Re: 1. What's the energy of an electron confined to a region the size of a nucl

Dear Camillo,

> 1. What's the energy of an electron confined to a region the size of 
>    a nucleus.

This is a well-known example problem from quantum mechanics called
"the particle in the spherical box".  Search any quantum mechanics
text or the web for background information on this interesting topic.

The short answer is that a particle bound by this potential has an
infinite number of quantized energy states.  You were probably
interested in the lowest energy state, called the "ground state".
The ground state energy is:

         h^2 / (8 m a^2)

where "h" is Planck's constant, 6.626075E-34 J.s;
"m" is the particle (in this case, the electron) mass, 9.109389E-31 kg;
and "a" is the radius of the confining sphere (in this case, the nucleus).
You did not specify the element.  The nuclear radius can range from
about 1E-15 meter for hydrogen to about 7.4E-15 meter for uranium.
The smaller the confining sphere, the larger the energy of the trapped
electron.

The range of typical electron energies is from 6E-8 to 1E-9 joules, or 
in particle physics units, from 3.8E11 to 6.9E9 eV (electron-Volts).

> 2. Why beta decay electrons never have more than a fraction of the 
>    calculated energy, even considering the energy taken away by the 
>    neutrino.

In beta decay, a neutron (either isolated or inside an unstable
nucleus) decays into a proton, an electron, and an electron
anti-neutrino.  The mass of a neutron is slightly larger than the sum
of the masses of the decay products.  This mass difference "m" is
converted through Einstein's famous relation E=mc^2 to energy, the
kinetic energies of the three decay products.

The ejected electron can have any kinetic energy from a minimum of
zero (when the proton and anti-neutrino alone share the kinetic
energy) to a maximum imposed by energy and momentum conservation laws.
The numerical value depends on the nucleus that is undergoing beta
decay, but typical maximum values are 1E4 to 2E6 eV.

This maximum ejected electron kinetic energy is about 300,000 times
SMALLER than the energy of an electron trapped in a spherical box the
size of the nucleus.

The point is that one should not think of the beta decay electron as
existing inside the nucleus before the decay.  The electron is not
bouncing around inside the nucleus waiting for its chance to escape.
Instead, the electron is created ex nihilo when the neutron decays.

--Randall J. Scalise    http://www.phys.psu.edu/~scalise/