Interpretations

Interpretation of the Wavefunction

Wavefunction $Ψ (\vec{x}, t)$ , w/ Probability Density $ρ (\vec{x}, t) = | Ψ (\vec{x}, t) |^{2}$ , $e ρ (\vec{x}, t)$ gives the charge density.

$\int_{V_{0}} d \vec{x} ρ (\vec{x}, t) =$ probability to find the particle in the volume.

Probability flux: $\vec{j} (\vec{x}, t) = - \frac{i ℏ}{2 m} (Ψ^{*} \vec{\nabla} Ψ - (\vec{\nabla} Ψ^{*}) Ψ) = - \frac{i ℏ}{2 m} 2 i ℑ [Ψ^{*} \vec{\nabla} Ψ] = \frac{ℏ}{m} ℑ [Ψ^{*} \vec{\nabla} Ψ]$ .

$\int_{R} d \vec{x} j (\vec{x}, t) = \frac{ℏ}{m} ℑ \int_{R} d \vec{x} Ψ (\vec{x}, t)^{*} \vec{\nabla} Ψ (\vec{x}, t) = \frac{1}{2 m} \int_{R} d \vec{x} Ψ^{*} (\hat{P} Ψ) + ({\hat{P}}^{*} Ψ^{*}) Ψ = \frac{1}{m} ℜ ⟨ \hat{P} ⟩_{t} = \frac{1}{m} ⟨ \hat{P} ⟩_{t}$ .

How is the probability density and flux related?

$\frac{\partial ρ}{\partial t} = \frac{\partial}{\partial t} Ψ^{*} Ψ = \frac{i}{ℏ} \hat{H} Ψ^{*} Ψ - i ℏ Ψ^{*} \hat{H} Ψ = \frac{i}{ℏ} [\frac{- ℏ^{2}}{2 m} (\vec{\nabla^{2}} Ψ^{*}) Ψ + (\hat{V} Ψ^{*}) Ψ - Ψ^{*} \frac{ℏ^{2}}{2 m} ({\vec{\nabla}}^{2} Ψ) - Ψ^{*} \hat{V} Ψ] = \frac{i ℏ}{2 m} (Ψ^{*} {\vec{\nabla}}^{2} Ψ - Ψ {\vec{\nabla}}^{2} Ψ) = - \frac{ℏ}{m} ℑ (Ψ^{*} {\vec{\nabla}}^{2} Ψ)$ .

The divergence of the probability flux is, $\vec{\nabla} \cdot \vec{j} (\vec{x}, t) = - \frac{i ℏ}{2 m} 2 i ℑ (Ψ^{*} {\vec{\nabla}}^{2} Ψ) = - \frac{\partial}{\partial t} ρ$ .

Continuity Equation - Probability Conservation Law

$\frac{\partial}{\partial t} ρ + \vec{\nabla} \cdot \vec{j} = 0$ .