spatialprobabilitydensity4.txt

Spatial Probability Density
===========================

假設 ρ 是 spatial probability density

∫∫∫ ρ dr (r dθ) (r sin(θ) dφ) = 1
∫∫∫ ρ (r^2 dr) (sin(θ) dθ) dφ = 1 ...(1)

let
S(n, f, t) = square wave function with frequency (n f), n = integer
S(n, f, t) = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) 2 π n f t)) ...(2)

Case 1:
∫∫∫ ρ (r^2 dr) (sin(θ) dθ) dφ = + S(n, f, t - t0) ...(3)

Case 2:
∫∫∫ ρ (r^2 dr) (sin(θ) dθ) dφ = - S(n, f, t - t0) ...(4)

(3) & (4):
∫∫∫ ρ (r^2 dr) (sin(θ) dθ) dφ = +/- S(n, f, t - t0) = +/- 1 ...(5)

define
{
∫ ρ_r (r^2 dr) = +/- S(n_r, f, t - t0) ...(6)
∫ ρ_θ (sin(θ) dθ) = +/- S(n_θ, 1, (θ - θ0)/(2 π)) ...(7)
∫ ρ_φ dφ = +/- S(n_φ, 1, (φ - φ0)/(2 π)) ...(8)
}

(6) * (7) * (8):
(∫ ρ_r (r^2 dr))(∫ ρ_θ (sin(θ) dθ))(∫ ρ_φ dφ) = +/- 1
(∫ ρ_φ dφ)(∫ ρ_θ (sin(θ) dθ))(∫ ρ_r (r^2 dr)) = +/- 1
(∫ ρ_φ dφ(∫ ρ_θ (sin(θ) dθ)(∫ ρ_r (r^2 dr)))) = +/- 1
(∫ ρ_φ dφ(∫∫ ρ_θ (sin(θ) dθ)(ρ_r (r^2 dr)))) = +/- 1
(∫∫∫ ρ_φ dφ( ρ_θ (sin(θ) dθ)(ρ_r (r^2 dr)))) = +/- 1
∫∫∫ ρ_r ρ_θ ρ_φ (r^2 dr) (sin(θ) dθ) dφ = +/- 1 ...(9)

(5) & (9):
ρ = ρ_r ρ_θ ρ_φ ...(10)

(2):
d/dt S(n, f, t) = d/dt (4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) 2 π n f (t - t0))))
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (((2 k + 1) 2 π n (df/dt (t - t0) + f) )/(2 k + 1) cos((2 k + 1) 2 π n f (t - t0)))
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (2 π n (df/dt (t - t0) + f) cos((2 k + 1) 2 π n f (t - t0))) ...(11)

d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (n (dω/dt (t - t0) + ω) cos((2 k + 1) n ω (t - t0)))
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (n (h_bar dω/dt (t - t0) + h_bar ω)/h_bar cos((2 k + 1) n ω (t - t0)))
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (n ((t - t0) d/dt (h_bar ω) + h_bar ω)/h_bar cos((2 k + 1) n h_bar ω/h_bar (t - t0)))
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (n ((t - t0) dE/dt + E)/h_bar cos((2 k + 1) n E/h_bar (t - t0)))
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (n ((t - t0) dE/dt + E)/h_bar (e^(i (2 k + 1) n E/h_bar (t - t0)) + e^(-i (2 k + 1) n E/h_bar (t - t0)))/2)

rename E as U:
d/dt S(n, f, t) = 4/π Σ_{k = 0..+oo} (n ((t - t0) dU/dt + U)/h_bar (e^(i (2 k + 1) n U/h_bar (t - t0)) + e^(-i (2 k + 1) n U/h_bar (t - t0)))/2) ...(12)

按等效座標理論
p x = E (t - t0)
p x - E t = -E t0
p x/E - t = -t0
d/dt (p x/E - t) = - d/dt t0
x/E dp/dt + p/E dx/dt - p x/E^2 dE/dt - 1 = 0
assume E <> 0
x E dp/dt + p E dx/dt - p x dE/dt - E^2 = 0
x F E + p v E - p x dE/dt - E^2 = 0
E^2 + (-x F - p v) E + (p x dE/dt) = 0
E = (-(-x F - p v) +/- ((-x F - p v)^2 - 4(p x dE/dt))^(1/2))/2
E = ((x F + p v) +/- ((x F + p v)^2 - 4 p x dE/dt)^(1/2))/2
3D 效果
E = ((r ． F + p ． v) +/- ((r ． F + p ． v)^2 - 4 (p ． r) dE/dt)^(1/2))/2

rename E as U:
U = ((r ． F + p ． v) +/- ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2

U = ((r ． F + p ． v) - ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2 ...(13)
or
U = ((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2 ...(14)

choose an arbitrary (new) origin
r = original vector representing the position
r_origin = vector representing the position of the origin
r_relative = vector representing the position relative to the origin
r = r_origin + r_relative
r_relative = (r_r, r_θ, r_φ) in spherical coordinates

(6), (7) & (8):
{
∫ ρ_r (r_r^2 dr_r) = +/- S(n_r, f, t - t0) ...(15)
∫ ρ_θ (sin(r_θ) dr_θ) = +/- S(n_θ, 1, (r_θ - r_θ0)/(2 π)) = +/- S(n_θ, 1/2, (r_θ - r_θ0)/π) ...(16)
∫ ρ_φ dr_φ = +/- S(n_φ, 1, (r_φ - r_φ0)/(2 π)) ...(17)
}

(15):
ρ_r (r_r^2 dr_r) = +/- dS(n_r, f, t - t0)
ρ_r  = +/- 1/(r_r^2 dr_r) dS(n_r, f, t - t0)
ρ_r^2  = (1/(r_r^2 dr_r) dS(n_r, f, t - t0))^2
ρ_r^2  = (1/(r_r^2 d/dt r_r) d/dt S(n_r, f, t - t0))^2 ...(18)

(12) & (18):
ρ_r^2  = (1/(r_r^2 d/dt r_r) (4/π Σ_{k = 0..+oo} (n_r ((t - t0) dU/dt + U)/h_bar (e^(i (2 k + 1) n_r U/h_bar (t - t0)) + e^(-i (2 k + 1) n_r U/h_bar (t - t0)))/2)))^2 ...(19)

(16):
ρ_θ (sin(r_θ) dr_θ) = +/- dS(n_θ, 1, (r_θ - r_θ0)/(2 π))
ρ_θ = +/- 1/(sin(r_θ) dr_θ) dS(n_θ, 1, (r_θ - r_θ0)/(2 π))
ρ_θ^2 = (1/(sin(r_θ) dr_θ) dS(n_θ, 1, (r_θ - r_θ0)/(2 π)))^2
ρ_θ^2 = (1/(sin(r_θ) d/dt r_θ) d/dt S(n_θ, 1, (r_θ - r_θ0)/(2 π)))^2 ...(20)

(11) & (20):
ρ_θ^2 = (1/(sin(r_θ) d/dt r_θ) 4/π Σ_{k = 0..+oo} (2 π n_θ (d(1)/d((r_θ - r_θ0)/(2 π)) ((r_θ - r_θ0)/(2 π)) + 1) cos((2 k + 1) 2 π n_θ (1) ((r_θ - r_θ0)/(2 π)))))^2
ρ_θ^2 = (1/(sin(r_θ) d/dt r_θ) 4/π Σ_{k = 0..+oo} (2 π n_θ (0 * ((r_θ - r_θ0)/(2 π)) + 1) cos((2 k + 1) 2 π n_θ (1) ((r_θ - r_θ0)/(2 π)))))^2
ρ_θ^2 = (1/(sin(r_θ) d/dt r_θ) 4/π Σ_{k = 0..+oo} (2 π n_θ (0 + 1) cos((2 k + 1) 2 π n_θ (1) ((r_θ - r_θ0)/(2 π)))))^2
ρ_θ^2 = (1/(sin(r_θ) d/dt r_θ) 4/π Σ_{k = 0..+oo} (2 π n_θ cos((2 k + 1) n_θ (r_θ - r_θ0))))^2 ...(21)

similarly
ρ_φ^2 = (1/(d/dt r_φ) 4/π Σ_{k = 0..+oo} (2 π n_φ cos((2 k + 1) n_φ (r_φ - r_φ0))))^2 ...(22)

Case 1, choose (13):
(19):
ρ_r^2  = (1/(r_r^2 d/dt r_r) (4/π Σ_{k = 0..+oo} (n_r ((t - t0) dU/dt + ((r ． F + p ． v) - ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2)/h_bar (e^(i (2 k + 1) n_r (((r ． F + p ． v) - ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2)/h_bar (t - t0)) + e^(-i (2 k + 1) n_r (((r ． F + p ． v) - ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2)/h_bar (t - t0)))/2)))^2 ...(23)

Case 2, choose (14):
(19):
ρ_r^2  = (1/(r_r^2 d/dt r_r) (4/π Σ_{k = 0..+oo} (n_r ((t - t0) dU/dt + ((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2)/h_bar (e^(i (2 k + 1) n_r (((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2)/h_bar (t - t0)) + e^(-i (2 k + 1) n_r (((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2)/h_bar (t - t0)))/2)))^2 ...(24)

F = m E_g + m v × B_g + q E + q v × B ...(25)