De Broglie-Bohm theory was first discovered by Louis de Broglie in 1927 and then later by David Bohm in 1952. It is often referred to as Bohmian mechanics, Pilot Wave model, or Hidden Variable Theory. It is also a part of what we call the hidden variable interpretation of Quantum Mechanics. De Broglie-Bohm’s theory introduces the position of particles as hidden variables and postulates that the actual configuration of particles exists even when it remains unseen or unobserved. De Broglie-Bohm Theory is a causal interpretation based on Classical and Quantum Mechanics.

The system of particles is described by a wave function that evolves according to the Schrodinger equation. But this provides only a partial description of the system. Configuration or actual positions of particles are specified through wave function, which evolves according to the guiding equation to give the velocities of particles in terms of the wave function. In simpler terms, the configuration of the system of particles is governed by this wave function. 

What is Non-Relativistic Quantum mechanics?

De Broglie-Bohm theory accounts for all phenomena governed by non-relativistic quantum mechanics. Non-relativistic quantum mechanics is nothing but the mathematical formulation of quantum mechanics for the particles that do not travel with the speed of light. 

Bell showed in 1964 the nonlocality of quantum mechanics, and it applies to all versions of quantum theory. Therefore, he proved that hidden variables in De Broglie-Bohm theory account for nonlocality. In De Broglie-Bohm theory, nonlocality arises from the fact that the velocity and acceleration of a particle depend on the instantaneous position of other particles.

History of De Broglie-Bohm Theory

According to the pilot wave model, the system is not described by its wave function only as per standard quantum theory but also by the set of some additional parameters or variables. If we talk about De Broglie-Bohm theory, those additional variables are the position of particles. Albert Einstein first used the pilot wave approach to explain the motion of photons influenced by the electromagnetic field, which turns out to be wrong. But this idea was later picked by Max Born to see whether wave function could act as a guiding field for a system of electrons or not.

Later on, Schrodinger discovered a Schrodinger equation of wave function in 1926. In the following year, De Broglie discovered Bohmian mechanics and found a guiding equation for scalar wave function describing the motion of a particle. After the Pauli objection concerning inelastic scattering, he and Max Born abandoned the idea of the pilot wave model. Later in 1952, David Bohm rediscovered the De Broglie-pilot wave theory by thoroughly understanding the idea and concept. Bohm collaborated with Jean-Pierre Vigier and Basil Hiley to make clear that this theory is non-deterministic. Later on, William Simpson suggested the hylomorphic interpretation of Bohmian mechanics.

De Broglie used Lagrangians and Hamilton-Jacobi equations and quantum potential as a model, whereas Bohm used the continuity equation and the guiding equation as its model. However, they are mathematically equivalent as Hamilton-Jacobi’s formulation applies in both of them.

Defining Equations of De Broglie-Bohm Theory

In De Broglie-Bohm theory, wave function obeying Schrodinger equation does not provide the complete description of the system of particles or quantum system. Quantum mechanics is about the behaviour of particles described by their positions. On the other hand, Bohmian mechanics describes how the behaviour of particles changes with time. In Bohmian mechanics, particles are given primary importance, and the wave function is considered secondary. However, the position of particles in Bohmian mechanics is considered a hidden variable. The theory is defined as combining two equations: Schrödinger’s equation and guiding equation.

1. Schrodinger equation: Schrodinger equation is a partial differential equation that time evolution of complex wave function. It can determine the position, trajectory, and energy of particle systems.

i t= H

Where is modified planck’s constant which is equal to h/2π,

represents the wavefunction, and 

H is the hamiltonian of the system.

2. Guiding Equation: It is used to represent the velocities of particles in terms of the wave function. In Bohmian Mechanics, instead of using Schrodinger Equation as the equation of motion for all particles, we use an additional Guiding Equation to define the actual positions of particles. Also, the Guiding equation can be derived from Schrodinger Equation.

dQkdt=mkIm 8k8(Q1, Q2,…………., QN)

Where Qk is representing the position of kth particle,

  is modified planck’s constant which is equal to h/2π,

mk is mass of kth particle, and

represents the wavefunction. 

The Two Slit Experiment

When a particle is sent through a two-slit apparatus, a slit through which it passes and its position on arrival on the photographic plate can be completely determined by its initial position and wave function.

Bohmian mechanics resolves the dilemma of the appearance of both wave and particle nature in one phenomenon by describing the motion of the system of particles guided by a wave. For this experiment, there is a family of Bohmian trajectories available. Each of these trajectories is passed through only one slit; however, waves pass through both slits to give an interference pattern which is similar to the pattern developed by the trajectories guided by a wave. De Broglie gave a detailed explanation of How the motion of a particle that travels through only one slit can be influenced by the waves that pass through both slits. It is influenced in such a way that the particle doesn’t go where both waves cancel each other out but is only attracted to the points where they both cooperate.

The most confusing aspect is the fact that if we try to determine the slit through which a particle passes by any means, then the interference pattern will be destroyed. It occurs due to the interaction with another system and must be included in Bohmian mechanical analysis.

De Broglie-Bohm Theory Importance

• The De broglie-Bohm theory’s importance lies in the fact that it gives similar results to quantum mechanics. It gives prime importance to wave function as it governs the motion of the particle.
• According to De Broglie-Bohm theory, there is a single wave function that guides the motion of all particles in the entire universe according to the Guiding Equation.
• It also proves Heisenberg’s uncertainty principle as uncertainty relation can be derived from De Broglie-Bohm theory, similar to other interpretations of quantum mechanics.
• De Broglie-Bohm theory also introduces the concept of ‘Non-Locality’ in quantum mechanics.

 De Broglie-Bohm Theory Examples

Some of de Broglie-Bohm theory examples are given below:

• Theory of Relativity
• Spin
• Quantum field theory
• Curved space
• Exploiting nonlocality.

Conclusion

It is the general thought that any quantum mechanical system is described by its wave function and quantum mechanics is all about wave functions. However, it’s not true. Quantum mechanics is about atoms, electrons, positrons, quarks, and all other subatomic particles. They are not identified with the wave function. Wave functions do not provide the complete description of physical reality or individual systems but provide the closest and most obvious interpretation. This is proven through De Broglie Bohm’s theory by using additional guiding equations.