PX262 - H1 - current

current can be analogised with conservation of mass in a fluid

flux through the surface is equal to the rate of decrease of mass inside it:

\iint \vec{j} \cdot d \vec{S} = - ∭_{V} \frac{\partial ρ}{\partial t} d V

where, $\vec{j}$ is the current density vector

using the divergence theorem:

\begin{matrix} (1) & \vec{\nabla} \cdot \vec{j} = - \frac{\partial ρ}{\partial t} \end{matrix}

in quantum mechanics, the density:

\begin{matrix} ρ = ψ^{*} (x, t) ψ (x, t) \\ ⟹ \frac{\partial ρ}{\partial t} = \frac{\partial}{\partial t} (ψ^{*} ψ) = ψ^{*} (x, t) \frac{\partial ψ (x, t)}{\partial t} + \frac{\partial ψ^{*} (x, t)}{\partial t} ψ (x, t) \end{matrix}

multiplying the schrödinger equation by $ψ^{*} :$

\begin{matrix} (2) & i ℏ ψ^{*} \frac{\partial ψ}{\partial t} = - \frac{ℏ^{2}}{2 m} ψ^{*} \frac{\partial^{2} ψ}{\partial x^{2}} + V (x) ψ^{*} ψ \end{matrix}

taking its complex conjugate:

\begin{matrix} (3) & - i ℏ ψ \frac{\partial ψ^{*}}{\partial t} = - \frac{ℏ^{2}}{2 m} ψ \frac{\partial^{2} ψ^{*}}{\partial x^{2}} + V (x) ψ ψ^{*} \end{matrix}

subtracting equation $(3)$ from $(2) :$

i ℏ (ψ^{*} \frac{\partial ψ}{\partial t} + ψ \frac{\partial ψ^{*}}{\partial t}) = i ℏ \frac{\partial}{\partial t} (ψ^{*} ψ) = - \frac{ℏ^{2}}{2 m} (ψ^{*} \frac{\partial^{2} ψ}{\partial x^{2}} - ψ \frac{\partial^{2} ψ^{*}}{\partial x^{2}})

in the $R H S :$

\begin{matrix} ψ^{*} \frac{\partial^{2} ψ}{\partial x^{2}} = \frac{\partial}{\partial x} (ψ^{*} \frac{\partial ψ}{\partial x}) - \frac{\partial ψ^{*}}{\partial x} \frac{\partial ψ}{\partial x} \\ ψ \frac{\partial^{2} ψ^{*}}{\partial x^{2}} = \frac{\partial}{\partial x} (ψ \frac{\partial ψ^{*}}{\partial x}) - \frac{\partial ψ}{\partial x} \frac{\partial ψ^{*}}{\partial x} \end{matrix}

therefore, it can be written as:

\begin{matrix} (4) & i ℏ \frac{\partial}{\partial t} (ψ^{*} ψ) = - \frac{ℏ^{2}}{2 m} \frac{\partial}{\partial x} (ψ^{*} \frac{\partial ψ}{\partial x} - ψ \frac{\partial ψ^{*}}{\partial x}) \end{matrix}

using the 1D form of equation $(1) :$

\frac{\partial \vec{j}}{\partial x} = - \frac{\partial ρ}{\partial t}

comparing it with equation $(4) :$

\vec{j} = \frac{ℏ}{2 i m} (ψ^{*} \frac{\partial ψ}{\partial x} - ψ \frac{\partial ψ^{*}}{\partial x})

in 3D:

\vec{j} = \frac{ℏ}{2 i m} (ψ^{*} \vec{\nabla} ψ - ψ \vec{\nabla} ψ^{*})

checking this result for a plane wave:

ψ (\vec{r}, t) = A \exp [i (\vec{k} \cdot \vec{r} - \frac{E t}{ℏ})]

the charge density:

\begin{matrix} \vec{j} = \frac{ℏ}{2 i m} A^{*} A [\exp (- \vec{k} \cdot \vec{r}) i k \exp (i \vec{k} \cdot \vec{r}) - \exp (i \vec{k} \cdot \vec{r}) {- i \vec{k} \exp (- i \vec{k} \cdot \vec{r})}] \\ ∴ \vec{j} = \frac{ℏ \vec{k}}{m} A^{*} A \end{matrix}

where, $\frac{ℏ \vec{k}}{m}$ is equivalent to the velocity, and $A^{*} A$ to the density as $\vec{j} = velocity \times density$