Relativistic Quantum Revivals

Quantum revivals are now a well-known phenomena within nonrelativistic quantum theory. In this Letter we display the effects of relativity on revivals and quantum carpets. It is generally believed that revivals do not occur within a relativistic regime. Here we show that while this is generally true, it is possible, in principle, to set up wave packets with specific mathematical properties that do exhibit exact revivals within a fully relativistic theory.

Figure 1

Contour maps of the density associated with the wave function of Eq. (2) as a function of time and space. The length of the orbit is chosen as one unit and $m_{\max }=1000$. This is an example of a quantum carpet. Left: The nonrelativistic limit $c=3\times {10}^5$. For later times the pattern repeats. The nature and fractality of such figures have been discussed in detail [5, 9, 20, 21]. Right: $c=30$; the pattern does not exactly repeat for later times. Note the diagonal lines that appear and which are described in the text. It can easily be seen that these are well defined in the nonrelativistic limit, but more “washed out” as relativity becomes more important.

Figure 2

The density associated with just two terms in the summation of the wave function of Eq. (2). The terms chosen are $m=p+1$ and $-p$ for (left) $p=2$ and (right) $p=6$. The length of the orbit is chosen as one unit and $c=3\times {10}^5$. The lines in the interference patterns with $p=2$ and $p=6$ are parallel and will undergo coherent interference when included in the full summation in Eq. (2).

Figure 3

The probability density contour map as a function of time and space. This was calculated with $m_{\min }=950$, $m_{\max }=1000$, and $c=1$.

Figure 4

The density associated with just two terms in the summation of the wave function (2). The terms chosen are $m=p+1$ and $-p$ for (left) $p=2$ and (right) $p=6$. The length of the orbit is chosen as one unit and $c=1$. The lines in the interference patterns with $p=2$ and $p=6$ are not parallel as they are in the nonrelativistic case and will not undergo the coherent interference when included in the full summation in Eq. (2).

Figure 5

The probability density contour map as a function of time and space with $c=5$ and 2 revival times. Left: $n_{\max }=1000$. The upper half of the figure does not repeat the lower half. The parameter values were chosen to emphasize that exact revivals do not, in general, occur within a relativistic regime. Right: Calculated for an orbit of length 60 Compton wavelengths. The wave packet was composed only of eigenstates with quantum numbers that form a Pythagorean triple with 60: i.e., $m=\pm 11$, 25, 32, 45, 63, 80, 91, 144, 175, 221, 297, 448, and 899. This probability density repeats exactly each relativistic revival time $hq/M{c}^2$.