In March 2023 our latest work experience student, Jacob, from The Priory Academy LSST in Lincoln joined us for three days of work experience. It was a pleasure to host Jacob for a few days. Aside from attending different university classes as part of a ‘peer observation’ activity that teaching staff regularly engage with to foster best teaching practices, Jacob also worked on writing a report on the de Broglie-Bohm formulation of quantum mechanics. We were very pleased with the outcome of this report, especially considering the difficulty of the subject and that Jacob is currently in Year 10. We reproduce his report (with slight modifications in places) below with his permission and hope it can inspire other school students interested in physics.

Bohmian mechanics

Jacob Evans

Introduction

Bohmian Mechanics is a theoretical formulation of Quantum Mechanics. It was devised as a realist view of quantum mechanics meaning that involves objects that have an existence independent to us observing them and it does exist before the act of measuring, or in the Copenhagen Interpretation prior to the waveform collapsing, contrasting the Copenhagen Interpretation which still remains the most popular foundation of quantum mechanics.

It was first developed in 1927 but was abandoned until 1952 when it was further developed by David Bohm. However due to the popularity of the Copenhagen Interpretation, Bohmian Mechanics till this day is seldom taught meaning that since most physicist do not know about Bohmian Mechanics it is also seldom worked on.

But why then had Born not told me of this ’pilot wave’? If only to point out:

“What was wrong with it? … Why is the pilot wave picture ignored in textbooks? Should it not be taught, not as the only way, but as an antidote to the prevailing complacency? To show that vagueness, subjectivity, and indeterminism, are not forced on us by experimental facts, but by deliberate theoretical choice?” [1]

History of Bohmian mechanics (pilot wave theory)

Bohmian mechanics was first presented in 1927 at the Solvay conference (in Belgium founded by Ernest Solvay) by Louis de Broglie. It was developed by him in collaboration with Schrodinger who developed the wave equations for the waveform denoted by the Greek letter psi which we will discuss further, this equation is the most fundamental equation for both foundations of quantum mechanics. There were many criticisms since it was deemed not to be compatible with theories at the time, such as the belief that all hidden variables are impossible which directly goes against Broglie’s theory which requires hidden causal variables. This then led to the disbandment of De Broglie’s theory for the next two decades. However, in 1952 David Bohm an American-British-Brazilian theoretical physicist who is carried on the work of Broglie’s pilot wave theory in his work ‘A Suggested Interpretation of the Quantum Theory in Terms Of “Hidden Variables”. I and II”

This extended Broglie’s theory to have a consistent theory of measurement unlike the Copenhagen Interpretation. The measurement problem is one of the largest flaws for our current understanding of the Copenhagen Interpretation. The measurement problem in the Copenhagen Interpretation refers to our inability to observe a waveform collapse directly. We shall describe how the De Broglie-Bohm theory attempts to solve this problem. It also addressed some of the criticisms of Bohmian mechanics that De Broglie had not been able to respond too. Bohm’s theory also removed the particle/wave duality paradox which is the idea that whilst light is a wave it can also be described as a particle, the photon, however it opened up another paradox which is nonlocality which we will explore later. Bohm also showed it to be deterministic meaning there are no elements of randomness involved. This was called the ‘De Broglie-Bohm Theory’ by Bell.

The De Broglie-Bohm Theory is an example of a hidden-variables theory, a theory which utilises unobservable variables. It was hoped that hidden variables could give a description that could eliminate many of the paradoxes of quantum mechanics such as Schrodinger’s cat.

The de Broglie-Bohm Theory was largely rejected, Albert Einstein who had suggested Bohm to search for a realist theory of the Copenhagen interpretation described the theory as ‘too cheap’ as he did not believe it to be a sufficient answer to quantum nonlocality. Heisenberg considered it as a ‘superfluous ideological superstructure.’ And even things that were completely unrelated to Bohm’s work caused people to not believe in his theory, namely his communist affiliations in his youth. In 1987 Bell defended the theory and produced several papers on hidden variable theories. Despite this it is still not largely accepted, the Copenhagen interpretation is the most widely accepted interpretation of quantum mechanics so is what is still taught . [2]

Bohmian Mechanics

The difference between the Copenhagen interpretation and Bohmian Mechanics

The Copenhagen interpretation is the most widely known formulation of quantum mechanics. This formulation of quantum mechanics, with others, is based on a notion called superposition. Superposition states that a quantum particle before observation does not exist in one state but may instead exist in a multiplicity of possible states at once and when observed, the wave function collapses and the particles superposition is lost, and the particle takes an exact, definite state. However due to the randomness of superposition it makes predicting quantum particles behaviour impossible when first observed. The most famous model of this is called Schrodinger’s cat. This is a thought experiment in which a cat is placed in a box with a vial of poison and a radioactive source connected to a Geiger counter. If the Geiger counter detects radioactivity, then it shatters the vial, and the cat dies. Since it is random whether the counter detects the radioactivity then whilst the box is closed then the cat can be thought as both alive and dead aka in a state of superposition but when the box is opened then the cat can only be in one of these states (representative of the waveform collapsing).

The wave function is a quantity that can describe the wave characteristics of a particle, the value of the wave function at a specific point in 4 dimensions (x, y, z, t coordinates) is related to the likelihood of the particle being in a specific point in 4-dimensional space.

For the mathematical wave function represented by the Greek letter psi, 𝜓, is essentially the amplitude of a De Broglie (particle) wave but for these waves’ amplitude is insignificant. However, |𝜓|² does have physical significance as it is proportional to the probability of finding the particles in a specific point at a given time.

Figure 1: graphical representation of the wave function for a single coordinate system.

The wave function graph. This is a probability density graph of finding the wave at a given point in a 4-dimensional Cartesian coordinate system. The graph never goes below the 0 on the y axis due to the fact it is impossible to have a negative probability. [3] [4]

The Schrodinger equation. By solving the equation for the waveform it gives the probability density for the waveform at a specific point in space and time. [5]

The above is Schrodinger’s equation for a non-relativistic particle in an external energy potential .

On the other hand, Bohmian Mechanics rejects the idea of superposition meaning that, unlike the Copenhagen interpretation, Bohmian Mechanics is completely deterministic, including during the measurement process. The way to predict the future of a system via Bohmian mechanics is by solving both the guiding equation (shown below) and the Schrödinger equation (shown above). This is because Bohmian mechanics describes the motion of physical particles as being guided by a real pilot wave given by the wave function. Meaning unlike the Copenhagen interpretation the particles can only take one trajectory unlike with the Copenhagen interpretation where the particles can take any trajectory when they are observed.

The guiding wave equation. – With Q representing the positions of the particles in the system [6]

Due to it being completely deterministic, once the initial state is given then the future positions of the state can be uniquely calculated. Which is one of the advantages of Bohmian mechanics over the Copenhagen interpretation.

Another advantage of Bohmian mechanics over the Copenhagen interpretation is the lack of particle wave duality. One of the largest paradoxes for the Copenhagen interpretation is due to the fact that in the Copenhagen interpretation system of particles can be completely described by the particles wave function, however this is somewhat an oxymoron since the definition of a particle is ‘a point-like entity whose most important feature is their position in space’. Since a complete description of a system of particles would have to include these positions because if it does not then the statement is too vague. [6] However, with Bohmian mechanics it rejects the idea of particle/wave duality meaning it doesn’t have this paradox as it describes real particles that are guided by a separate entity, a real wave called the pilot wave.

Problem of non-locality and the solution to this

One of the reasons which caused and still causes doubt today that we mentioned earlier is that the problem of quantum nonlocality. Quantum nonlocality is a phenomenon in which quantum particles can essentially know the state of another quantum particle no matter the distance simultaneously after the waveform has collapsed. [7] Quantum nonlocality violates one of the most important of Einstein’s laws of relativity, the law being that nothing can travel faster than the speed of light. Quantum nonlocality violates this law due to the fact that the information about the other particle has been measured to travel faster than the speed of light. In both the Copenhagen interpretation and Bohmian mechanics have quantum nonlocality though the Copenhagen Interpretation does not give any explanation to quantum nonlocality or the apparent violation of special relativity.

The way that Bohmian mechanics attempts to solve the problem between non-locality and Einstein’s theory of relativity is by an idea of wholism. Essentially, the wave function spreads throughout all of time and space, and since all particles share a common wave function (the pilot wave or universal wave function), all particles behaviour is derived from this single wave function meaning that all particles are linked. These means that the reason two entangled particles seem to ‘know’ each other’s properties when observed is due to the fact that they are both part of the same thing, the universal wave function. Also, by the use of non-local hidden variables no information is sent meaning it does not break Einstein’s law that nothing can travel faster than the speed of light. [8]

Hidden Variables

We have already briefly talked about hidden variables in the De Broglie-Bohm theory. Hidden variables were first used in the EPR paradox published by Einstein, Podolsky and Rosen. This paper was published in response to the lack of determinism in quantum mechanics, namely nonlocality which we mentioned above. [9] Whilst the solution given to this paradox in the paper is false as it requires local hidden variables which we know from the history of Bohmian mechanics have been shown not to exist by Bell in Bells Theorem, some elements of the paper such as non-local hidden variables are still used in quantum mechanics today. Whilst we stated what the definition of hidden variable theories (such as the De Broglie-Bohm Theory) were before just to refresh they are theories that provide explanations of quantum mechanical phenomena through the use of unobservable variables. The EPR paradox states that when two entangled particles first interact with each other to gain their entangled state then information somehow is shared, and they both decide what to do when one’s wave form collapses. Essentially, they believed that the particles decided which spin too take when one is observed. [10] The equation for the probability of each pair of entangled particles adds to 1, when one particle takes a spin then the other must choose the opposite spin, this is also shows that neither particle can be thought of as separate as they are both entangled and what state one is in affects what state the other particle must be in.

Linking this to the De Broglie-Bohm theory we can see how these non-local variables are used in terms of entanglement. In order to be able to explain nonlocality like Bohmian Mechanics does whilst not breaking the law that nothing can travel faster than the speed of light then the way that Bohmian mechanics can pass information faster than the speed of light is by these non-local hidden variables.

The flaws of Bohmian Mechanics

One of the biggest flaws of Bohmian Mechanics is how it contradicts Quantum Field Theory. QFT is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. It describes the behaviour between subatomic particles and their interactions with forces. It is used in particle physics to create models and has many branches such as quantum electrodynamics which describes particles interactions with electromagnetic fields. Another one of the major fields is quantum chromodynamics which describes the interaction of quarks and the strong nuclear force. These theories are used in many branches of physics and is considered to be one the most important quantum theories we have to date. However, this gives rise to one of if not the biggest problem with Bohmian mechanics in that it would be very hard if not impossible to make QFT to work with Bohmian mechanics. There have been attempts at this however some unwanted consequences of special relativity occur such as the possibility of particle creation and annihilation during an interaction process. This is because QFT requires superposition to be true in order for it too work due to the fact that all the fields exist in superposition at all times. However Bohmian Mechanics rejects the idea of superposition and 1 particle or field existing at multiple points in space at the same time meaning Bohmian Mechanics cannot work with our biggest theory to do with quantum mechanics. [11]

Another major flaw of Bohmian mechanics is that it goes against newton’s third law of motion. This is due to the fact that real waves are pushing real particles, but these particles do not push back equally which is another major flaw to do with Bohmian Mechanics. [11]

The double slit experiment

The double slit experiment was first done with electrons in 1969, this experiment led to the first theory of quantum mechanics, namely the Copenhagen Interpretation in 1925. This is due to the fact that it is impossible to describe the results of this experiment by the means of classical mechanics. The double split experiment is an experiment in which a beam of electrons is fired at a plate with two slits equally distant from the centre. In the Copenhagen interpretation then the result is explained by the particle/wave duality of particles. Due to the wave like nature of particles the interference produces bands of light and dark on the screen, something that could not occur in the Copenhagen interpretation without particle/wave duality. On the screen though the light is absorbed in specific points as particles meaning that if the Copenhagen interpretation is correct then particles can behave as waves and vice versa. Also, the way that electrons appear on the screen is exactly the same as due to the fact that the wave form to the power of 2 is the probability of the electron being in that specific point at time meaning that is random which slit the particle goes through as when observed the waveform collapses and the particles superposition is lost however in the Copenhagen interpretation this randomness can be seen as proof for superposition. [12]

In Bohmian mechanics as we mentioned before there is no such thing as particle wave duality so how can the light and dark bands be formed without any interference with waves? In Bohmian mechanics the particles do only go through one of the slits as there is no superposition. However, the wave function will be defined on both sides of the slit meaning there are set trajectories for the electron to follow, due to the fact that the wave is defined on both sides of the slit it creates the interference pattern on the screen meaning to explain this experiment there is no need for superposition nor particle/wave duality. [11]

Figure 2: paths followed by particles in a double slit experiment in Bohmian quantum mechanics [12]

The figure above shows the possible routes that each electron can travel though due to the fact there is no superposition then the electron can only follow one of these paths. [12]

This shows that both the Copenhagen interpretation and Bohmian Mechanics are not distinct theories rather interpretations. We define a theory as something that can be distinctly proven via an experiment however since the same experiment can prove either of these two, they are interpretations of the effects of quantum mechanics and how to solve these interpretations of the double-slit experiment.

Conclusion

Whilst it would be very difficult to devise an experiment to prove whether the Copenhagen Interpretation or the De Broglie-Bohm Theory is the correct foundation for quantum mechanics both of these two theories do have drawbacks and advantages over one another, Bohmian Mechanics should be taught equally as much as the Copenhagen Interpretation due to some of the major advantages which Bohmian mechanics has over the Copenhagen interpretation.

Is it not clear from the smallness of the scintillation on the screen that we have to do with a particle? And is it not clear, from the diffraction and interference patterns, that the motion of the particle is directed by a wave? De Broglie showed in detail how the motion of a particle, passing through just one of two holes in screen, could be influenced by waves propagating through both holes. And so influenced that the particle does not go where the waves cancel out, but is attracted to where they cooperate. This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored.” (Bell 1987) [6]

Bohmian mechanics allows us to solve some of the biggest flaws that have arisen due to the Copenhagen interpretation such as the reason for quantum nonlocality and the it makes up for the lack of determinism in the Copenhagen interpretation whilst still giving the same valid results for experiments. Whilst it does solve many of the underlying problems and paradoxes of the Copenhagen interpretation it does still have some drawbacks showing that whilst it solves many issues it also creates some that don’t exist in the Copenhagen interpretation meaning that it is not possible to know with our current understanding the true foundation of quantum mechanics in both of their current states.