Skript / lecture notes - Universität Paderborn

Prof. Dr. Wolf Gero Schmidt
Universität Paderborn, Lehrstuhl für Theoretische Physik
Bem: Um die Zustände mit negativer Energie physikalisch interpretieren zu können,
nahm Dirac an, daß alle Zustände negativer Energie vollständig besetzt seien. Nach
dem Pauli’schen Ausschließungsprinzip finden damit in dem sogenannten “Dirac–
See” keine weiteren Teilchen Platz, und es ist ausgeschlossen, daß ein Elektron durch
das Fallen in immer tiefere Zustände unendlich viel Strahlung emittiert. Anregung
eines Teilchens aus dem Dirac–See erzeugt ein reales Elektron mit negativer Ladung,
dem ein effektiv positiv geladenes Loch im Dirac–See gegenübersteht. Dieses
Loch wird “Positron” genannt und stellt das Antiteilchen zum Elektron dar. Durch
Paarvernichtung können Elektronen und Positronen annihilieren.
Note: The DE is extremely successful in describing electron dynamics. However, it possesses
a peculiar feature: for each quantum state possessing a positive energy E,
there is a corresponding state with energy −E. This is not a big difficulty when
an isolated electron is considered, because its energy is conserved and negativeenergy
electrons may be left out. However, difficulties arise when effects of the
electromagnetic field are considered, because a positive-energy electron would be
able to shed energy by continuously emitting photons, a process that could continue
without limit as the electron descends into lower and lower energy states. Real
electrons clearly do not behave in this way. Dirac’s solution to this was to turn to
the Pauli exclusion principle. Electrons are fermions, and obey the exclusion principle,
which means that no two electrons can share a single energy state within an
atom (if spin is ignored). Dirac hypothesized that what we think of as the ”vacuum”
is actually the state in which all the negative-energy states are filled, and none of
the positive-energy states. Therefore, if we want to introduce a single electron we
would have to put it in a positive-energy state, as all the negative-energy states
are occupied. Furthermore, even if the electron loses energy by emitting photons it
would be forbidden from dropping below zero energy. Dirac also pointed out that
a situation might exist in which all the negative-energy states are occupied except
one. This ”hole” in the ”Dirac sea” of negative-energy electrons would respond to
electric fields as though it were a positively-charged particle. Initially, Dirac identified
this hole as a proton. However, Robert Oppenheimer pointed out that an
electron and its hole would be able to annihilate each other, releasing energy on
the order of the electron’s rest energy in the form of energetic photons; if holes were
protons, stable atoms would not exist. Hermann Weyl also noted that a hole should
act as though it has the same mass as an electron, whereas the proton is about two
thousand times heavier. The issue was finally resolved in 1932 when the positron
was discovered by Carl Anderson, with all the physical properties predicted for the
Dirac hole.
97