Friday, July 9, 2010

More Quantum Mechanics: Electron Spin

So far, we've seen three different quantum numbers: n (principal), l (angular momentum), and m_l (magnetic). We now examine how the fourth quantum number (m_s) or electron spin came about.

By the early 1920s, quantum mechanics had developed to the point that theorists realized an electron magnetic moment ought to exist. In any external magnetic field B, the magnetic moment ought to experience a force similar to what a magnetic compass needle experiences in Earth's magnetic field. According to quantum mechanics, the assumed values should be quantized, so the magnetic moment can assume only certain values, given by:

2l + 1

Thus, the orientation number will depend on the second quantum number, l. The angular momentum (L) would therefore be:

L = h/2 pi[l(l +1)]^1/2

In 1922, the German physicists Otto Stern and Walther Gerlach passed a beam of electrons through a non-uniform magnetic field B as shown in the top sketch of the diagram (C denotes collimator and D, detector plate). A non-uniform field meant the field was stronger on one side of the beam than the other. As predicted from theory, the force on the magnetic moment of the electrons is such that the field ought to deflect the beam according to the orientation of its moment.

The field should therefore split the beam into 2l + 1 parts (according to 2l +1 orientations). Stern and Gerlach found their beam of hot silver atoms split into TWO parts. At first this appeared surprising and at odds with theory but later work showed the conflict could be resolved if the electrons going into the opposing (up and down) beams, had their own spins, or intrinsic angular momenta. In this case, the upper beam would feature electrons with "spin up" angular momentum of (+1/2) while the lower beam would feature electrons with "spin down" angular momentum of (-1/2) . These are shown at the lower section of the diagram. Note that in each case, spin up or down, the orientation accords with the direction of rotation: clockwise or anti-clockwise.

The refining experiment was actually done by Phipps and Taylor using a beam of heated H-atoms. They began by defining the magnetic dipole moment such that:

u(lz) = - g(l) u(b) m_l

where, as before, m_l = -l, -l +1, 0, ....+l

and u(b) is the Bohr Magneton: u(b) = eh/4pi m

where m is the mass of the H-electron.

Meanwhile, g(l) is known as the "orbital g-factor".

From their low oven temperature, Phipps-Taylor recognized that l = 0 for the electrons coming off and entering the magnetic field. Therefore, m_l = 0. Since this was so, then:

u(lz) = 0

Phipps -Taylor therefore assumed the beam would be unaffected or not split at all. Yet, it split into two smmetric components, e.g. deflected symmetrically. Given the earlier Stern-Gerlach experiment, plus their own, they therefore had to expect the electron had its own magnetic dipole moment..

They assumed this to be u(s), due to the electron having an intrinsic angular momentum S (its spin) analogous to L (angular momentum) so that:

S = [s(s + 1)]^1/2 (h/2pi) and S(z) = m_s(h-bar) {z-component of angular momentum)

where h-bar = h/ 2pi

Then m_s is the electron spin or spin quantum number, 1/2 or - 1/2,

Then the electron magnetic dipole moment would be:

u(s) = - g(s)u(b)/ h-bar [S]

To nail down the basis quantitatively, Phipps-Taylor knew that the net force felt by the electrons traversing the field would be:

F(z) = - (dB(z)/dz) u(b) g(s) m_s

where m_s is the putative spin quantum number.

Since they knew the Bohr magneton u(b) = 9.27 x 10^-24 J/T

and dB/dz could be measured (e.g. the difference in the B-field over the collimation width dz)
then :

g(s)m_s = F(z)/ - (dB(z)/dz) u(b) = +/- 1

as measured.

Since g(s) = 2, then by deduction:

m_s = +/- 1/2

Thus, Phipps and Taylor discovered that the splitting effectively showed two possible values for the orientational potential energy, or Diff(E) = - u(s) B = +/- g(s)u(b)B/2. As an example, let's compute S for the value m_s = 1/2.

S = [s(s+ 1)]^1/2 h/2 pi= [1/2(1/2 + 1)^1/2 (h/ 2pi) =

[1/2(3/2)]^1/2 (h/2pi) = [3/4]^1/2 h/2pi = (3/2) h/2pi = 3 h-bar/2

1 comment:

Divine Advanced Human Beings said...

The Electron… E-Space

Article Written by:J.S. Thompson • Saturday, July 03rd, 2010

Space is broken up into geometrical patterns formed by the intersection of particles. These pieces of space contain their own individual and unique pressure, which exists within a spectrum.

Every particle which exists in “natural space,” the space that existed before the creation of the universe (altered space) and, which exists as infinite space outside of the universe, exists within a multitude of spectrums, are limited in mass and density.

In other words, particles can only be so small and so large and pressures can only be so great and so little. First, let’s examine the most massive particle that exists in space, the “macro particle” and all of its working parts. Like everything that exists in both natural and altered space, everything must adhere to the law of physics. Nothing can become too heavy nor can it become too large; if you put one too many bricks, the structure will collapse. The same principle applies to particles in space. In the case of space, its fluidity conforms to a natural configuration, just like water. Space is a sea of particles one linked to the other. These particles are pushed in all directions into one another as far as the pressure that opposes them will allow. “A macro particle” is a particle that exists in atomic space and so would be as small as what contemporary science calls an atom and can be as big as one half the size of the universe (altered space). These two massive particles that engulf the universe would be separated at the periphery and the center by a “quantum degree.” Quantum degree (qd) represents the smallest point in space, a space within a space. These qd’s form quantum structures that crisscross through space within which electrons flow. A “Quantum Structure” is a conduit within, which is contained pressure that is so intense, that it cannot become any greater “at the periphery” of the center. However, the pressure at the “center” of this structure is much less than the pressure at the periphery as one particle has been pushed through the center of another providing a space of “relative ambiance” at the very center. This space or Electron space (E. space) is broken up in the same way that natural space and altered space are broken up into particles that conform to the law of physics in relative pressure…electrons! This configuration of space at the greatest relative pressure provides the frequency signals that make matter what it is. The electrons flowing through these conduits in signature configurations are a computer program that makes this universe or altered space work. Matter is the final product of these signature configurations of electrons that vector up from the E. space. Matter is a chain reaction that begins with the Electron.

The electron conforms to the pressure that surrounds it on eight sides. Quantum conduits are octagonal at the center surrounded by four less octagonal larger particles at the periphery. Space is octagonal becoming less pronounced and more geometrically round as it vectors up to the limits of altered space until…one quantum degree at the periphery of the largest two macro particles is obtained.

Motion is a product of a natural imbalance that exists in space, an imbalance that takes us from the largest to the smallest. All structures in space reflect this imbalance. This imbalance causes particles within larger particles to be less dense at the periphery and most dense at the center. All particles are being pushed toward the center of a larger particle becoming denser as they move into the particle and eventually are pushed out of the center by larger more massive particles. This natural motion of particles from the periphery to center constitutes “particle orbit”.

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