Sep 03 2008

## The Graviton as a Momentum Operator

The fundamental commutator relation [x, p] = iћ, between the operators of coordinate and momentum, provides a way to show how a graviton can add linear momentum to an electron.
An energy relation for a synchronous encounter by a graviton with an oncoming electron in an atomic orbital can start with:
[x, pg]2 = i2ћ2 = -ћ2,
The added momentum, as shown by the minus sign, is in the opposite direction of that in which the graviton was traveling at the speed of light in a vacuum before it was absorbed by the electron.
Internal to the electron we can use the {N} representation to form the basis of a set of wavefunctions forming orthonormal vectors: │0 >, │1 >, …, │n >, …, with eigenvalues of N: 0, 1, …, n, … [Messiah, XII.16, pg 436].  The graviton in the process of being absorbed by an electron in a quantum atomic orbital can then be seen as a raising operator, where
aφn = φn+1 and aφn+1 = φn+2 ,
and the release of a graviton a lowering operator, with
aφn = φn-1 and aφn-1 = φn-2 .
The Hamiltonian for such a system is represented as:
H φn = ћω0(aa + ½) φn        [Liboff, Section 7.2],
with energy eigenvalues
En = ћω0(n + ½)        n = (0, 1, 2, …, 68, …nmax).
Here 68 represents the average n value at the face of the earth, and nmax depends on the orbital.
If the mass of the electron diminishes as the gravitational field diminishes, the characteristic wavenumber β of the electron also diminishes.  For each graviton internal to the electron βi2 = miω0/ћ, and for the mass of the electron at the face of the earth we have Σmi = me = 9.1095 x 10-31 kg.

Sep 01 2008