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Kepler Problem

Alternate names

Classical mechanics

S(t_1, t_2) = integral_(t_1)^(t_2) (1/2 m x'(τ)^2 - λ/abs(x(τ))) dτ

L = 1/2 m x'(t)^2 - λ/abs(x(t))

m x''(t) = (λ x(t))/abs(x(t))^3

ℋ = p(t)^2/(2 m) + λ/abs(x(t))

x'(t) = p(t)/m | p'(t) = (λ x(t))/abs(x(t))^3

p(t) = m x'(t)

Quantum mechanics

i ℏ (dΨ(x, t))/(dt) = -ℏ^2/(2 m) (d^2 Ψ(x, t))/(dx^2) + λ/abs(x) Ψ(x, t)

-ℏ^2/(2 m) ψ''(x) + λ/abs(x) ψ(x) = ℰ ψ(x)

ψ = (sqrt(2) (-1)^(n - 1) m^(3/2) (-λ)^(3/2))/(n^(5/2) ℏ^3) abs(x) exp((m λ abs(x))/(n ℏ^2)) L_(n - 1)^1(-(2 m λ abs(x))/(n ℏ^2)) for λ<0

E = -(m λ^2)/(2 ℏ^2 n^2) for λ<0

Variables

Notable contributors

Charles-Augustin de Coulomb | Erwin Schrödinger | Johannes Kepler

System classification