Schrödinger's Cat: Schrödinger Equation

Ψ(t) = a(t)|alive cat> + b(t)|dead cat> λ = h / pWhere λ is the wavelength, h is Planck's constant, and p is the momentum. E = h * ν for frequency ν E = ħ * ω for angular frequency ωWhere E is energy, ħ (h-bar) is the reduced Planck's constant, and ω is the angular frequency. ψ(x, t) = A * exp(i / ħ * (p * x - E * t))Where ψ(x, t) is the wavefunction, A is the amplitude, i is the imaginary unit, ħ is the reduced Planck’s constant, p is the momentum, x is the position, E is the energy, and t is time. Energy operator: E = i * ħ * ∂/∂t Momentum operator: p = -i * ħ * ∇ Ψ(t) = a(t)|alive cat> + b(t)|dead cat> Time-independent Schrödinger Equation The wavefunction depends only on spatial coordinates, not on time： -h^2/2m * ∇^2ψ(r) + V(r)ψ(r) = Eψ(r) Time-dependent Schrödinger Equation The wavefunction depends on both spatial coordinates and time: iℏ * ∂/∂t ψ(r, t) = [-ℏ^2/2m * ∇^2 + V(r, t)] ψ(r, t) Ψ(t) = a(t)|alive cat> + b(t)|dead cat> One-dimensional Schrödinger Equation A special case of the Schrödinger equation for systems with only one spatial dimension: -h^2/2m * d^2/dx^2 ψ(x) + V(x)ψ(x) = Eψ(x) Time-independent Schrödinger Equation The wavefunction depends only on spatial coordinates, not on time： -h^2/2m * ∇^2ψ(r) + V(r)ψ(r) = Eψ(r) Time-dependent Schrödinger Equation The wavefunction depends on both spatial coordinates and time: iℏ * ∂/∂t ψ(r, t) = [-ℏ^2/2m * ∇^2 + V(r, t)] ψ(r, t) Ψ(t) = a(t)|alive cat> + b(t)|dead cat> One-dimensional Schrödinger Equation A special case of the Schrödinger equation for systems with only one spatial dimension: -h^2/2m * d^2/dx^2 ψ(x) + V(x)ψ(x) = Eψ(x) Multidimensional Schrödinger Equation For systems with two or more spatial dimensions (here shown for three dimensions): -h^2/2m * (∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2)ψ(r) + V(r)ψ(r) = Eψ(r) Ψ(t) = a(t)|alive cat> + b(t)|dead cat> Ψ(t) = a(t)|alive cat> + b(t)|dead cat> Multidimensional Schrödinger Equation For systems with two or more spatial dimensions (here shown for three dimensions): -h^2/2m * (∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2)ψ(r) + V(r)ψ(r) = Eψ(r) Ψ(t) = a(t)|alive cat> + b(t)|dead cat> [Outro] Ψ(t) = a(t)|alive cat> + b(t)|dead cat> [end]