Autumn · project 7 of 20

Relativistic tremor and pressure

Topic: Relativistic quantum mechanics

  1. Let the state of a free electron at \(t=0\) be given by \[\psi(0,\boldsymbol{x}) = \frac{1}{(2\pi d^2)^{3/4}}e^{-\frac{\boldsymbol{x}^2}{4d^2}} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix}.\] Find \(\psi(t,\boldsymbol{x})\) for \(t>0\).

  2. Obtain an analytic expression for the mean current as an integral over the 3-momentum: \[\boldsymbol{J}=\int d^3x \psi^\dagger(x)\,\boldsymbol{\alpha}\,\psi(x).\]

  3. Compute this integral and make an animation of \(\boldsymbol{J}(t)\).

  4. Does the tremor of the current last forever?

  5. Compute the pressure \(P\) of the field and make an animation of \(P(t)\).

  6. Explain the results.

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