Download Feynman's Thesis: A New Approach to Quantum Theory by Laurie M. Brown Richard Phillips Feynman PDF

By Laurie M. Brown Richard Phillips Feynman

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0 δq1 (t) δq2 (t) δqN (t) (7) δA = 0, as if there were only one (We shall often simply write δq(t) variable). That is to say if all the derivatives of A , with respect to qn (t), computed for the functions q¯m (σ) are zero for all t and all n, then q¯m (σ) describes a possible mechanical motion for the systems. We have given an example, in equation (5), for the usual one dimensional problem when the action is the time integral of a Lagrangian (a function of position and velocity, only). As another example consider an action function arising in connection with the theory of action at a distance: A = ∞ −∞ 2 m(x(t)) ˙ ˙ x(t ˙ + T0 ) dt .

The time it takes for light to reach the mirror from the particle is assumed constant, and equal to T 0 /2. The quantity August 31, 2005 10 15:31 WSPC/Book Trim Size for 9in x 6in feynman Feynman’s Thesis — A New Approach to Quantum Theory k2 depends on the charge on the particle and its distance from the mirror. If we vary x(t) by a small amount, λ(t), the consequent variation in A is, ∞ δA = −∞ ˙ ˙ x(t {mx(t) ˙ λ(t) − V (x(t))λ(t) + k 2 λ(t) ˙ + T0 ) ˙ + T0 )x(t)}dt ˙ + k 2 λ(t ∞ = −∞ {−m¨ x(t) − V (x(t)) − k 2 x ¨(t + T0 ) ¨(t − T0 )}λ(t)dt , − k2 x by integrating by part so that, according to our definition (4), we may write, δA ¨(t + T0 ) − k 2 x ¨(t − T0 ) .

S= (21 ) T Taking an infinitesimal time interval t to t + δt, we see that (q t+δt |qt ) iLδt . This result gives probably the most fundacorresponds to e mental quantum analogue for the classical Lagrangian function. It is preferable for the sake of analogy to consider the classical Lagrangian as a function of the coordinates at time t and the coordinates at time t+δt, instead of a function of the coordinates and velocities at time t. There is an important action principle in classical mechanics concerning Hamilton’s principal function (21).

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