We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on $${\mathbb{R}^n}$$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in $${\hbar}$$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the $${t \to 0}$$ limit approaches the δ distribution. As such, our construction gives explicitly the full $${\hbar\to 0}$$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.

arXiv:1003.1156Letters in Mathematical PhysicsSep 29, 20108 citations

The Motion of a Dirac Electron in a Magnetic Field

L. D. Huff

Physical ReviewAug 1, 193137 citations

Annals of Physics

Solution of the Schrödinger equation in terms of classical paths

Roger Balian, Claude Bloch

Published Jun 1, 1974• 380 citations

Publisher

DOI: 10.1016/0003-4916(74)90421-7

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Theo Johnson-Freyd

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on ${\mathbb{R}^n}$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in ${\hbar}$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the ${t \to 0}$ limit approaches the δ distribution. As such, our construction gives explicitly the full ${\hbar\to 0}$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on $${\mathbb{R}^n}$$ , with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in $${\hbar}$$ , given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the $${t \to 0}$$ limit approaches the δ distribution. As such, our construction gives explicitly the full $${\hbar\to 0}$$ asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.

arXiv:1003.1156Letters in Mathematical PhysicsSep 29, 20108 citations

The Motion of a Dirac Electron in a Magnetic Field

L. D. Huff

Solutions of the Dirac wave equation representing a free electron moving in a uniform magnetic field are obtained. The functions are similar to those obtained by Landau and by Uhlenbeck and Young as solutions of the Schr\"odinger equation. A wave packet is constructed representing a beam of electrons passing through a slit. The results agree with the classical predictions to terms of the order of the de Broglie wave-length of the electron divided by the radius of curvature of its classical path. For experimental cases this ratio is of the order ${10}^{\ensuremath{-}8}$ to ${10}^{\ensuremath{-}10}$. Hence it is concluded that the difference between magnetic deflection measurements of $\frac{e}{m}$ and other determinations cannot be explained as a quantum effect.Solutions of the Dirac wave equation representing a free electron moving in a uniform magnetic field are obtained. The functions are similar to those obtained by Landau and by Uhlenbeck and Young as solutions of the Schr\"odinger equation. A wave packet is constructed representing a beam of electrons passing through a slit. The results agree with the classical predictions to terms of the order of the de Broglie wave-length of the electron divided by the radius of curvature of its classical path. For experimental cases this ratio is of the order ${10}^{\ensuremath{-}8}$ to ${10}^{\ensuremath{-}10}$. Hence it is concluded that the difference between magnetic deflection measurements of $\frac{e}{m}$ and other determinations cannot be explained as a quantum effect.

Physical ReviewAug 1, 193137 citations

We describe the “Feynman diagram” approach to nonrelativistic quantum mechanics on

{\mathbb{R}^n}

, with magnetic and potential terms. In particular, for each classical path γ connecting points q 0 and q 1 in time t, we define a formal power series V γ (t, q 0, q 1) in

{\hbar}

, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V γ ) satisfies Schrödinger’s equation, and explain in what sense the

{t \to 0}

limit approaches the δ distribution. As such, our construction gives explicitly the full

{\hbar\to 0}

asymptotics of the fundamental solution to Schrödinger’s equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman’s path integral in diagrams.

\frac{e}{m}

and other determinations cannot be explained as a quantum effect.