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For Klein-Gordon equation, the probability current density is given by :

  • A.\(\frac{i\hbar c}{2m}(\psi\nabla\psi^{*} - \psi^{*}\nabla\psi)\)
  • B.\(\frac{i\hbar}{2m}(\psi\nabla\psi^{*} - \psi^{*}\nabla\psi)\)
  • C.\(\frac{i\hbar}{2m}(\psi\nabla\psi^{*} + \psi^{*}\nabla\psi)\)
  • D.\(\frac{i\hbar}{2mc}(\psi\nabla\psi^{*} - \psi^{*}\nabla\psi)\)

Answer: B

The Klein-Gordon probability current is \(\frac{i\hbar}{2m}(\psi\nabla\psi^{*} - \psi^{*}\nabla\psi)\).

Multiply the Klein-Gordon equation by \(\psi^{*}\), its conjugate by \(\psi\), and subtract. The result is a continuity equation \(\partial\rho/\partial t + \nabla\cdot\vec{j} = 0\), with

\[ \begin{aligned} \rho &= \frac{i\hbar}{2mc^{2}}\left(\psi^{*}\frac{\partial\psi}{\partial t} - \psi\frac{\partial\psi^{*}}{\partial t}\right) \\ \vec{j} &= \frac{i\hbar}{2m}(\psi\nabla\psi^{*} - \psi^{*}\nabla\psi) \end{aligned} \]

The current has the same form as the Schroedinger current; it is the density that differs and can become negative. The factors c and 1/c in the first and last options make the dimensions wrong, and a plus sign between the terms gives a real quantity, \(\nabla|\psi|^{2}\), which does not describe flow.

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