spatialprobabilitydensity3.txt

Spatial Probability Density
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假設 ρ 是 spatial probability density

∫∫∫ ρ dx dy dz = 1 ...(1)

let
f(t) = square wave function
f(t) = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) 2 π f t)) ...(2)

Case 1:
∫∫∫ ρ dx dy dz = f(t)

∫∫∫ ρ dx dy dz = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) 2 π f t))
∫∫∫ ρ dx dy dz = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) ω t))
∫∫∫ ρ dx dy dz = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) h_bar ω t/h_bar))
∫∫∫ ρ dx dy dz = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) sin((2 k + 1) E t/h_bar))
∫∫∫ ρ dx dy dz = 4/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar))/(2 i)
∫∫∫ ρ dx dy dz = -2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar))
ρ = d^3/(dx dy dz) (-2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar)))
ρ = dt/dx dt/dy dt/dz d^3/dt^3 (-2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar)))
ρ = 1/(dx/dt) 1/(dy/dt) 1/(dz/dt) d^3/dt^3 (-2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar)))
ρ = 1/v_x 1/v_y 1/v_z d^3/dt^3 (-2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar)))
ρ = 1/(v_x v_y v_z) d^3/dt^3 (-2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar)))
ρ = -1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar))) ...(3)

Case 2:
∫∫∫ ρ dx dy dz = -f(t)
ρ = 1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar))) ...(4)

(3) & (4):
ρ^2 = (1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) E t/h_bar) - e^(-i (2 k + 1) E t/h_bar))))^2 ...(5)

按等效座標理論
p x = E (t - t0) ...(6)
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

因為下面有用到 E field，以 U 代替 E ：
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 ...(7)
or
U = ((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2 ...(8)

同理，以 U 代替 E :
(5):
ρ^2 = (1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) U t/h_bar) - e^(-i (2 k + 1) U t/h_bar))))^2 ...(9)

(7) & (9):
ρ^2 = (1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) (((r ． F + p ． v) - ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar) - e^(-i (2 k + 1) (((r ． F + p ． v) - ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar))))^2 ...(10)

(8) & (9):
ρ^2 = (1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) (((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar) - e^(-i (2 k + 1) (((r ． F + p ． v) + ((r ． F + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar))))^2 ...(11)

F = m E_g + m v × B_g + q E + q v × B ...(12)

(10) & (12):
ρ^2 = (1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) (((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v) - ((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar) - e^(-i (2 k + 1) (((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v) - ((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar))))^2 ...(13)

(11) & (12):
ρ^2 = (1/(v_x v_y v_z) d^3/dt^3 (2 i/π Σ_{k = 0..+oo} (1/(2 k + 1) (e^(i (2 k + 1) (((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v) + ((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar) - e^(-i (2 k + 1) (((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v) + ((r ． (m E_g + m v × B_g + q E + q v × B) + p ． v)^2 - 4 (p ． r) dU/dt)^(1/2))/2) t/h_bar))))^2 ...(14)