aimusic.one

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]