Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations

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It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the ... continued below

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11 pages

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Weinstein, Marvin & /SLAC February 12, 2009.

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Description

It is apparent to anyone who thinks about it that, to a large degree, the basic concepts of Newtonian physics are quite intuitive, but quantum mechanics is not. My purpose in this talk is to introduce you to a new, much more intuitive way to understand how quantum mechanics works. I begin with an incredibly easy way to derive the time evolution of a Gaussian wave-packet for the case free and harmonic motion without any need to know the eigenstates of the Hamiltonian. This discussion is completely analytic and I will later use it to relate the solution for the behavior of the Gaussian packet to the Feynman path-integral and stationary phase approximation. It will be clear that using the information about the evolution of the Gaussian in this way goes far beyond what the stationary phase approximation tells us. Next, I introduce the concept of the bucket brigade approach to dealing with problems that cannot be handled totally analytically. This approach combines the intuition obtained in the initial discussion, as well as the intuition obtained from the path-integral, with simple numerical tools. My goal is to show that, for any specific process, there is a simple Hilbert space interpretation of the stationary phase approximation. I will then argue that, from the point of view of numerical approximations, the trajectory obtained from my generalization of the stationary phase approximation specifies that subspace of the full Hilbert space that is needed to compute the time evolution of the particular state under the full Hamiltonian. The prescription I will give is totally non-perturbative and we will see, by the grace of Maple animations computed for the case of the anharmonic oscillator Hamiltonian, that this approach allows surprisingly accurate computations to be performed with very little work. I think of this approach to the path-integral as defining what I call a guided numerical approximation scheme. After the discussion of the anharmonic oscillator I will turn to tunneling problems and show that the instanton can also be though of in the same way. I will do this for the classic problem of a double well potential in the extreme limit when the splitting between the two lowest levels is extremely small and the tunneling rate from one well to another is also very small.

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11 pages

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  • Journal Name: PoS LC2008:012,2008; Conference: Contributed to Light Cone 2008 Relativistic Nuclear and Particle Physics, 7-11 Jul 2008, Mulhouse, France

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  • Report No.: SLAC-PUB-13541
  • Grant Number: AC02-76SF00515
  • Office of Scientific & Technical Information Report Number: 947567
  • Archival Resource Key: ark:/67531/metadc897363

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  • February 12, 2009

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  • Sept. 27, 2016, 1:39 a.m.

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  • Nov. 30, 2016, 7:12 p.m.

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Weinstein, Marvin & /SLAC. Quantum Calisthenics: Gaussians, The Path Integral and Guided Numerical Approximations, article, February 12, 2009; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc897363/: accessed August 22, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.