Quantum mechanics is often praised as a theory which unifies classical mechanics and classical wave theory. Quanta are said to behave either as particles or waves, depending on the type of experiment they are subjected to. But where in the standard formalism can a pointlike particle structure actually be found? Perhaps only to some degree in the reduction postulate applied to position measurements. In reaction to this unsatisfactory state of affairs, Bohmian mechanics introduces a mathematically precise particle concept into quantum mechanical theory. The fuzzy wave functions are supplemented by sharp particle world-lines. Through this additional structure some quantum phenomena like the double slit experiment have lost their mystery.
Quantum mechanics is often praised as a theory which unifies classical mechanics and classical wave theory. Quanta are said to behave either as particles or waves, depending on the type of experiment they are subjected to. But where in the standard formalism can a pointlike particle structure actually be found? Perhaps only to some degree in the reduction postulate applied to position measurements. In reaction to this unsatisfactory state of affairs, Bohmian mechanics introduces a mathematically precise particle concept into quantum mechanical theory. The fuzzy wave functions are supplemented by sharp particle world-lines. Through this additional structure some quantum phenomena like the double slit experiment have lost their mystery.
The popular impression of Bohmian mechanics is that it is standard quantum mechanics with the addition of some extra gadgets---exact particle positions and a guiding equation for particle trajectories---the advantages being that the gadgets pave the way for a resolution of the measurement problem that eschews state vector reduction while restoring the determinism lost in standard quantum mechanics. In fact, the Bohmian mechanics departs in significant ways from standard quantum mechanics. By itself this is not a basis for criticism; indeed, it makes Bohmian mechanics all the more interesting. But Bohmian mechanics is not, as the popular impression would have it, empirically equivalent to standard quantum mechanics in terms of probabilistic predictions for the outcomes of measurements of quantum observables. Indeed, in physically important applications to systems for which standard quantum mechanics delivers empirically well-confirmed probabilistic predictions, the sophisticated form of Bohmian mechanics designed to prove the global existence of Bohmian particle trajectories fails to deliver unequivocal predictions---of even a probabilistic variety---for the future behavior of said systems. Possible responses to this lacuna are discussed.
This thesis deals with the theoretical and experimental details of trying to understand \nand infer the quantum potential from Bohmian mechanics using Bohmian trajectories. In \nthis work, some of the key components of the history of Bohmian mechanics is given, a \ntheoretical explanation of how to measure the Bohmian potential, followed by an overview \nof the experimental apparatus and its components. Some tests of the experimental apparatus \nare then presented followed by a discussion of the results. Lastly, some interesting \nfollow up experiments are de ned and discussed. \n \nBohmian mechanics, while completely agreeing with quantum mechanics could not be \nmore di erent from the standard notions such as the fact that particles retain their classical \nidentity of a point like object that have de nite and causal trajectories. In this realm, rst \ndeveloped by de Broglie, these particles move and are guided by a wave, which turns out \nto be the wave function itself. Typically, the velocity of the particles anywhere in space \nis given by spatial derivatives of the phase of this wave and the energy content of the \nparticles is given by the temporal derivative. The trajectories can bend even when there \nare no outside (classical) forces that are acting on the objects. This is due to the fact that \nthese observations of Bohmian mechanics must conform to the standard measurements in \nquantum mechanics. \n \nOne detail that has gone unmeasured (in the sense of inference) in a laboratory setting, \nis the quantum potential (the mysterious force that moves particle to make sure they \nreproduce the measurements of quantum mechanics). Using the technique of measuring \nthe trajectories of photons, the question was asked, can the quantum potential for single \nphotons in the double slit be inferred? A setup has been built in order to try and answer this \nquestion using a single photon detecting camera and a heralding spontaneous parametric \ndown conversion source. Unfortunately, the quantum potential still goes unmeasured due \nto noise in the camera pictures destroying the integrity of the signal and an interesting \nproblem of trying to stitch together the quantum potential at the varying z-planes.
Bohmian mechanics is a theory that provides a consistent explanation of quantum phenomena in terms of point particles whose motion is guided by the wave function. In this theory, the state of a system of particles is defined by the actual positions of the particles and the wave function of the system; and the state of the system evolves deterministically. Thus, the Bohmian state can be compared with the state in classical mechanics, which is given by the positions and momenta of all the particles, and which also evolves deterministically. However, while in classical mechanics it is usually taken for granted and considered unproblematic that the state is, at least in principle, measurable, this is not the case in Bohmian mechanics. Due to the linearity of the quantum dynamical laws, one essential component of the Bohmian state, the wave function, is not directly measurable. Moreover, it turns out that the measurement of the other component of the state — the positions of the particles — must be mediated by the wave function; a fact that in turn implies that the positions of the particles, though measurable, are constrained by absolute uncertainty. This is the key to understanding how Bohmian mechanics, despite being deterministic, can account for all quantum predictions, including quantum randomness and uncertainty.
We show that it is possible to explain the quantum measurement process within the framework of quantum mechanics without any additional postulates. The key concept of the theory is decoherence, which appears as an inherent characteristic of quantum mechanics and results from the uncertainty relation. In contrast to environment-induced decoherence, this decoherence exists prior to a measurement being made. To clarify our idea, we examine three elemental experiments: a Stern-Gerlach-like experiment, the Einstein-Podolsky-Rosen-Bohm (EPR-Bohm) experiment, and the double-slit experiment. By considering the first experiment, we explain how the uncertainty relation between position and momentum introduces decoherence prior to measurement. Consideration of the EPR-Bohm experiment leads us to conclude that the correlation of the EPR pair is not a consequence of what is known as the collapse of the wave function. Our theory of decoherence can also be applied to experiments with continuous eigenvalues, such as the double-slit experiment. Consideration of the double-slit experiment leads us to understand how pure quantum mechanics describes the fact that quanta behave as interfering particles.
Bohm’s mechanics was built for explaining individual results in measurements, and mainly for getting rid of the enigmatic reduction postulate. Its main idea is that particles have at any time definite positions and velocities. An additional axiom is that particles follow continuous trajectories that admit the first derivative in time, the velocity. In the quantum theory, if the position of a quantum object is well-defined at some time, a Δt time later the object may be found anywhere in space, so, the velocity defined as Δx/Δt is completely undefined. This incompatibility is regarded in standard quantum theory as nature’s property. The disagreement between quantum and Bohm’s mechanics is particularly strong in wave-like phenomena, e.g. interference. For a particle traveling through an interference fringe, Bohm’s velocity formula shows a dependence of the time-of-flight on the fringe length. Such a dependence is not supported by the quantum theory. Thus, for deciding which prediction is correct one has to measure times-of-flight. But this is a problem. If one detects a particle at two positions and records the detection times, the time difference is meaningless, because the first position measurement disturbs the particle’s Bohm velocity (if exists). This text suggests a way around: instead of measuring positions and times, the particles are raised to an excited, unstable level, by passing them through a laser beam. The unstable level will decay in time, s.t. the density of probability of the excited atoms will indicate the time elapsed since excitation. For comparing the Bohmian and quantum predictions, this text proposes in continuation to send the beam of excited particle upon a mirror. Bohm’s velocity leads to anomalies in the reflected wave.
We discuss a recently proposed extension of Bohmian mechanics to quantum field theory. For more or less any regularized quantum field theory there is a corresponding theory of particle motion, which, in particular, ascribes trajectories to the electrons or whatever sort of particles the quantum field theory is about. Corresponding to the nonconservation of the particle number operator in the quantum field theory, the theory describes explicit creation and annihilation events: the world lines for the particles can begin and end.