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By Steven M. Christensen
The 33 essays during this quantity jointly supply a necessary evaluate of the development in the direction of a idea of quantum gravity, and provide succinct remedy of: quantum box concept in curved space-times, supergravity, cosmology, black holes, quantum geometry, supersymmetry and similar phenomena
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3. 5 A quantum particle is observed at first at initial position xi at time ti and a second time at final position xf at time tf . The quantum particle’s path being indeterminate means that the single particle simultaneously exists in all the allowed paths. 24 The mathematical structure of quantum mechanics What is the description of the quantum particle making a transition from xi , ti to xf , tf when it is not observed at an intermediate time t? The following is a summary of the conclusions: • The quantum indeterminacy of the degree of freedom, together with the linearity of the Schrödinger equation, leads to the conclusion that the path of the quantum particle is indeterminate.
1 Continuous degree of freedom Continuous and discrete degrees of freedom occur widely in quantum mechanics. An in-depth analysis of a discrete degree of freedom is presented in Chapter 8. In this chapter, the focus is on analysis of a continuous degree of freedom and its state space and operators. 1 Discretization of a continuous degree of freedom space F = . emerge naturally by taking the continuum limit of an underlying system consisting of a discrete degree of freedom. Consider a quantum particle that can be detected by the position projection operators at any point of space; to simplify the discussion suppose the particle can move in only one dimension and hence can be found at any point x ∈ [−∞, +∞] = .
From Eq. 10 d (b − a) = δ(b − a). db A representation of the delta function based on the Gaussian distribution is δ(x − a) = lim √ σ →0 1 2π σ 2 exp − 1 (x − a)2 . 12) Moreover 1 μ exp −μ|x − a| . μ→∞ 2 δ(x − a) = lim The definition of Fourier transform yields a representation of the Dirac delta function. It can be shown that δ(x − a) = +∞ −∞ dp ip(x−a) . 13) A proof of Eq. 13 is found in the book on quantum mechanics by Landau and Lifshitz (2003). One can perform the following consistency check of Eq.