SCHRÖDINGER EQUATION DERIVATION
Key to symbols Unit
A c e E f ℎ ℏ
i k m KE p t v V 𝑥 ψ
λ π ω ∂
m/s J Hz J/Hz J/Hz/rad rad/m kg J kg m/s s m/s J m m rad/s
1 𝐾𝐸 = 𝑚𝑣 2 2
𝐾𝐸 =
Meaning Wave amplitude Wave velocity = λf = ω/k Euler’s number = 2.718281828 Total energy Wave frequency Planck constant Reduced Planck constant = ℎ/2π Imaginary number = ξ−1 Wave number = 2π/λ Mass Kinetic energy Momentum Time Velocity Potential energy Position Wave function ≡ ψ (𝑥, 𝑡) Wavelength Pi = 3.141592654 Angular frequency = 2πf Partial derivative
𝑝2 2𝑚
𝑝 = 𝑚𝑣
Maxwell-de Broglie Plane wave
Ψ = 𝐴𝑒 𝑖(𝑘𝑥−𝜔𝑡)
∂Ψ = −𝑖𝜔Ψ ∂𝑡
∂2 Ψ = −𝑘 2 Ψ ∂𝑥 2
𝑘2 = −
𝜔=𝑖
1 ∂Ψ Ψ ∂𝑡
ℏ2 𝑘 2 𝐾𝐸 = 2𝑚
𝐸 ℏ𝜔 𝑝= = = ℏ𝑘 𝑐 𝑐
1 ∂2 Ψ Ψ ∂𝑥 2
𝐸 = ℎ𝑓 = ℏ𝜔
ℏ𝜔 =
ℏ2 𝑘 2 +𝑉 2𝑚
Planck-Einstein
ℏ𝜔 =
𝑖ℏ
ℏ2 1 ∂2 Ψ ቆ− ቇ+𝑉 2𝑚 Ψ ∂𝑥 2
1 ∂Ψ ℏ2 1 ∂2 Ψ =− +𝑉 Ψ ∂𝑡 2𝑚 Ψ ∂𝑥 2
𝒊ℏ
𝛛𝚿 ℏ𝟐 𝛛𝟐 𝚿 =− + 𝑽𝚿 𝛛𝒕 𝟐𝒎 𝛛𝒙𝟐 1-D time-dependent Schrödinger equation
𝐸 = 𝐾𝐸 + 𝑉