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

{exp(Ki.X)}

{Delta(X-X0)}

<oj|oi>= (0 | i!=j; 1 i==j)

<oi|oi>!=0 ; <oi|oj>=0 (i!=j)

<O> = <a|O|a>/<a|a>=<a|Oa>/<a|a>=(sum(|a_o|^2*o<o|o>)/sum(|a_z|^2<z|z>))

<p|q>=delta(p-q)

"IMPORTANT COMMENT:
The usefulness of the contiuous bases which we have just introduced is revealed more clearly in what follows. However, we must not lose sight of the following point: a physical state MUST ALWAYS correspond to a square-integrable wave function. In no case can |p> or |x> ‏¹ represent the state of a particle."

|<p|E>|^2

|<x|E>|^2 >1

p(x in [x1,x2]) = (int(x1,x2)(<E|x><x|E>dx))/(int(x1,x2)(|x><x|dx))

<x|En> = a*sin(nx)

|b> = Sum(n=1,infinity)[|En><En|b>]

0 = <x1|x2> = <x1|Sum(n=1,infinity)[|En><En|x2>] =
= a^-2 *Sum(n=1,infinity)[sin(n*x1)sin(n*x2)]

a_{1,5,9,13,17,21,...} = 1/2
a_{2,6,10,14,18,22,...}= -1
a_{3,7,11,15,19,23,...} = 1/2
a_{4,8,12,16,20,24,...} = 0

sum(n=1,infinity)[a_n] = sum(m=0,infinity/4)[sum(q=1,4)[a_{m*4+q}]] = sum(m=0,infinity)[sum(q=1,4)[a_{m*4+q}]] = sum(m=0,infinity)[1/2-1+1/2+0] = sum(m=0,infinity)‏⁰=0

Sum({k})[|k><k|]

<x1|x2> = Delta(x1-x2)

Delta(x-x0)

a(x) <=> <x|a>

Sum({k})[|k><k|]

E~ = Sum({a})[Ea|Ea><Ea|]

E~|f> = 0

const*f"(x) = Ea*f(x)

f(x) (= <x|f>) =0

S1={0,0,0,0,0,...}

S2={1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,...}

S2={1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,1/2-1+1/2+0,...}

S3={sum(q=1,4)[a_q],sum(q=1,4)[a_q],sum(q=1,4)[a_q],sum(q=1,4)[a_q],sum(q=1,4)[a_q],sum(q=1,4)[a_q],sum(q=1,4)[a_q],...}

sum(m=0,infinity)[sum(q=1,4)[a_{m*4+q}]]=0

m(m=0,infinity)[sum(q=1,4)[a_{m*4+q}]]!=0

1. sum(m=0,infinity)[sum(q=1,4)[a_{m*4+q}]]!=sum(m=0,infinity)[1/2-1+1/2+0]
2. sum(m=0,infinity)[1/2-1+1/2+0]!=sum(m=0,infinity)‏⁰
3. sum(m=0,infinity)‏⁰!=0

(1) sum(m=0,infinity)[sum(q=1,4)[a_{m*4+q}]]

(2) sum_{i=0}^{\infty}a_i

sum(m=0,infinity)[sum(q=1,4)[a_{m*4+q}]]=0