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Free-Particle Solutions and Wave Packets

Because of the completeness of plane-wave solutions, we may superimpose plane waves to construct localized wave packets. The wave packets are also solutions to the Dirac equation since it is linear.

A positive-energy wave packet solution can be written as

\psi^{(+)}(\vec{x},t) = \int \frac{d^3p}{(2\pi\hbar)^{3/2}}
...{mc^2}{E}} \sum_{\pm s} b(p,s) u(p,s) e^{-ip\cdot x/\hbar} ,
\end{displaymath} (5.269)

where $b(p,s)$ is a complex scaler function. Normalized to unit probability gives

    $\displaystyle \int d^3x{\psi^{(+)}}^\dagger(\vec{x},t)\psi^{(+)}(\vec{x},t)$  
  $\textstyle =$ $\displaystyle \int\frac{d^3xd^3pd^3p^\prime}{(2\pi\hbar)^3}
e^{i(p-p^\prime)\cdot x/\hbar}$  
  $\textstyle =$ $\displaystyle \int d^3p \frac{mc^2}{E} \sum_{\pm s,\pm s^\prime}
  $\textstyle =$ $\displaystyle \int d^3p \frac{mc^2}{E} \sum_{\pm s,\pm s^\prime}
  $\textstyle =$ $\displaystyle \int d^3p \sum_{\pm s} \vert b(p,s)\vert^2 = 1 .$ (5.270)

The average current for such a wave packet is given by the expectation value of the velocity operator

$\displaystyle \vec{j}^{(+)}$ $\textstyle =$ $\displaystyle \int d^3x \psi^{(+)^\dagger} c\vec{\alpha} \psi^{(+)}
= \int d^3x \overline{\psi}^{(+)} c\vec{\gamma} \psi^{(+)}$  
  $\textstyle =$ $\displaystyle \int\frac{d^3xd^3pd^3p^\prime}{(2\pi\hbar)^3}
...-p^\prime)\cdot x/\hbar}
\overline{u}(p,s) c\vec{\gamma} u(p^\prime,s^\prime) .$ (5.271)

The space part of the Gordon decomposition

c\overline{\psi}_2\gamma^\mu\psi_1 = \frac{1}{2m} [\overline...
...c{i}{2m} \hat{p}_\nu (\overline{\psi}_2\sigma^{\mu\nu}\psi_1)
\end{displaymath} (5.272)

can be used. The Gordon decomposition decomposes the Dirac current density into a convection current-density term, similar to the nonrelativistic case, and an additional spin current-density term. We can now write

$\displaystyle j_k^{(+)}$ $\textstyle =$ $\displaystyle \int\frac{d^3xd^3pd^3p^\prime}{(2\pi\hbar)^3}
...\pm s,\pm s^\prime}
e^{i(p-p^\prime)\cdot x/\hbar}$  
  $\textstyle \cdot$ $\displaystyle \overline{u}(p,s) \frac{1}{2m} \left[ (p_k + p_k^\prime) +
\sigma_k^{\ \nu} (p_\nu - p_\nu^\prime) \right] u(p^\prime,s^\prime)$  
$\displaystyle \vec{j}^{(+)}$ $\textstyle =$ $\displaystyle \int d^3p \frac{\vec{p}c^2}{E} \sum_{\pm s}
\vert b(p,s)\vert^2 .$ (5.273)

Using the normalization condition, the current can be written

\vec{j}^{(+)} = \langle c\vec{\alpha} \rangle_+ = \left\langle
\frac{c^2\vec{p}}{E} \right\rangle .
\end{displaymath} (5.274)

Thus the average current for an arbitrary packet formed of positive energy solutions is just the classical group velocity. In the Schrödinger theory the velocity operator $\hat{v}=\hat{p}/m$ is proportional to the momentum, but this is not the case in the Dirac theory. In the Dirac theory the velocity operator for a free particle $c\hat{\alpha}$ is no longer a constant. We see that wave packets consisting of plane waves with only positive energy have the expectation value of the velocity $\vert\langle
c\hat{\alpha}\rangle\vert \sim \vert\langle c^2\vec{p}/E \rangle\vert < c$, whereas the eigenvalues of $c\hat{\alpha}$ are exactly $\pm c$. This motivates us to consider a wave packet containing both positive and negative energy solutions:

\psi(\vec{x},t) = \int \frac{d^3p}{(2\pi\hbar)^{3/2}}
...^{-ip\cdot x/\hbar} +
d^*(p,s) v(p,s) e^{+ip\cdot x/\hbar}
] ,
\end{displaymath} (5.275)

where $b(p,s)$ and $d^*(p,s)$ are complex scalar functions. Normalized to unit probability, we have

    $\displaystyle \int d^3x \psi^\dagger(\vec{x},t) \psi(\vec{x},t)$  
  $\textstyle =$ $\displaystyle \int \frac{d^3xd^3pd^3p^\prime}{(2\pi\hbar)^{3}}
...ime) u^\dagger(p,s) u(p^\prime,s^\prime)
e^{i(p-p^\prime)\cdot x/\hbar} \right.$  
  $\textstyle +$ $\displaystyle b^*(p,s) d^*(p^\prime,s^\prime) u^\dagger(p,s) v(p^\prime,s^\prime)
e^{-i(p+p^\prime)\cdot x/\hbar}$  
  $\textstyle +$ $\displaystyle d(p,s) b(p^\prime,s^\prime) v^\dagger(p,s) u(p^\prime,s^\prime)
e^{i(p+p^\prime)\cdot x/\hbar}$  
  $\textstyle +$ $\displaystyle \left.
d(p,s) d^*(p^\prime,s^\prime) v^\dagger(p,s) v(p^\prime,s^\prime)
e^{-i(p-p^\prime)\cdot x/\hbar} \right]$  
  $\textstyle =$ $\displaystyle \int d^3p \frac{mc^2}{E} \sum_{\pm s,\pm s^\prime} \left[
b^*(p,s) b(p,s^\prime) u^\dagger(p,s) u(p,s^\prime) \right.$  
  $\textstyle +$ $\displaystyle b^*(p,s) d^*(-p,s^\prime) u^\dagger(p,s) v(-p,s^\prime)
e^{-2ip_0 x_0/\hbar}$  
  $\textstyle +$ $\displaystyle d(p,s) b(-p,s^\prime) v^\dagger(p,s) u(-p,s^\prime)
e^{2ip_0 x_0/\hbar}$  
  $\textstyle +$ $\displaystyle \left. d(p,s) d^*(p,s^\prime) v^\dagger(p,s) v(p,s^\prime)
  $\textstyle =$ $\displaystyle \int d^3p \frac{mc^2}{E} \sum_{\pm s,\pm s^\prime}
\left[ b^*(p,s...
+ d(p,s) d^*(p,s^\prime) \frac{E}{mc^2} \delta_{ss^\prime} \right]$  
  $\textstyle =$ $\displaystyle \int d^3p \sum_{\pm s} \left[ \vert b(p,s)\vert^2 + \vert d(p,s)\vert^2 \right] = 1 .$ (5.276)

A short calculation shows the current of the wave packet is

$\displaystyle j^k$ $\textstyle =$ $\displaystyle \int d^3x \psi^\dagger(\vec{x},t) c\hat{\alpha}_k
  $\textstyle =$ $\displaystyle \int d^3p \left\{ \sum_{\pm s} \left[ \vert b(p,s)\vert^2 + \vert d(p,s)\vert^2 \right]
\frac{p^kc^2}{E} \right.$  
  $\textstyle +$ $\displaystyle ic \sum_{\pm s,\pm s^\prime} b^*(-p,s)d^*(p,s^\prime)
e^{2ix_0p_0/\hbar} \overline{u}(-p,s) \sigma^{k0} v(p,s^\prime)$  
  $\textstyle -$ $\displaystyle \left. ic \sum_{\pm s,\pm s^\prime} b(-p,s)d(p,s^\prime)
e^{-2ix_0p_0/\hbar} \overline{v}(p,s) \sigma^{k0} u(-p,s^\prime)
\right\} .$ (5.277)

The first term represents the time-independent group velocity that appeared before for positive energy only wave packets. The second and third terms are interferences of the solutions with positive and negative energy, which oscillate time-dependently because of the factors $\exp(\pm 2ip_0x_0/\hbar)$. The frequency of this Zitterbewegung is

\frac{2p_0c}{\hbar} > \frac{2mc^2}{\hbar} \approx 2\times 10^{21}
\textrm{s}^{-1} .
\end{displaymath} (5.278)

next up previous contents index
Next: Klein Paradox for Spin-1/2 Up: Dirac Equation Previous: Combined PCT Symmetry
Douglas M. Gingrich (gingrich@