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Deuteron with a Nuclear Strong Force that is an Exponentially Weighted Inverse Distance Squared Law |
Consider two particles of equal mass m with their centers separated by a distance s. Let ω be the rate of rotation of the particles about their center of mass. The particles will travel in circular orbits of radius r equal to s/2. The angular momentum of the system is then 2m(ωr)r, which is equal to mωs²/2. Angular momentum is quantized so
where n is an integer, called the principal quantum number, and h is Planck's constant
divided by 2π.
Thus the angular rate of rotation is given by
The kinetic energy K of the system is 2(½m(ωr)²) which reduces to mω²s²/4. If the expression above for ω² is substituted into this formular the result is
This formula will be used later.
For circular orbits of radius r the centrifugal force on each particle is mω²r which is equivalent to mω²s/2. Thus a balance requires
Equating the two expressions found above for ω² gives
where λ = h²/(mHs_{0}).
The function σ*
exp(−σ) has the form
This is the quantization condition for σ which leads to the quantization of ω and the other variables.
The separation distance s_{1} in the physical deuteron is about 2.252 fermi (2.252×10^{-15} m), Thus s_{1}/s_{0} is equal to 1.4796, a pure number. When this value is substituted into the LHS of the above equation the result is
This is essentially at the maximum of the function σ*exp(−σ) so n can only possibly have the value of 1.
Values for all quantities in the quantization condition are known except for the constant H.
If s_{1} is the actual separation distance then
The potential energy function V(s) is given for s_{1} by
If the above expression for H is substituted into this formula the result is
The function exp(s_{1}/s_{0}) may be taken inside the integral to give
The variable of integration may be changed from z to y=z-s_{1} to give
Another change of the variable of integration to p=y/s_{0} gives
Since K(s_{1})=n²h²/(ms_{1}²)
The integral in the above equation can be approximated by numerical integration. For s_{1}/s_{0} equal to 1.4796 its value is 0.24683. This makes the ratio V(s_{1})/K(s_{1}) equal to −0.73042.
Since the magnitude of V(s_{1}) is less than the magnitude of K(s_{1}) it is physically impossible for the two particles to come together with a principlal quantum number of 1 without an input of energy. The base state has to be one of zero angular momentum. From this base state of principal quantum number of zero a state with a high principal quantum number would require an input of energy.
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