PHYS212 : Lectures

Subsections

• 7.3.1 Defined
• 7.3.2 Plane Waves
• 7.3.3 Infinite Square Wells

7.3 Continuity and Probability Current

7.3.1 Defined

• Mass conservation is expressed with an equation of continuity identical to that describing conservation of charge,
  

  $$\frac{\partial \varrho}{\partial t} + \vec{\nabla} \cdot \vec{J} = 0$$ (7.34)

  
  
  where $\varrho$ is the probability density in units of particles per unit volume and $\vec{J}$ is the probability current with units of particles per unit area per unit time.^7.3

• Integrating Eq. 7.34 over a volume $V$, and applying the divergence theorem, we have
  

  $$\frac{\partial}{\partial t} \int_V \varrho d\vec{r} + \int_A \vec{J} \cdot d\vec{a} = 0$$

  $$\frac{\partial N}{\partial t} + \int_A \vec{J} \cdot d\vec{a} = 0$$ (7.35)

  
  
  where $A$ represents the area bounding volume $V$, and $N$ is the number of particles described by $\varrho$ in $V$.

• Eq. 7.35 translates nicely into words as

  The rate of change of the number of particles in volume $V$ is due entirely to the particle current flowing through the area $A$ bounding $V$.

• In one dimension, the equation of continuity takes the form
  

  $$\frac{\partial \varrho}{\partial t} + \frac{\partial J}{\partial x} = 0$$ (7.36)

  
  
  where the units of $\varrho$ are particles per unit length, and the units of $J$ are particles per time (a true probability current).

• The Schrödinger equation (Eq. 7.9) enables us to evaluate the first term in Eq. 7.36, yielding
  
  $$\frac{1}{i \hbar} \left[ -\frac{\hbar^2}{2m} \Psi^* \frac{\... ...}{\partial x^2} - V(x) \left\vert \Psi \right\vert^2 \right]$$
  
  
  which simplifies to
  

  $$\frac{\partial J}{\partial x} = \frac{i\hbar}{2m} \left[ \P... ...l x^2} - \Psi^* \frac{\partial^2 \Psi}{\partial x^2} \right]$$ (7.37)

  
  

• The form of the probability current can be deduced from Eq. 7.37 if one thinks about the product rule a bit. A less enjoyable approach is integration by parts. We find
  
  $$J = \frac{i\hbar}{2m} \left[ \Psi \frac{\partial \Psi^*}{\partial x} - \Psi^* \frac{\partial \Psi}{\partial x} \right]$$
  
  

7.3.2 Plane Waves

• A plane wave (Eq. 7.19) has probability current
  

  $$J = \frac{i\hbar}{2m} \left[ -ik \Psi \frac{\partial \Psi^*... ...x} \right] = \frac{\hbar k}{m} \left\vert \Psi \right\vert^2$$ (7.38)

  
  

  which can be written alternatively as
  

  $$J = \frac{p}{m} \left\vert C \right\vert^2 = v_{classical} \left\vert C \right\vert^2$$ (7.39)

  
  
  where $v_{classical}$ is the velocity of a classical particle with momentum $\hbar k$.

• Eq. 7.39 suggests a way of setting the normalization constant $C$ for a continuous beam of particles. In order to give a sensible current density, $\vert C\vert^2$ must be the number of particles per unit length. Hence,
  

  $$\Psi_{beam}(x,t) = \sqrt{\mu} e^{i(kx - \omega t)}$$ (7.40)

  
  
  where $\mu$ is the linear particle density.

7.3.3 Infinite Square Wells

• The position dependent part of the infinite square well wavefunctions (Eq. 7.32) are real functions.

• Consider the general case
  

  $$\Psi(x,t) = \psi_{Re}(x) e^{-i\omega t}$$ (7.41)

  
  
  where $\psi_{Re}(x)$ is a real function.

• If we evaluate the probability current, we find that the time dependence disappears ( $e^{i\omega t}e^{-i\omega t} = 1$), and
  

  $$J = \frac{i\hbar}{2m} \left[ \psi_{Re} \frac{\partial \psi_... ...- \psi_{Re} \frac{\partial \psi_{Re}}{\partial x} \right] = 0$$ (7.42)

  
  
  the complex conjugation of $\psi_{Re}$ has no effect, giving a cancellation of the two terms in $J$.

• Hence, if a solution $\psi(x)$ to the time independent Schrödinger equation (Eq. 7.16) is real, it has no probability current.

• In fact, identical reasoning leads to the conclusion that purely imaginary wavefunctions also have zero probability current.

Copyright © 2003-2009, Lewis A. Riley

Updated Wed Jan 18 09:51:28 2006