Answer to Question #238394 in Quantum Mechanics for Matimu

Question

. Consider the Schrödinger equation for a particle in the presence of the potential of the form 𝑉(𝑥) = 𝑈(𝑥) + 𝑖𝑊(𝑥) where 𝑈 and 𝑊 are real functions of 𝑥. What form does the conservation equation take?

Answer 1

let us consider the Schrödinger equation with potential $𝑉(𝑥) = 𝑈(𝑥) + 𝑖𝑊(𝑥)$:

$$i\hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi+( 𝑈(𝑥) + 𝑖𝑊(𝑥))\psi\quad (1)$$

The conjugate equation

$$-i\hbar \frac{\partial \psi*}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\psi^*+( 𝑈(𝑥) - 𝑖𝑊(𝑥))\psi^*\quad (2)$$

After multiplying Eqn 1 by $\psi^*$, Eqn 2 by $\psi$ and subtracting, we get

$$i\hbar\left( \frac{\partial \psi}{\partial t}\psi^*+\frac{\partial \psi^*}{\partial t}\psi\right)=$$

$$-\frac{\hbar^2}{2m}\left(\psi^*\nabla^2\psi-\psi\nabla^2\psi^*\right)+2iW(x)\psi\psi^*$$

or

$$\frac{\partial \rho}{\partial t}+\nabla {\bf j}-\gamma\rho=0$$

Here

$$\rho=\psi\psi^*$$

$${\bf j}=\frac{\hbar}{2mi}\left(\psi^*\nabla\psi-\psi\nabla\psi^*\right)$$

$$\gamma=2W(x)/\hbar$$