spatialprobabilitydensity1.txt

Spatial Probability Density
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假設 ρ 是 spatial probability density

∫∫∫ ρ dx dy dz = 1 ...(1)

∫∫∫ ρ dx dy dz = e^(i 2 π n), n = integer
∫∫∫ ρ dx dy dz = e^(i 2 π n n_t), n_t = integer
∫∫∫ ρ dx dy dz = e^(i 2 π n 1/τ n_t τ), τ = period

設 f = 1/τ, t = n_t τ
∫∫∫ ρ dx dy dz = e^(i 2 π n f t)
∫∫∫ ρ dx dy dz = e^(i 2 π n h f/h t)
∫∫∫ ρ dx dy dz = e^(i 2 π n E/h t)
∫∫∫ ρ dx dy dz = e^(i n E/(h/2 π) t)
∫∫∫ ρ dx dy dz = e^(i n E/h_bar t)
ρ = d^3/(dx dy dz) e^(i n E/h_bar t)
ρ = (d^3/dt^3 e^(i n E/h_bar t)) dt/dx dt/dy dt/dz
ρ = (d^3/dt^3 e^(i n E/h_bar t)) 1/(dx/dt) 1/(dy/dt) 1/(dz/dt)
ρ = 1/(v_x v_y v_z) (d^3/dt^3 e^(i n E/h_bar t))
ρ = 1/(v_x v_y v_z) (d^3/dt^3 e^(i n 1/h_bar E t)) ...(2)

按等效座標理論
p x = E t ...(3)
3D 效果
p ． r = E t ...(4)

(2) & (4):
ρ = 1/(v_x v_y v_z) (d^3/dt^3 e^(i n 1/h_bar (p ． r)))
ρ = 1/(v_x v_y v_z) d^2/dt^2 (e^(i n 1/h_bar (p ． r)) (i n 1/h_bar (dp/dt ． r + p ． dr/dt)))
ρ = 1/(v_x v_y v_z) d^2/dt^2 (e^(i n 1/h_bar (p ． r)) (i n 1/h_bar (F ． r + p ． v))) ...(5)

F = m E_g + m v × B_g + q E + q v × B ...(6)

(5) & (6):
ρ = 1/(v_x v_y v_z) d^2/dt^2 (e^(i n 1/h_bar (p ． r)) (i n 1/h_bar ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v))) ...(7)

(4):
d/dt (p ． r) = d/dt (E t)
dp/dt ． r + p ． dr/dt = dE/dt t + E
F ． r + p ． v = t d/dt (h f) + E
F ． r + p ． v = h t df/dt + E
E = F ． r + p ． v - h t df/dt ...(8)

(6) & (8):
E = (m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt ...(9)

(2) & (9):
ρ = 1/(v_x v_y v_z) (d^3/dt^3 e^(i n 1/h_bar ((m E_g + m v × B_g + q E + q v × B) ． r + p ． v - h t df/dt) t)) ...(10)