🚀 go-pugleaf

There continues to be discussion in this group about the
existence of "hydrinos", proposed by John Farrell and others.  Hydrinos
are supposed to be states of the hydrogen atom having total binding 
energies significantly greater than the binding energy
of the conventional ground state, 13.6 eV.  Since states of the conventional
hydrogen atom have mean radii varying inversely as the binding energy, in good
approximation, it has been suggested that hydrinos are correspondingly 
smaller.  I find these ideas rather silly, and I will try to explain why
in this submission.

The hydrogen atom, that is, a system of a hydrogen nucleus and electron
bound by the Coulomb force, is one quantum system which we understand 
very well.  The wavefunctions and energies of all physically admissible 
states follow directly from the solution of the Schroedinger wave 
equation.   These states are characterized by quantum numbers, the most
important of which is known as the principal quantum number n.  It is found
that the total energy of a state is given by  E(n)= -13.6 eV/n**2, and the
mean radius is given (roughly) by r(n)= n**2 a/0/, in which a/0/ is the Bohr 
radius, 0.053 nanometers.  The only solutions found are for positive, 
integer values of n.  The mathematics is not especially difficult to follow 
but is inappropriate for this forum.  Interested persons may read further in
an undergraduate text such as Quantum Physics, by R. Eisberg and R. Resnick.

In the remainder of this submission, I will try to explain qualitatively why
hydrinos cannot exist using an argument based on the wave-particle duality 
of matter and Heisenberg's uncertainty relation.  The argument is independent 
of any detailed solutions.  We shall find the radius and energy of the 
ground state of the hydrogen atom are given by the conventional n=1 of the
Bohr model. 
  
Matter has either wave or particle attributes, depending on the circumstances
under which it is observed.  Thus, for example, Debroglie's relation tells us
that a particle with momentum p has an associated wavefunction whose
wavelength l given by  p= h/l, in which h is Planck's constant.  A point 
that I wish to emphasize is that a confined electron, or any other particle, 
can only have a wavefunction built up out of a superposition of wavelengths 
equal to or smaller than the range of confinement, which I will call
(dx) along one direction in space.  Each wavelength component has an
associated momentum.  If we take the maximum wavelength to be equal to (dx), 
then there must be a minimum momentum (dp), in the root-mean-square sense,
of the confined particle equal to (dp)= h/(dx).  The quantity (dp) can
be considered the minimum uncertainty in the momentum of the particle,
and the momentum can be no smaller than this uncertainty.  

The rough connection just found between the uncertainties in position and 
momentum of a matter-wave is more precisely expressed in Heisenberg's 
uncertainty relation:

	(dx)(dp) >= hbar/2. 						(1)

Here, hbar is Planck's constant divided by 2pi= 6.2830.  We will now use
Heisenberg's relation to estimate the minimum kinetic energy of the electron.  
The argument is qualitative but will reveal an essential dependence of the 
minimum energy of the atom on atomic size.  The total energy of the system 
is equal to the sum of the potential and kinetic energies:

	E= U+ K,							(2)

in which U= -k e**2/r is the Coulomb potential energy and K= p**2/(2m) is
the kinetic energy of the electron (we neglect the kinetic energy of the
much heavier nucleus).  Here r is the distance of the electron from the
nucleus, p, e and m are the electron's momentum, charge and mass, and k is 
a measure of the Coulomb interaction stength.  For simplicity, we will 
assume that the particle is confined by equal lengths (dx) along the three 
directions in space.  Then, summing the p**2 contributions for the three 
independent directions in space, the minimum value of K is found to be

	K/min/=  3(dp)**2 /2m =  3 hbar**2/ (8 m (dx)**2),		(3)

using eqation 1 to obtain for the second equality.  To obtain an order-of-
magnitude estimate of the minimum value of the TOTAL energy we set (dx) 
equal to the atomic radius, r, in the above equation, and add K/min/ to 
the potential energy, U, obtaining

	E/min/=  -k e**2/r + 3 hbar**2/ (8 m r**2).			(4)

Equation 4 shows how the minimum energy of a hydrogen atom varies as a 
function of r, the atomic radius.  No assumptions have been made about 
quantum numbers.  For large values of r, E/min/ goes to 
zero (the atom becomes unbound).  As r goes to zero, E/min/ diverges to 
positive infinity as 1/r**2.  In between, E/min/(r) takes on negative 
values with a global minimum which can be obtained by setting the 
derivative of E/min/(r) with respect to r equal to zero.  Doing this, 
we find that the following value of r minimizes the function E/min/(r):

	r= 3/4 [ hbar**2/ (m k e**2)].					(5)

The factor in square brackets is just the Bohr radius, a/0/, whereas the
the prefactor of 3/4 is a consequence of the particular assumption made 
here about (dx).  

According to the Bohr model, the energy and radius of a hydrogen atom
are given in terms of a quantum number n by  E= -13.6 eV/n**2  and 
r= n**2 a/0/.  Thus, we can interpret the result in eq. 5 as showing that
the minimum-energy state corresponds to n= 1.   Equation 4 shows that
both smaller and larger atoms have GREATER energies.  If a state satisfying 
r= n**2 a/0/ with fractional quantum number n= 1/210 were to exist, for
example, its minimum energy computed from eq. 4 would be equal to 
about +10**10 eV!   Thus, we find that fractional quantum numbers do not 
lead to "sub-ground-level" energies and can not be used to explain reports
of excess heat production in electrolysis experiments.  

Yours,  Gary S. Collins  (collins@cougar.csc.wsu.edu)