Abstract In Non-Relativistic Quantum Mechanics, the Schr¨odinger equation describes the evolution of a quantum system as a superposition of different states, but cannot explain why the system is only ever measured to be in one of these states. Here we show a new dynamical wave equation for the evolution of the wave function, and then apply this equation to a variety of different physical cases, showing how the wave function changes when we move from one potential to another. We then show how a measurement in quantum mechanics is an example of a potential changing the possible eigenvalues associated with a quantum system. The dynamical process of a measurement that evolves the wave function not only returns a single eigenvalue for the basis it is measuring, but also returns a new Hamiltonian with corresponding eigenvalues. This shows, contrary to some interpretations, that the wave function is continuous after measurement.
Abstract In Non-Relativistic Quantum Mechanics, the Schr¨odinger equation describes the evolution of a quantum system as a superposition of different states, but cannot explain why the system is only ever measured to be in one of these states. Here we show a new dynamical wave equation for the evolution of the wave function, and then apply this equation to a variety of different physical cases, showing how the wave function changes when we move from one potential to another. We then show how a measurement in quantum mechanics is an example of a potential changing the possible eigenvalues associated with a quantum system. The dynamical process of a measurement that evolves the wave function not only returns a single eigenvalue for the basis it is measuring, but also returns a new Hamiltonian with corresponding eigenvalues. This shows, contrary to some interpretations, that the wave function is continuous after measurement.
The measurement problem in quantum mechanics can be characterized as explaining the question of definite outcomes, where a measurement of a superposed quantum state only gives a single definite outcome. According to quantum mechanics, the wave function, ψ, defines the state of the system, while the Schrödinger equation is a linear superposition of states that evolves deterministically. When a measurement is made, the outcome always uncovers the physical system as being in a definite state, and the future evolution of the wave function is amended based on the state that was measured. This change is often referred to as the “collapse” of the wave function. The problem concerns the question of what and how the measurement affected the system and why this changed the wave function. This paper addresses the measurement problem by separating it into five separate questions that are resolved through linking measurements to an entropy transfer from the superposed state to the environment. The underlying premise is the wave function is descriptive and not prescriptive, and entropy—the metric of quantum uncertainty and uncertainty in general—is the fundamental property rather than the wave function.
Background and Objectives: There are a few different ways to change the quantum mechanical wave function. One is the continuous evolution according to the Schrodinger equation; the others are the collapse provided by strong measurements and the weak continuous measurement. The question arises: Can the strong measurement be considered as a dynamical process. Methods: To answer the question, a dynamic model is proposed to simulate numerically the measurement of the coordinates of quantum particles. By measuring the coordinate, we mean the capture of a particle in a narrow box and the excitation of the stationary wave function in it. We consider the amplitude of the probability of transition to a stationary state as a wave function, and the non-stationary Schrodinger equation is solved using this initial function. The transition to a stationary state was modeled as a gated process. Results and Conclusion: It is shown that when measuring coordinates, along with the instant collapse, there are cases when the measurement takes a short but finite time. In particular, the projective measurement of the coordinate of one of the entangled particles takes some time and is accompanied by the correlation in the probability density of the particles coordinates. Moreover, the correlation is lost over time.
The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities (or eigen states).This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation.However, when a measurement is carried out on the system to determine the value of that property (say position), the system instantaneously transforms to one of the eigen states and thus we get only a single value as an outcome of the measurement.Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation.In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real-world and will identify the root cause behind the stochastic nature of events.Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement.In the process of development of model, we will explain Schrodinger cat paradox and will show how Born's rule for probability becomes a natural consequence of the measurement process.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
The foundations of quantum mechanics are reconsidered in relation to the measurement problem. We define two kinds of quantities. The type A quantities are observables obtained experimentally from a single measurement. A dynamical variable is a type A quantity whose measurement selects a set of eigenvalues (EVs) as c-numbers. The type B quantities represent theoretical quantities related to probabilities. State vectors, the wave function, and the probability distribution (PD) are type B quantities. They are derived from an eigenvalue equation such as the Schrödinger equation. A quantum jump (QJ) is the measurement of a dynamical variable to select a set of EVs. The wave function is a theoretical quantity that persists throughout an experiment and therefore does not collapse. After a QJ, a selected set of EVs persists until it becomes macroscopic. However, an eigenstate is a theoretical notion irrelevant to a single measurement and does not exist in reality after a QJ. One experiment is complete when an ensemble of EVs has been obtained and compared with the theoretically predicted PD. To obtain an ensemble, repeated measurements are required in general, but there are some exceptions, like a Bose-Einstein condensate system, for which a single measurement yields a real ensemble. We comment on the relation between the reality of the wave function and the derivation of non-relativistic quantum mechanics from relativistic quantum field theory.
The measurement problem of quantum mechanics concerns the question as to under which circumstances coherent wave evolution becomes disrupted to produce eigenstates of observables, instead of evolving superpositions of eigenstates. The problem already needs to be addressed within wave mechanics, before second quantization, because low-energy interactions can be dominated by particle-preserving potential interactions. We discuss a scattering array of harmonic oscillators, which can detect particles penetrating the array through interaction with a short-range potential. Evolution of the wave function of scattered particles, combined with Heisenberg’s assertion that quantum jumps persist in wave mechanics, indicates that the wave function will collapse around single oscillator sites if the scattering is inelastic, while it will not collapse around single sites for elastic scattering. The Born rule for position observation is then equivalent to the statement that the wave function for inelastic scattering amounts to an epistemic superposition of possible scattering states, in the sense that it describes a sum of probability amplitudes for inelastic scattering off different scattering centers, whereas, at most, one inelastic scattering event can happen at any moment in time. Within this epistemic interpretation of the wave function, the actual underlying inelastic scattering event corresponds to a quantum jump, whereas the continuously evolving wave function only describes the continuous evolution of probability amplitudes for scattering off different sites. Quantum jumps then yield definite position observations, as defined by the spatial resolution of the oscillator array.