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Example text

A(T − tm ) where in the sum we shall write Q for q m+1 , and T for tm+1 . In the limit as we take ﬁner and ﬁner subdivisions of the interval t 0 to T and thus make an ever increasing number of successive integrations, the expression on the right side of (47) becomes equal to ψ(Q, T ). T ˙ q)dt with the integral The sum in the exponential resembles t0 L(q, written as a Riemann sum. 32 Feynman’s Thesis — A New Approach to Quantum Theory In a similar manner we can compute ψ(q 0 , t0 ) in terms of the wave function at the later time T = tm+1 , by the equation; ψ ∗ (q0 , t0 ) = ψ ∗ (qm+1 , tm+1 ) ··· × exp i m L i=0 qi+1 − qi , qi+1 · (ti+1 − ti ) ti+1 − ti √ √ gm+1 dqm+1 · · · g1 dq1 .

12 Feynman’s Thesis — A New Approach to Quantum Theory The conservation of a physical quantity is of considerable interest because in solving problems it permits us to forget a great number of details. The conservation of energy can be derived from the laws of motion, but its value lies in the fact that by the use of it certain broad aspects of a problem may be discussed, without going into the great detail that is often required by a direct use of the laws of motion. To compute the quantity I(t), of equation (11), for two diﬀerent times, t1 and t2 that are far apart, in order to compare I(t 1 ) with I(t2 ), it is necessary to have detailed information of the path during the entire interval t1 to t2 .

F = −V (x)), then the integrand is a perfect diﬀerential, and may be integrated to become 1 2 + V (x(t)). A comparison of I for two times, t and t , ˙ 1 2 2 m(x(t)) now depends only on the motion in the neighborhood of these times, all of the intermediate details being, so to speak, integrated out. We therefore require two things if a quantity I(t) is to attract our attention as being dynamically important. The ﬁrst is that it be conserved, I(t1 ) = I(t2 ). The second is that I(t) should depend only locally on the path.

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