# Bohmian Mechanics and the two slit experiment

I am trying to understand the essentials of the Bohmian interpretation of quantum mechanics (BI),and my (many) difficulties start with his description of the two slit experiment.As I understand it,Bohm asserts that every quantum particle has associated with it a quantum potential or guiding wave,which I assume to be the same concept as in the original deBroglie pilot wave theory.What I have not been able to find is a clear and concise explanation of the nature and origin of this quantum potential.

In the two slit experiment,BI assumes that the guiding wave always passes through both slits,after which it forms an interference pattern of channels.The particle on the other hand always goes through only one or the other slit,after which it is channelled into those regions of constructive interference by the guiding wave,ultimately producing the recorded interference pattern.

So,I have three questions.
1) What is the nature and origin of the guiding wave/quantum potential ?
Presumably it can't be electromagnetic,as neutral entities such as neutrons,atoms,and even large molecules such as C60fullerene have been shown to exhibit interference effects.
2)What is the nature of the "force" which directs particles into the constructive channels between the slits and the detecting system ?.
There must be such a force because if a stream of particles is directed at the two slits,each particle must individually pass through only one slit,but thereafter all the particles are "pulled across" into channels that result in the observed interference pattern.
3)If every particle in a quantum system always has its accompanying guiding wave,this must presumably emanate in all directions in space (Hilbert space ?) even if the particle is at rest.When particles such as electrons,protons and neutrons combine to form neutral atoms and then molecules,how do the quantum potentials combine ?.

I would like some clear answers to these questions

Related Quantum Interpretations and Foundations News on Phys.org
See Lecture 1 of http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken].

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I don't seem to be able to open this link;any helpful suggestions ?
I can email it to you

Demystifier
Gold Member
1) What is the nature and origin of the guiding wave/quantum potential ?
Presumably it can't be electromagnetic,as neutral entities such as neutrons,atoms,and even large molecules such as C60fullerene have been shown to exhibit interference effects.
Electromagnetic potential is a classical potential. By contrast, the nature of the quantum potential is not classical. This is someting new that does not exist in classical physics.

2)What is the nature of the "force" which directs particles into the constructive channels between the slits and the detecting system ?.
There must be such a force because if a stream of particles is directed at the two slits,each particle must individually pass through only one slit,but thereafter all the particles are "pulled across" into channels that result in the observed interference pattern.
Just like the quantum potential in 1), the nature of the quantum force resulting from the quantum potential is not classical. That's something new.

3)If every particle in a quantum system always has its accompanying guiding wave,this must presumably emanate in all directions in space (Hilbert space ?) even if the particle is at rest.When particles such as electrons,protons and neutrons combine to form neutral atoms and then molecules,how do the quantum potentials combine ?.
It is not true that every particle always has its accompanying guiding wave. Instead, there is only one "big" guiding wave for ALL particles in the universe. This wave satisfies the superposition principle, which determines how the quantum potential combines.

I can email it to you
Thanks.I've now found the article;it looks interesting and I'll report back when I've digested it.

Electromagnetic potential is a classical potential. By contrast, the nature of the quantum potential is not classical. This is someting new that does not exist in classical physics.

Just like the quantum potential in 1), the nature of the quantum force resulting from the quantum potential is not classical. That's something new.

It is not true that every particle always has its accompanying guiding wave. Instead, there is only one "big" guiding wave for ALL particles in the universe. This wave satisfies the superposition principle, which determines how the quantum potential combines.
You just say that the quantum potential is non-classical,something new.Well I know that,but your answer doesn't tell me what the nature and origin of the force is.Is it multidirectional,does it vary with distance,does it exist irrespective of whether or not the particle is at rest or in motion.
If this is a fundamentally new force concept then physicists must have given much thought to it,rather than just dismissing it as "something new".
Can you explain further your assertion that "it is not true that-----".

The quantum potential in pilot-wave theory/Bohmian mechanics is a totally superfluous concept.

In standard QM, only waves exist (or depending on your point of view - nobody knows or cares what exists but the equations of QM make probabilistic predictions about what will happen in particular kinds of experiment).

The basic idea of pilot-wave theory/Bohmian mechanics is that (a) both particles and waves exist, and (b) the particles are pushed around by the wave (which - taking the non-relativistic case for simplicity - is identified with the usual QM wave function evolving according to the Schroedinger equation) as well as by each other (via the electromagnetic field). The wave function is thus postulated to be a real objectively-existing universal field which acts as a 'guiding field' for all particles. The wave guides the particles so that they have a trajectory given by velocity ${\bf v}=\nabla S/m$ where S is the phase of the wave (this equation just follows from translating the usual formula for the quantum probability current over the density). That's it. That's all you need to know. All the predictions of standard QM follow but now one sees it is no longer an absolute requirement that Nature must be inherently probabilistic or that particles don't exist unless an 'observer' is looking at them; rather one can think (if you like) of the whole of QM as just the perfectly ordinary statistical mechanics of particles with a particular non-classical dynamics. This has various interesting advantages, not least of which is the comedy value of watching the genuine annoyance this can provoke when pointed out.

Now, if you want to make the above trajectory equation look a bit like the equation for Newton's second law, you can just take the time derivative of it, and then mass times acceleration = time derivative of $\nabla S$. Work through the math and this can be made to look like force $=-\nabla(V+Q)$ where the quantum potential Q is just some potential function derived from the curvature of the wave function amplitude. The only real reason for doing this is so you can show that QM reduces to the classical limit in a situation when Q tends to zero (remember this can't be done in standard QM, where the existence of the classical world is assumed as a postulate). Other than this the quantum potential is utterly superfluous and only serves to make the issue less understandable (as you have clearly discovered). In particular it turns the nice linear Schroedinger equation into a pair of hideous non-linear ones. I don't know why people do this. It's probably Bohm's fault - as this is the way he did it in his 1950s paper. When de Broglie introduced the theory in 1927 he gave the usual first order form - all Bohm actually did was take the time derivative of de Broglie (and clarify a little how measurement worked).

All this is explained in boring detail in the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken] I gave you above.

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Demystifier
Gold Member
You just say that the quantum potential is non-classical,something new.Well I know that,but your answer doesn't tell me what the nature and origin of the force is.Is it multidirectional,does it vary with distance,does it exist irrespective of whether or not the particle is at rest or in motion.
If this is a fundamentally new force concept then physicists must have given much thought to it,rather than just dismissing it as "something new".
Can you explain further your assertion that "it is not true that-----".
In the post above Camboy have written a nice explanation which, I hope, indirectly answers some of your questions. Let me just give some short direct answers to your questions.
I don't know what is a "multidirectional force". It does vary with position of the particle. It does not depend on the particle velocity. Since there is no a separate wave function for each particle, the same is true for the potential.

The quantum potential in pilot-wave theory/Bohmian mechanics is a totally superfluous concept.

In standard QM, only waves exist (or depending on your point of view - nobody knows or cares what exists but the equations of QM make probabilistic predictions about what will happen in particular kinds of experiment).

The basic idea of pilot-wave theory/Bohmian mechanics is that (a) both particles and waves exist, and (b) the particles are pushed around by the wave (which - taking the non-relativistic case for simplicity - is identified with the usual QM wave function evolving according to the Schroedinger equation) as well as by each other (via the electromagnetic field). The wave function is thus postulated to be a real objectively-existing universal field which acts as a 'guiding field' for all particles. The wave guides the particles so that they have a trajectory given by velocity ${\bf v}=\nabla S/m$ where S is the phase of the wave (this equation just follows from translating the usual formula for the quantum probability current over the density). That's it. That's all you need to know. All the predictions of standard QM follow but now one sees it is no longer an absolute requirement that Nature must be inherently probabilistic or that particles don't exist unless an 'observer' is looking at them; rather one can think (if you like) of the whole of QM as just the perfectly ordinary statistical mechanics of particles with a particular non-classical dynamics. This has various interesting advantages, not least of which is the comedy value of watching the genuine annoyance this can provoke when pointed out.

Now, if you want to make the above trajectory equation look a bit like the equation for Newton's second law, you can just take the time derivative of it, and then mass times acceleration = time derivative of $\nabla S$. Work through the math and this can be made to look like force $=-\nabla(V+Q)$ where the quantum potential Q is just some potential function derived from the curvature of the wave function amplitude. The only real reason for doing this is so you can show that QM reduces to the classical limit in a situation when Q tends to zero (remember this can't be done in standard QM, where the existence of the classical world is assumed as a postulate). Other than this the quantum potential is utterly superfluous and only serves to make the issue less understandable (as you have clearly discovered). In particular it turns the nice linear Schroedinger equation into a pair of hideous non-linear ones. I don't know why people do this. It's probably Bohm's fault - as this is the way he did it in his 1950s paper. When de Broglie introduced the theory in 1927 he gave the usual first order form - all Bohm actually did was take the time derivative of de Broglie (and clarify a little how measurement worked).

All this is explained in boring detail in the http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html" [Broken] I gave you above.
Thank you for your helpful reply.I have now got Mike Towler's lecture notes and although they look formidably mathematical,I will try and work my way through them.

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