sumF = M * a

(1): sumF = P'

F = -gradU

Fx = -dU/dx

P` = -gradU

P` + gradU = 0

T = M(V^2)/2

T = P^2/(2*M)

(2) L = T - U

(3) gradL = gradT - gradU

(4) dL/dV = dT/dV - dU/dV

dT/dV = d/dV [MV^2/2] = MV = P

d/dt(dL/dV) = d/dt(dT/dV) - 0 = P'

(5) d/dt(dL/dV) = gradL

(1) P' = sumF

W' = dW/dt

Pw = M * dL/dW'

L = T - U = Pw^2 / 2M - U(W)

dL/dW` = dT/dW` = Pw/M

(6) Pw = (ih/2pi) * d/dW

PwK = (ih/2pi) *dK/dW

gradF = (dF/dx, dF/dy, dF/dz)

(7) T K = (1/2M) * P^2 K

P^2 = (ih/2pi)^2 * grad^2

T K = - (h/2pi)^2 * (1/2M) * grad^2K

grad^2 K = d^2K/dx^2 + d^2K/dy^2 + d^2K/dz^2

H = W* Pw - L(w)

H(X,Y,Z,U,V,W, Px,Py,Pz,Pu,Pv,Pw) = XPx + YPy + ZPz +UPu + VPv +WPw - L(...)

H = T + U

H K = TK + UK = UK - (h/2pi)^2 * (1/2M) * grad^2K

(9) dK/dt = (ih/2pi) HK

dW/dt = dH/dPw

dPw/dt = -dH/dW

psi=A*exp[i*2*pi*S/h]

S is the Action

S=int(L)dt (1)

S = E * t - p * x

dS/st = E

dS/dp = -x ; dS/dx = -p

dS = 0

E = h * w / 2pi

P = h * k / 2pi

E * t - P * r = (h/2pi) * (wt - kr)

psi = A * exp [i(wt-kr)]

<psi|psi> = |A|^2