spatialprobabilitydensity2.txt

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

假設 ρ 是 spatial probability density

∫∫∫ ρ dx dy dz = 1 ...(1)

let
ρ = ρ_c + ρ_s ...(2)

set
∫∫∫ ρ_c dx dy dz = (cos(ω t))^2 ...(3)
∫∫∫ ρ_s dx dy dz = (sin(ω t))^2 ...(4)

檢查：
(3) + (4):
∫∫∫ (ρ_c + ρ_s) dx dy dz = (cos(ω t))^2 + (sin(ω t))^2
=> ∫∫∫ ρ dx dy dz = 1
符合 (1) 式。

(3):
∫∫∫ ρ_c dx dy dz = (1 + cos(2 ω t))/2
∫∫∫ ρ_c dx dy dz = (1 + (e^(i 2 ω t) + e^(-i 2 ω t))/2)/2
∫∫∫ ρ_c dx dy dz = (1 + (e^(i 2 h_bar ω t/h_bar) + e^(-i 2 h_bar ω t/h_bar))/2)/2
∫∫∫ ρ_c dx dy dz = (1 + (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2 ...(5)

(4):
∫∫∫ ρ_s dx dy dz = (1 - cos(2 ω t))/2
∫∫∫ ρ_s dx dy dz = (1 - (e^(i 2 ω t) + e^(-i 2 ω t))/2)/2
∫∫∫ ρ_s dx dy dz = (1 - (e^(i 2 h_bar ω t/h_bar) + e^(-i 2 h_bar ω t/h_bar))/2)/2
∫∫∫ ρ_s dx dy dz = (1 - (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2 ...(6)

(5):
ρ_c = d^3/(dx dy dz) (1 + (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2
ρ_c = dt/dx dt/dy dt/dz d^3/dt^3 (1 + (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2
ρ_c = 1/(dx/dt) 1/(dy/dt) 1/(dz/dt) d^3/dt^3 (1 + (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2
ρ_c = 1/v_x 1/v_y 1/v_z d^3/dt^3 (1 + (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2
ρ_c = 1/(v_x v_y v_z) d^3/dt^3 (1 + (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2 ...(7)

similarly
(6):
ρ_s = 1/(v_x v_y v_z) d^3/dt^3 (1 - (e^(i 2 E t/h_bar) + e^(-i 2 E t/h_bar))/2)/2 ...(8)

按等效座標理論
p x = E t ...(9)
3D 效果
p ． r = E t
因為下面有用到 E field，以 U 代替 E ：
p ． r = U t ...(10)

同理
(7):
ρ_c = 1/(v_x v_y v_z) d^3/dt^3 (1 + (e^(i 2 U t/h_bar) + e^(-i 2 U t/h_bar))/2)/2 ...(11)

(8):
ρ_s = 1/(v_x v_y v_z) d^3/dt^3 (1 - (e^(i 2 U t/h_bar) + e^(-i 2 U t/h_bar))/2)/2 ...(12)

F = m E_g + m v × B_g + q E + q v × B ...(13)

(10):
d/dt (p ． r) = d/dt (U t)
dp/dt ． r + p ． dr/dt = dU/dt t + U
F ． r + p ． v = t d/dt (h f) + U
F ． r + p ． v = h t df/dt + U
U = F ． r + p ． v - h t df/dt ...(14)

(13) & (14):
U = (m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt ...(15)

(11) & (15):
ρ_c = 1/(v_x v_y v_z) d^3/dt^3 (1 + (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2 ...(16)

(12) & (15):
ρ_s = 1/(v_x v_y v_z) d^3/dt^3 (1 - (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2 ...(17)

(2), (16) & (17):
ρ = 1/(v_x v_y v_z) (d^3/dt^3 (1 + (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2) +
1/(v_x v_y v_z) (d^3/dt^3 (1 - (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2) ...(18)

(10) & (12):
ρ_s = 1/(v_x v_y v_z) d^3/dt^3 (1 - (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/2)/2 ...(19)

(2), (16) & (19):
ρ = 1/(v_x v_y v_z) (d^3/dt^3 (1 + (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2) +
1/(v_x v_y v_z) (d^3/dt^3 (1 - (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/2)/2)

ρ = 1/(v_x v_y v_z) (d^3/dt^3 (+1 (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/4)) +
1/(v_x v_y v_z) (d^3/dt^3 (-1 (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/4)) ...(20)

(10) & (11):
ρ_c = 1/(v_x v_y v_z) d^3/dt^3 (1 + (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/2)/2 ...(21)

(2), (17) & (21):
ρ = 1/(v_x v_y v_z) (d^3/dt^3 (1 - (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2) +
1/(v_x v_y v_z) (d^3/dt^3 (1 + (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/2)/2)

ρ = 1/(v_x v_y v_z) (d^3/dt^3 (-1 (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/2)/2) +
1/(v_x v_y v_z) (d^3/dt^3 (+1 (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/2)/2)

ρ = -(1/(v_x v_y v_z) (d^3/dt^3 (+1 (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/4)) +
1/(v_x v_y v_z) (d^3/dt^3 (-1 (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/4))) ...(22)

(20) & (22):
ρ^2 = (1/(v_x v_y v_z) (d^3/dt^3 (+1 (e^(i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar) + e^(-i 2 ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t/h_bar))/4)) +
1/(v_x v_y v_z) (d^3/dt^3 (-1 (e^(i 2 (p ． r)/h_bar) + e^(-i 2 (p ． r)/h_bar))/4)))^2 ...(23)