In this article we will demonstrate that the detection of electrons in an interference experiment with electrons is a transition between two Hilbert spaces caused by interactions with the detecting screen. The first Hilbert space is that of the free electrons before detection and the second is that of the bound electrons in the screen after detection. This transition can be described by unitary quantum-mechanical perturbation theory where the grand Hilbert space required to describe the electrons is the direct sum of these two Hilbert spaces. We can say an electron is detected at the screen only after the interaction between the two has caused an observable effect like an atomic transition. Detection of the electrons does not solely indicate modification of the quantum states taking place in the Hilbert space used to describe the electron before detection. It cannot be used to imply a non-unitary transformation of the quantum states belonging to this Hilbert space only. Similarly, emission of electrons from a material to free space is a transition from the set of bound states to the set of free states. This transition can be described by using time-dependent perturbation theory without referring to any non-unitary transformation. In both the cases, relevant transitions would be given by the time evolution of exact solutions of the interacting theory and rescind the requirement of introducing any non-unitary transformation. This interpretation may be extended to other measurement processes. We will conclude the article with a brief discussion on non-locality in quantum mechanics. This is based on the necessity of complex functions in quantum theory and perception of human brain on imaginary numbers.
In this article we will demonstrate that the detection of electrons in an interference experiment with electrons is a transition between two Hilbert spaces caused by interactions with the detecting screen. The first Hilbert space is that of the free electrons before detection and the second is that of the bound electrons in the screen after detection. This transition can be described by unitary quantum-mechanical perturbation theory where the grand Hilbert space required to describe the electrons is the direct sum of these two Hilbert spaces. We can say an electron is detected at the screen only after the interaction between the two has caused an observable effect like an atomic transition. Detection of the electrons does not solely indicate modification of the quantum states taking place in the Hilbert space used to describe the electron before detection. It cannot be used to imply a non-unitary transformation of the quantum states belonging to this Hilbert space only. Similarly, emission of electrons from a material to free space is a transition from the set of bound states to the set of free states. This transition can be described by using time-dependent perturbation theory without referring to any non-unitary transformation. In both the cases, relevant transitions would be given by the time evolution of exact solutions of the interacting theory and rescind the requirement of introducing any non-unitary transformation. This interpretation may be extended to other measurement processes. We will conclude the article with a brief discussion on non-locality in quantum mechanics. This is based on the necessity of complex functions in quantum theory and perception of human brain on imaginary numbers.
In order to resolve the measurement problem of Quantum Mechanics, non-unitary time evolution has been derived from the unitarity of standard quantum formalism. New wave functions of free and non-free quantum systems follow from Schroedinger equation after inserting an ansatz. Quantum systems show up as probability waves before measurement. A pure entangled state of a composite system evolves non-unitarily, only to disentangle itself into a definite state after reduction at the measurement point. A classical space-time point is created momentarily in this event. Unitarity is restored at that point. The non-Hermitian observables defined in the domain of rigged Hilbert space transform into Hermitian ones at the measurement point. The problem of preferred basis is resolved by the requirement of specifying the position of measurement point. Two theorems prove that time is a non-Hermitian operator, thus placing space and time on an equal footing. Bound states are found to need discrete space-time, which supports its use in loop quantum gravity. Non-unitarity in the theory helps buttress the no-boundary proposal; and uncertainty relation makes a leeway to singularity-free Quantum Cosmology. Quantum Mechanics also accommodates complex and negative probabilities.
The Quantum Mechanics is interpreted, in this article, in a simple and direct way. By combining the unitary evolution with the quantum condition that observations require the state vector to be an eigenstate of the observable, a discontinuity in evolution (the state vector reduction) seems to be mandatory. Thus, for each such discontinuity, new initial conditions for the time evolution state vector are needed, and they are obtained by measurements. Delayed-choice experiments suggest that these new initial conditions are specified after the discontinuity takes place. Consequently, because it needs initial conditions that can be specified with a delay, the time evolving state vector is semi-realistic (in the sense that it is not completely specified until the measurement is performed), and not entirely realistic. The collapse of the wave function, especially when it is combined with the entanglement, seems to be a non-local phenomenon. In fact, the non-locality is present only as a consistency requirement for the initial conditions needed to select a solution of the evolution equation. The Direct Interpretation is intended to provide to our intuition a physical background, for helping us thinking about quantum phenomena. It identifies the main counterintuitive parts of the Quantum Mechanics in the discontinuity and the delayed initial conditions. Because it makes minimal assumptions, it is compatible with the main interpretations of Quantum Mechanics. Two principal unclear points of Quantum Mechanics are identified in the discontinuities, and the measurement problem. Both problems will be approached in subsequent articles.
The basic requirement that, in quantum theory, the time evolution of any state is determined by the action of a unitary operator, is shown to be the underlying cause for certain ``exact'' results that have recently been reported about the time dependence of transition rates in quantum theory. Departures from exponential decay, including the ``quantum Zeno effect,'' as well as a theorem by Khalfin about the ratio of reciprocal transition rates, are shown to follow directly from such considerations. At sufficiently short times, unitarity requires that reciprocity must hold, independent of whether T invariance is valid. If T invariance does not hold, unitarity restricts the form of possible time dependence of reciprocity ratios. \textcopyright{} 1996 The American Physical Society.
The causal quantum mechanics (i.e. Bohmian or de Broglie-Bohm or Bohm-de Broglie quantum mechanics) has made possible to calculate the trajectories of electrons in a typical double-slit experiment [C. Philippidis et al., Il Nuovo Cimento, 52 B, 15-28 (1979)]. The trajectories do not correspond to an uniform movement but to an accelerated one. The acceleration is caused by the quantum potential. From the quantum theoretical point of view, there is a probability for the electron to emit photons, with a certain emission power, during its movement from the slits to the screen. We find a quantum general formula for the emission power of photons, valid independently of the interpretation. Then, according to the Copenhagen interpretation, this formula gives a strictly zero value for the emission power because the electron moves as a free particle after it leaves the slit and before reach the screen. Then, there is no emission of photons. In the case of the causal interpretation, the emission power results, for a concrete real experiment, in a very tiny but not a zero value, driven by the square of the quantum potential gradient. We give an idea of the type of spectrum that could be measured. A brief idea of a possible experimental arrangement in order to detect this effect, is given.
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.
The presence of the imaginary unit i in quantum mechanics is frequently treated as a computational postulate. We demonstrate that the complex structure of Hilbert space is not a decorative choice but a structural necessity derived from the first principles of modern physics: continuous symmetry, the requirement of unitarity to preserve probability, and the correspondence between generators and observables. By examining the Lie algebra of the unitary group, we prove that a real-valued theory fails to map self-adjoint observables to generators of continuous evolution without sacrificing local tomography or the existence of non-trivial phases. We conclude that i is the unique logical bridge between symmetry and observability.