終極物理（二）
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按等效座標理論
x dx = c^2 t dt
x dx/dt = c^2 t
x v = c^2 t
x m v = m c^2 t
x p = m c^2 t

if x = c t
x p = E t
p x = E t

p λ = E τ, λ = wavelength, τ = period
p λ = E τ = h
p λ/h = E τ/h = 1
p λ/h_bar = E τ/h_bar = 2 π

p (x - floor(x/λ) λ) = E (t - floor(t/τ) τ)

consider θ = p x/h_bar = E t/h_bar
e^(i θ) = e^(i p x/h_bar) = e^(i E t/h_bar)

e^(i p x/h_bar) = e^(i E t/h_bar)

change m to M = n-by-n matrix => E = n-by-n matrix.
change p to P, where P = n-by-n matrix.

e^(i P x/h_bar) = e^(i E t/h_bar)
e^(i P x/h_bar) [s1_0; s2_0; ...; sn_0] = e^(i E t/h_bar) [s1_0; s2_0; ...; sn_0]
e^(i P (x - x0)/h_bar) e^(i P x0/h_bar) [s1_0; s2_0; ...; sn_0] = e^(i E (t - t0)/h_bar) e^(i E t0/h_bar) [s1_0; s2_0; ...; sn_0]
e^(i P (x - x0)/h_bar) [s1_x0; s2_x0; ...; sn_x0] = e^(i E (t - t0)/h_bar) [s1_t0; s2_t0; ...; sn_t0]

P = P_x for x direction

如果同時考慮 x, y, z 三個方向，前面左方的項全部轉為相關於 x, y, z 三個方向的項
按等效座標理論
Σ x_i dx_i = c^2 t dt
即
x dx + y dy + z dz = c^2 t dt

Σ p_i x_i = E t

Π e^(i p_i x_i/h_bar) = e^(i E t/h_bar)

e^(i P_x (x - x0)/h_bar) *
e^(i P_y (y - y0)/h_bar) *
e^(i P_z (z - z0)/h_bar) [s1_x0y0z0; s2_x0y0z0; ...; sn_x0y0z0] = e^(i E (t - t0)/h_bar) [s1_t0; s2_t0; ...; sn_t0]