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P.A.M. Dirac's Theory of Electrons and Positrons |
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This is a presentation of Paul Adrien Maurice Dirac's theory of the electron and positron as he described it in his acceptance speech upon receiving the Nobel Prize in Physics in 1933.
In classical (nonrelativistic) mechanics the total energy of a particle is equal to the sum of its kinetic energy K and its potential energy V. The kinetic energy is ½m_{0}v² where m_{0} is the mass of the particle and v is its velocity. The linear momentum p of the particle is mv and therefore kinetic energy can be expressed in terms of momentum as p²/(2m_{0}). For a particle not tied to a potential field, a so-called free particle, the kinetic energy E is then
In relativistic classical mechanics the mass of a particle depends upon its velocity according to the equation
where c is the speed of light. This leads to the formula for total energy E of
Total energy includes rest-mass energy m_{0}c² as well as kinetic energy.
When velocity v is small compared with c
but in general it is not true that kinetic energy is equal or closely approximated by ½m_{0}v².
Note that relativistic momentum p is given by
Note also that (m_{0}c²)² is equal to m_{0}²c^{4}. Thus
On the other hand
Thus
This is known as the Relativistic Momentum-Energy Equation.
Dirac starts with the equation
which looks like the Relativistic Momentum-Energy Equation but with kinetic energy replacing total energy, which includes rest-mass energy as well as kinetic energy.
Dirac notes that according to standard procedure the above would lead to a quantum mechanical equation for the wave function ψ of the form
with K converted to the operator ih(∂/∂t) where i is the imaginary unit, h is Planck's constant and t is time. Momentum p is converted to the operator −ih(∂/∂x).
Dirac notes that this wave equation will not do because the wave equation must be linear in (∂/∂t). Furthermore for relativistic invariance the wave equation must be linear in (∂/∂x). Dirac proposed a wave equation of the form
where α_{r} is a vector of some sort to match the vector of momenta p_{r} and the zeroeth component term (m_{0}c). The term α_{r}p_{r} represents the sum of the products of the components of p and α for r=1 to 3.
Dirac worked by proposing equation and then looked for variables that would satisfy the equation. In this case the α vector turned out to be a vector of matrices. These component matrices have elements which are in turn matrices.
The properties which the α_{r} components had to satisfy are
Further details of the nature of the α variables are given in Dirac's electron.
Experimentally it was found in the 1920's that an electron has a spin of half a quantum and a magnetic moment of one Bohr magnetron in reverse direction to the spin. Dirac asserts that the α_{r} variables imply the spin and magnetic moment found empirically for an electron.
Dirac also asserts that the α_{r} variables give rise to what Erwin Schrödinger called Zitterbewegung, a trembling motion of electrons at high frequency and low amplitude which involves electrons moving at the speed of light.
(To be continued.)
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