot-wave theories are parallel-universes theories in a state of chronic denial.”
Bohm himself, interestingly enough, seems to have changed his attitude towards his own theory. He originally thought it would in some cases give predictions different from quantum mechanics. I only learned this recently from a Biography of Bohm written by David Peat. Peat writes
“Bohm told Einstein… his only hope was that conventional quantum theory would not apply to very rapid processes. Experiments done in a rapid succession would, he hoped, show divergences from the conventional theory and give clues as to what lies at a deeper level.”
However, Bohm had pretty much the whole community against him. After a particularly hefty criticism by Heisenberg, Bohm changed course and claimed that his theory made the same predictions as quantum mechanics. But it did not help. After this, they just complained that the theory did not make new predictions. And in the end, they just ignored him.
So is Bohmian mechanics in the end just a way of making you feel better about the predictions of quantum mechanics? Depends on whether or not you think the “quantum equilibrium hypothesis” is always fulfilled. If it is always fulfilled, the two theories give the same predictions. But if the equilibrium is actually a state the system must first settle in, as the name certainly suggests, then there might be cases when this assumption is not fulfilled. And then, Bohmian mechanics is really a different theory. Physicists still debate today whether such deviations from quantum equilibrium can happen, and whether we can therefore find out that Bohm was right."" This video was sponsored by Brilliant which is a website that offers interactive courses on a large variety of topics in science and mathematics. I always try to show you some of the key equations, but if you really want to understand how to use them, then Brilliant is a great starting point. For this video, for example, I would recommend their courses on differential equations, linear algebra, and quantum objects. To support this channel and learn more about Brilliant, go to Brilliant.org/Sabine and sign up for free. The first 200 subscribers using this link will get 20 percent off the annual premium subscription.
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