Pfadintegral verstanden (Zee)

30/11/2007 - 22:42 von Inge Bein | Report spam
Einige von uns sollten die Lesart des Pfadintegrals
noch verstehen lernen bevor sie Feynman allzusehr
bewundern, die Crux sind unendlich viele Schirme, jeder
mit unendlich vielen Löchern, auf die alle der Befund
für das Doppelspaltexperiment angewendet wird,

A.Zee schreibt also dazu:

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Long ago, in a quantum
mechanics class, the professor droned on and on about the double-slit
experiment, giving the standard treatment. A particle emitted from a
source S (Fig. 1.2.1) at time t = 0 passes through one or the other
of two holes, A 1 and A2, drilled in a screen and is detected at time
t = T by a detector located at O. The amplitude for detection
is given by a fundamental postulate of quantum mechanics,
the superposition principle, as the sum of the amplitude
for the particle to propagate from the source S through the
hole A1 and then onward to the point O and the amplitude for the
particle to propagate from the source S through the hole A2 and
then onward to the point O, the amplitude for the particle
to propagate from the source S through the hole A 2 and then
onward to the point O, and the amplitude for the particle to
propagate from the source $ through the hole A 3 and then
onward to the point O."

The professor was just about ready to continue when Feynman
interjected again, "What if I drill a fourth and a fifth hole
in the screen?" Now the professor is visibly losing his patience:
"All right, wise guy, I think it is obvious to the whole class
that we just sum over all the holes."
To make what the professor said precise, denote the amplitude
for the particle to propagate from the source $ through the
hole A i and then onward to the point O as A(S --> A i > 0).
Then the amplitude for the particle to be detected at the
point O is

A(detected at O)= E A(S > A i > O) (1)

But Feynman persisted, "What if we now add another screen (Fig. 1.2.2)
with some holes drilled in it?" The professor was really losing his
patience: "Look, can't you see that you just take the amplitude to go
from the source $ to the hole A i in the first screen, then to the hole
Bj in the second screen, then to the detector at O, and then sum over
all i and j .9"
Feynman continued to pester, "What if I put in a third screen, a fourth
screen, eh? What if I put in a screen and drill an infinite number of
holes in it so that the screen is no longer there?" The professor sighed,
"Let's move on; there is a lot of material to cover in this course."

But dear reader, surely you see what that wise guy Feynman was driving at.

I especially enjoy his observation that if you put in a screen and
drill an infinite number of holes in it, then that screen is not
really there. Very Zen! What Feynman showed is that even if there
were just empty space between the source and the detector, the amplitude
for the particle to propagate from the source to the detector is the
sum of the amplitudes for the particle to go through each one of the
holes in each one of the (nonexistent) screens. In other words, we have
to sum over the amplitude for the particle to propagate from the source
to the detector following all possible paths between the source and the
detector (Fig. 1.2.3).

A(particle to go from S to O in time T) =
E A (particle to go from S to O in time T following a particular path) (2)
(paths)

Now the mathematically rigorous will surely get anxious over how E(paths)
is to be defined. Feynman followed Newton and Leibniz: ... ...

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Lesen sie die antworten

#1 Hendrik van Hees
01/12/2007 - 07:15 | Warnen spam
Inge Bein wrote:

Einige von uns sollten die Lesart des Pfadintegrals
noch verstehen lernen bevor sie Feynman allzusehr
bewundern, die Crux sind unendlich viele Schirme, jeder
mit unendlich vielen Löchern, auf die alle der Befund
für das Doppelspaltexperiment angewendet wird,



Oh, bitte nicht. Zees QFT-Buch war die größte Enttàuschung der letzten
Jahre auf dem Lehrbuchmarkt. Das müssen wir hier nicht diskutieren.

Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:

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