theories:quantum_mechanics:path_integral
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theories:quantum_mechanics:path_integral [2018/05/04 11:14] jakobadmin [FAQ] |
theories:quantum_mechanics:path_integral [2022/09/12 21:33] (current) 207.34.115.128 |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | <blockquote>In this theory, the quantum particle (be it a photon, a positron, an electron, and so on) goes from an initial event to a (nearby) later event along each and every path - there’s an infinity of them (shades of Huygens Principle). The final amplitude is found by adding up the contributions from all these alternative paths. However, as the different paths have different lengths, then the phase-advance will be different and not necessarily equal to a whole number of wavelengths, and so the waves will not arrive instep. Only a very few nearby paths will add coherently and contribute significantly to the final amplitude; most paths will add destructively and yield no net contribution. (As Freeman Dyson reported it: “Dick Feynman told me. . . “The electron does anything it likes. . . It just goes in any direction at any speed. . . however it likes, and then you add up the amplitude and it gives you the wave-function.” I said to him, “You’re crazy.” But he wasn’t.”) Feynman’s beautifully simple theory not only agrees very precisely with experiment but explains away the teleological objection - that the particle doesn’t ‘know’ which path to follow - wrong, it follows all paths. Also, the theory goes over into the classical limit: for massive particles, say, a football moving at 10 ms$^{–1}$, the wavelength is so miniscule (around $10^{-34}$ m) that there isn’t the remotest hope for paths to interfere constructively - only one path survives, the very path predicted by the Principle of Least Action. | ||
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| + | <cite>The Lazy Universe by Coopersmith</cite></blockquote> | ||
| + | |||
| + | ---- | ||
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| + | **Origin of the Path Integral** | ||
| The path integral idea emerges naturally when we consider what happens when we have many [[experiments:double_slit_experiment|double slit experiments]] in a row: | The path integral idea emerges naturally when we consider what happens when we have many [[experiments:double_slit_experiment|double slit experiments]] in a row: | ||
| + | |||
| + | {{ :theories:quantum_mechanics:pathintegral.png?nolink&800 |}} | ||
| <blockquote>Recall the famous double-slit experiment | <blockquote>Recall the famous double-slit experiment | ||
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| </blockquote> | </blockquote> | ||
| - | [{{ :advanced_tools:ff570b79-257d-46ec-9672-f9f2a9d88518.jpeg?nolink |source: http://www.pd.infn.it/statphys_venice_2013/talks_stat_fin_2013/tempere.pptx}}] | ||
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| Each amplitude is basically a [[basic_tools:complex_analysis|complex number]] and thus can be represented by an arrow in the complex plane. | Each amplitude is basically a [[basic_tools:complex_analysis|complex number]] and thus can be represented by an arrow in the complex plane. | ||
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| * http://qu-bit.narod.ru/texts/feynman.pdf | * http://qu-bit.narod.ru/texts/feynman.pdf | ||
| * Shankar, Principles of QM, the Chapter "Path integrals revisited" is especially useful for students interested in condensed matter physics. | * Shankar, Principles of QM, the Chapter "Path integrals revisited" is especially useful for students interested in condensed matter physics. | ||
| + | * Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets by H. Kleinert is the bible on path integrals. | ||
| + | * [[https://fondationlouisdebroglie.org/AFLB-312/aflb312m504.pdf|Majorana and the path-integral approach to | ||
| + | Quantum Mechanics]] by S. Esposito, Annales de la Fondation Louis de Broglie, Volume 31 no 2-3, pp. 207-225, 2006 | ||
| ---- | ---- | ||
| - | The path integral formalism is what we get when we apply the [[formalisms:lagrangian_formalism|Lagrangian framework]] to [[theories:quantum_mechanics:canonical_quantum_mechanics|quantum mechanics]]. This is described, for example, [[http://www.physics.utah.edu/~starykh/phys7640/Lectures/FeynmansDerivation.pdf.|here]]. | + | The path integral formalism is what we get when we apply the [[formalisms:lagrangian_formalism|Lagrangian framework]] to [[theories:quantum_mechanics:canonical|quantum mechanics]]. This is described, for example, [[http://www.physics.utah.edu/~starykh/phys7640/Lectures/FeynmansDerivation.pdf.|here]]. |
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | The path integral is extremely useful to understand global properties of a system in [[theories:quantum_field_theory|quantum field theory]]. | + | The path integral is extremely useful to understand global properties of a system in [[theories:quantum_field_theory:canonical|quantum field theory]]. |
| In addition to these standard applications, the path integral is also somewhat popular in quantitative finance. | In addition to these standard applications, the path integral is also somewhat popular in quantitative finance. | ||
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| of viewing quantum mechanics: anyone who can understand Young’s double slit experiment | of viewing quantum mechanics: anyone who can understand Young’s double slit experiment | ||
| in optics should be able to understand the underlying ideas behind path integrals. Secondly, | in optics should be able to understand the underlying ideas behind path integrals. Secondly, | ||
| - | the [[theories:classical_theories|classical limit]] of [[theories:quantum_mechanics:canonical_quantum_mechanics|quantum mechanics]] can be understood in a particularly clean way via | + | the [[theories:classical_theories|classical limit]] of [[theories:quantum_mechanics:canonical|quantum mechanics]] can be understood in a particularly clean way via |
| path integrals. | path integrals. | ||
| - | It is in [[theories:quantum_field_theory|quantum field theory]], both relativistic and nonrelativistic, that path integrals | + | It is in [[theories:quantum_field_theory:canonical|quantum field theory]], both relativistic and nonrelativistic, that path integrals |
| (functional integrals is a more accurate term) play a much more important role, for several reasons. They provide a relatively easy road to [[advanced_tools:quantization|quantization]] and to expressions for Green’s | (functional integrals is a more accurate term) play a much more important role, for several reasons. They provide a relatively easy road to [[advanced_tools:quantization|quantization]] and to expressions for Green’s | ||
| functions, which are closely related to amplitudes for physical processes such as scattering | functions, which are closely related to amplitudes for physical processes such as scattering | ||
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| <blockquote> | <blockquote> | ||
| - | The path integral formulation of [[theories:quantum_mechanics:canonical_quantum_mechanics|quantum mechanics]] is the one in which | + | The path integral formulation of [[theories:quantum_mechanics:canonical|quantum mechanics]] is the one in which |
| one can see in the most natural and transparent way that the quantization of a given classical dynamical system depends often in a crucial way upon the global | one can see in the most natural and transparent way that the quantization of a given classical dynamical system depends often in a crucial way upon the global | ||
| topological properties of the underlying configuration space. Indeed, the | topological properties of the underlying configuration space. Indeed, the | ||
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| See https://physics.stackexchange.com/questions/303984/particle-picture-in-the-path-integral-formulation-of-quantum-field-theory | See https://physics.stackexchange.com/questions/303984/particle-picture-in-the-path-integral-formulation-of-quantum-field-theory | ||
| + | <-- | ||
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| + | -->Are the path integral formalism and the operator formalism inequivalent?# | ||
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| + | [[https://physics.stackexchange.com/questions/252213/are-the-path-integral-formalism-and-the-operator-formalism-inequivalent]] | ||
| + | |||
| <-- | <-- | ||
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| <-- | <-- | ||
| + | --> What's the physical Interpretation of the Integrand of the Feynman Path Integral?# | ||
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| + | see [[https://physics.stackexchange.com/questions/61139/physical-interpretation-of-the-integrand-of-the-feynman-path-integral/202298#202298]] | ||
| + | <-- | ||
| <tabbox History> | <tabbox History> | ||
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