1
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)
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
| description | physical quantity | basic dimensions | D λ | strength of the potential | inverse distance potential coupling constant | [length]^3 [mass] [time]^-2 | R m | mass of a test particle | mass | [mass] | R^+
Charles-Augustin de Coulomb | Erwin Schrödinger | Johannes Kepler
classical physical systems | continuous spectrum physical systems | discrete spectrum physical systems | exactly solvable physical systems | non-dissipative physical systems | quantum physical systems | time-dependent physical systems
