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Motion of Nucleons in Nuclei,
Quantum Mechanics and
the Uncertainty Principle

The Copenhagen Interpretation of Quantum Mechanics holds that electrons in atoms do not have motion. The argument based upon the Uncertainty Principle that electrons in atoms do not have definite trajectories or even motion of any kind goes as follows. The outer limits of atoms are less than 10-8 cm apart. The uncertainty of the location of an electron, σx, therefore cannot more than 10-8 cm. By the Uncertainty Principle the product of the uncertainties of location and momentum must be greater than or equal to Planck's constant divided by 4π. Since Planck's constant divided by 4π is 5.275×10-28 erg-sec the uncertainty of momentum must be at least 5.275×10-19 g-cm/s.

An electron's mass is 9.11×10-28 g. Therefore the uncertainty in electron velocity, σv, must be at least 5.79×108 cm/s. This is translated naively into an uncertainty in the kinetic energy as

½mσv² = 1.522×10-10 ergs
= 95.33 electron-volts (eV).

The correct procedure for computing the uncertainty in the kinetic energy would give an even higher value. The above value however is asserted to be so large compared with the magnitudes of 15 eV for potential energy and kinetic energy of electrons that no location and velocity for an electron in an atom is meaningful.

If the above argument concerning electrons in atoms then it might apply in spades for nucleons in nuclei which are about one ten thousandth of the scale of atoms. The diameter of a small nucleus is on the order of 10-12 cm. With an uncertainty of location of this amount the uncertainty of momentum must be at least 5.275×10-15 g-cm/s. The mass of a proton is 1.674×10-24 g. This means the uncertainty of the velocity of a nucleon is at least 3.165×109 cm/s. This is a bit more than one tenth of the speed of light. If the uncertainty in the kinetic energy of a proton is computed as was done above for an electron the result is

σK = ½(1.674×10-24)(3.165×109)² = 8.384×10-6 ergs.
in eV this is
σK = 5.23×106 eV = 5.23 MeV

The kinetic and potential energies are of the same order of magnitude so the above figure would also be the order of magnitude of the uncertainty of the potential energy. Potential energy and binding energy are closely related for nuclei. The binding energy of a deuteron is 2.225 MeV. Therefore the uncertainty of energy in a deuteron is of the same order of magnitude as its energy. For larger nuclei the uncertainties in the energies could be a lower order of magnitude than the energies themselves. This means that apparently the principle of uncertainty argument does not preclude nucleons having definite trajectories. The scale of a nucleus is about one ten thousandth of that of an atom but the mass of a nucleon is about two thousand times that of an electron. The binding energies of nuclei are on the order of a million times greater than the ionization energies of electrons in atoms.The relationships are

K = p²/(2m)
so, at least roughly
σK = σp²/(2m)
but
σph/(4π σx)
and thus
σKh²/(32π²σx²m)

Thus the 104 ratio of scale get squared to 108 but the ratio of masses of 2×103 reduces this to ½×105. The ratio of 106 of the energies means that the ratio of the uncertainty of energies to the level of the energies for nuclei is fraction of that of electrons in atoms.

Evidence of Nuclear Rotation

There is ample evidence rotation at the nuclear level, both natural and induced. The research on this topic started with the work of Aage Bohr, the son of Neils Bohr. [Aage is pronounced like the English word owe with a little uh at the end.] Later he was joined by Ben Mottleson, a Danish-American physicist. The two, along with James Rainwater who published the first article on the topic, were awarded the Nobel Prize in Physics in 1975. Bohr and Mottelson published a two volume treatise on their Collective Model of nuclear structure and rotation; Volume I came out in 1969 and Volume II in 1975.

Bohr and Mottelson found that the energy levels En of nuclear rotation are given by the formula

En = h²n(n+1)/(2J)

where n is an integer, called the principal quantum number, h is Planck's constatnt divided by 2π, and J is the moment of inertia of the rotating system. Consider the integers that are n(n+1): {2, 6, 12, 20, 30, 42, …} These are familar figures in the numerology of nuclear structure. Several correspond to filled shells or subshells of nuclei.

Physicists had been working with theories and models that involved a quantization of such things as energy and angular momenta since Neils Bohr developed his model of the electronic structure of atoms. It was not until about 1950 that anyone thought of the quantization of the shapes of nuclei. Information about the shape of nuclei is obtained from the measurements of dipole and quadrapole moments of nuclei. In 1950 James Rainwater of Columbia University published an article entitled, "Nuclear Level Arguments for a Spheroidal Nuclear Model." Rainwater relied on previous work by Townes, Foley and Low that made the case that quadrpole moments provided evidence for a nuclear shell structure. ("Nuclear Quadrapole Moments and Nuclear Shell Structure," Physical Review,, Vol. 57 (November 1949), pp. 1415-1416.)

(To be continued.)

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