The free particle: probability current

Required math: calculus

Required physics: Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.19.

The rate of change of probability of a particle in a given range of {x} can be written as the difference in probability current at the two ends. The current is defined as

\displaystyle  J(x,t)\equiv\frac{i\hbar}{2m}\left(\frac{\partial\Psi^*}{\partial x}\Psi-\frac{\partial\Psi}{\partial x}\Psi^*\right) \ \ \ \ \ (1)

For the free particle, a stationary state is given by

\displaystyle  \Psi(x,t)=Ae^{ikx}e^{-i\hbar k^{2}t/2m} \ \ \ \ \ (2)

The probability current for this state is found by working out the derivative:

\displaystyle   \frac{\partial\Psi}{\partial x} \displaystyle  = \displaystyle  ikAe^{ikx}e^{-i\hbar k^{2}t/2m}\ \ \ \ \ (3)
\displaystyle  \displaystyle  = \displaystyle  ik\Psi \ \ \ \ \ (4)

So we get

\displaystyle   J(x,t) \displaystyle  = \displaystyle  \frac{i\hbar k}{2m}\left(-i\left|\Psi\right|^{2}-i\left|\Psi\right|^{2}\right)\ \ \ \ \ (5)
\displaystyle  \displaystyle  = \displaystyle  \frac{\hbar k}{m}\left|A\right|^{2} \ \ \ \ \ (6)

(The complex exponentials cancel out in {\left|\Psi\right|^{2}}.) Since the current is positive, it ‘flows’ in the positive {x} direction. Note that the current is independent of {x}, so the probability of a particle being found in any given range of {x} is constant. (Actually, as we’ve seen, a free particle can’t exist in a single stationary state since such a state cannot be normalized.)

2 thoughts on “The free particle: probability current

  1. Pingback: Finite step potential – scattering « Physics tutorials

  2. George

    If I understood the theory well, a free particle cannot exist in a stationary state so we should probably have the integral over k form of the wave equation? Are we allowed to use stationary states as a mathematically useful trick which does not have any physical meaning, and getting right results? The integral form shouldn’t have given the same result?

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