By Laurie M. Brown Richard Phillips Feynman
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Extra resources for Feynman's Thesis: A New Approach to Quantum Theory
That for the oscillator is, m¨ x + mω 2 x = [Iy (t) + Iz (t)] . (25) The solution of this last equation is, where we have written γ = I y +Iz x(t) = x(0) cos ωt + x(0) ˙ t sin ωt 1 + ω mω γ(s) sin ω(t − s)ds . (26) 0 This may be expressed in other ways, for example, x(t) = 1 sin ω(T − t) x(0) − sin ωT mω + 1 sin ωt x(T ) − sin ωT mω t sin ωsγ(s)ds 0 T t sin ω(T − s)γ(s)ds , (27) August 31, 2005 15:31 WSPC/Book Trim Size for 9in x 6in feynman The Principle of Least Action in Quantum Mechanics 19 or again, x(t) = 1 [RT sin ωt + R0 sin ω(T − t)] sin ωT + − 1 2mω 1 2mω t sin ω(t − s)γ(s)ds 0 T sin ω(t − s)γ(s)ds , (28) t where we have written R0 = sin ωT 1 x(0) + x(T ) cos ωT − x(T ˙ ) 2 ω RT = sin ωT 1 x(T ) + x(0) cos ωT − x(0) ˙ 2 ω (29) and .
We will suppose, for simplicity that there is only one coordinate, q(σ), instead of the N coordinates qn (σ).
A m 2x2 (44) The left hand side of this, for very small ε approaches ψ(x, t) so that for the equality to hold we must choose, ψ(x, t + ε) = e ψ(x, t) + A(ε) = 2π εi . m (45) Expanding both sides of (44) in powers of ε up to the ﬁrst, we ﬁnd, ψ(x, t) + ε iε iε ∂ 2 ψ ∂ψ(x, t) = ψ(x, t) − V (x)ψ(x, t) + , ∂t 2m ∂x2 and therefore, − 2 ∂2ψ ∂ψ =− + V (x)ψ i ∂t 2m ∂x2 which is just Schr¨odinger’s equation for the system in question. This conﬁrms the remark that equation (41) is equivalent to Schr¨ odinger’s diﬀerential equation for the wave function ψ.
Feynman's Thesis: A New Approach to Quantum Theory by Laurie M. Brown Richard Phillips Feynman