FEYNMAN'S PATH INTEGRAL

 

Mathematics Learning

1 ημ. 

·

FEYNMAN'S PATH INTEGRAL 

Developed in the late 1940s by Richard Feynman, the path integral formulation is a revolutionary approach to quantum mechanics that calculates probability amplitudes by summing over all possible particle paths, rather than just the classical trajectory. Inspired by Dirac’s work and Wiener’s Brownian motion studies, Feynman introduced this concept to describe quantum electrodynamics (QED) and provide a more intuitive Lagrangian view.

Richard Feynman's favorite integral trick 

The path integral devised in 1948 by Richard Feynman...

Feynman’s integral trick is a method of evaluating integrals by differentiating them with respect to a parameter that is introduced artificially. The idea is to find a function of two variables, F(x, t), such that the original integral is equal to F(x, 0) for some x. Then, by applying the chain rule, we can obtain a differential equation for F(x, t) and solve it using initial or boundary conditions. The solution will give us the value of the original integral.

• Introduce a parameter t and define a function F(x, t) such that the original integral is equal to F(x, 0) for some x.

• Differentiate F(x, t) with respect to t and obtain a differential equation for F(x, t).

• Solve the differential equation using initial or boundary conditions and find F(x, t) as a function of t.

• Plug in t = 0 and get the value of the original integral as F(x, 0).

In his autobiography Surely You're Joking Mr. Feynman!, he mentions,

I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. It showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. If guys at MIT or Princeton had trouble doing a certain integral, then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.

Today (May 11) is the 108th BIRTHDAY Anniversary to RICHARD FEYNMAN ..... 

Δείτε λιγ

ότερα