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Relativistic corrections to the energy spectra of completely confined particles
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Relativistic corrections to
the energy spectra of
completely confined particles
Shang Yuan Ren
Department of Physics, Peking University
Beijing 100871, People’s Republic of China
1
Abstract
An analytical expression for the relativistic corrections to the energy spectra of par-
ticles completely confined in an one-dimensional limited length in real space is given,
based upon the wave property of particles, the relativistic energy-momentum relation
and two mathematical equations.
PACS numbers: 03.65.-w,03.65.Pm.
2
The quantum confinement is one of the most fundamental problems in low dimen-
sional physics. The expression for the energy spectra of a non-relativistic particle of
mass m confined in an one-dimensional limited length in real space between x = ?L/2
and x = L/2 has been given in almost any standard quantum mechanics textbook as a
classical example of solving the one dimensional Schro?dinger differential equation with
infinite potential barriers and is well known as
Ej =
j2h?2π2
2mL2
, (1)
where j is a postive integer. This result is widely used in condensed matter physics, as a
theoretical basis of many potential applications of low-dimensional quantum confinement
devices[1].
However, as the energy of the confined particle increases, the relativistic effect will
show up and the equation (1), which was based on the solution of the non-relativistic
Schro?dinger differential equation, must be modified. As the energy further increases to
very large then there could be even creations of new particles. Here we are interested
in the energy range of the particle that the relativistic effect could show up but no new
particle’s creations.
The usual way of obtaining (1) - solving the Schro?dinger differential equation with in-
finite potential barriers - is not easy to extend to the relativistic case straightforwardly[2].
In the following we use a different approach to give the result of equation (1).
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