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Register a new userUnified Description of Matrix Mechanics and Wave Mechanics II
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In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.
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In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.
This article is the second in a series of two presenting the Scale Relativistic approach to non-differentiability in mechanics and its relation to quantum mechanics. Here, we show Schroedingers equation to be a reformulation of Newtons fundamental relation of dynamics as generalized to non-differentiable geometries in the first paper cite{paper1}. It motivates an alternative interpretation of the other axioms of standard quantum mechanics in a coherent picture. This exercise validates the Scale Relativistic approach and, at the same time, it allows to identify macroscopic chaotic systems considered at time scales exceeding their horizon of predictability as candidates in which to search for quantum-like structuring or behavior.
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the study of separability and entanglement for states of composite quantum systems.
It is demonstrated how quantum mechanics is generated by stochastic momentum kicks from the force carriers, transmitting the fundamental interactions between the point particles. The picture is consistent with quantum field theory and points out that the force carriers are the only quantum particles. Since the latter are waves in the coordinate space, they are responsible for the wavy character of quantum mechanics.
Large $N$ matrix quantum mechanics is central to holographic duality but not solvable in the most interesting cases. We show that the spectrum and simple expectation values in these theories can be obtained numerically via a `bootstrap methodology. In this approach, operator expectation values are related by symmetries -- such as time translation and $SU(N)$ gauge invariance -- and then bounded with certain positivity constraints. We first demonstrate how this method efficiently solves the conventional quantum anharmonic oscillator. We then reproduce the known solution of large $N$ single matrix quantum mechanics. Finally, we present new results on the ground state of large $N$ two matrix quantum mechanics.
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