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SEARCH FOR QUANTUM GYROSCOPES
Abhishek Singh Visen
032, Clippinger Physics Lab,
Ohio University, Athens, OH 45701.
[email protected]
Abstract: This paper is primarily a review of developments in Quantum Mechanics
pertaining to spin related phenomena so far, with particular emphasis on exploring
gyroscopic effects in the published spin related papers. The paper indeed aims at
exposing the existing vacuum in the incipient field of Rotational Quantum Dynamics –
which possesses the potential to generate dramatic quantum phenomena.
The paper begins with exposing the ongoing misinterpretation of Stern-Gerlach
experiment[1]. Then the paper exhaustively explores reported spin-precessions. The
paper also indulges into an interesting discussion “impossibility of arbitral flip” of
electrons. Attempts will be made to explore gyroscopic effects in scattering of the
particles, explain geo-magnetism and bio-chirality, peculiar torque experienced by
magnets etc..
Keywords: Stern-Gerlach, Spin precession, Larmour Precession, Thomas precession
I. Spin Precession
This part of the paper primarily aims at analyzing various spin related precessions to
arrive at vital conclusion of : “Quantization of spin precession” or “impossibility of
existence of arbitral flip” (which has been pompously used in the JJ Sakurai text), which
may have dramatic influence on our way of notion of spin and may give cosmic
background / relicit radiation style avenue for exploration of bigbang.
Before I begin let me reveal what Quantization of angular momentum during flip is : It
states that “when even a flip occurs then the angular momentum is nћ”.
Let’s begin with precession of classical charged sphere in a magnetic field. Assuming
that at time t = 0, there is no magnetic field but a charged sphere is spinning in space with
spin pointing towards positive êx axis. Then the inertia tensor is given by:
Since the matrix is already digitalized, the kinetic energy of the sphere is given by:
1/5 MR2 ωx2
If the sphere is having charge q, then the magnetic moment of sphere is given by:
1/3 QωR2
As the magnetic field is switched on in êz direction
then, the top instead of acquiring minimum electromagnetic energy configuration (i.e., magnetic
moment aligned with the magnetic field), however,
the sphere precesses under the effect of gyroscopic
torque which is proportional to (Ix - Iy) ωxωz. This
phenomena is not very lucidly understood.
However the rate of precession is:
ω = -qB/2m = γB
where, γ is gyro-magnetic ratio
Such precessions are referred as Larmor precession.
Such precessions are now converted into technology. MRI, Paramagnetic Resonance,
Ferromagnetic resonances (Liftshift & Landau) are fantastic manifestations of Larmor
precession.
Now let’s come on to the quantum mechanical version of precession. Actually JJ Sakurai
(revised edition) contains over generalized description of spin precession.
Actually, as the electron is in the magnetic field, with its spin inclined to angle θ is given
by:
hS1(t)i =
While Larmor frequency is given by:
Now it can be seen that precession does depends upon orientation of the particle with the
magnetic field.
The (possible) dependence of Hamiltonians on the angle of inclination may have
important technological importance.
Actually, such precession has been aptly applied for various technological purposes like:
MRI, NMR, ESR etc..
II. Semi-classical explanation of Stern-Gerlach experiment
Usually Stern-Gerlach experiment is thought of as ‘bizarre’ Quantum Mechanical
phenomena. In the following lines attempts are made to re-interpret is semi-classically.
(space left)
III. Quantization of Precession
For proper understanding of quantum phenomena, I think following should be enunciated
as established principles:
1. For a fermionic system, the spin precession is quantized.
2. It is impossible to have any stable state other than eigen-values of the Fermion.
Above principles can help us in understanding various phenomena like:
1.
2.
3.
4.
5.
Nuclear Magnetic Resonance
ESR
Magnetic Resonance Imaging
Precession clocks
Ferromagnetic resonances etc..
IV. Trajectory of the precessing electrons
The classically predicted trajectory of electron ignores the interaction of the spin of the
electron with the magnetic field. More elaborate studies of trajectories of electrons in
magnetic field is required.
V. Certain Gyroscopic experiments
There are certain gyroscopic experiments which I want to propose. These experiments are
very simple and affordable, but are important because they offer a unique synergy of
classical and quantum mechanics.
a) Torque on a magnetic bar: If we take a magnetic bar and an iron bar of same
dimensions, and rotate them. The magnetic bar will experience a torque. This is
because, there are magnetic moments inside the magnets, which when rotated will
generate an torque. For first initial estimate. Assuming about 1023 atoms/cm3 are
inclined in a same direction. Then the approximate torque is given by:
τ ≈ ω me r2 (1015) * 1023 ≈ 10-3 (This calculation may be wrong)
b) Demagnetization of magnet bar: Using the gyroscopic torque the magnet can be
demagnetized by revolving it.
c) Magnetization of rotating fluid: It is predicted that molten metals can be
magnetized by rotating it. This may be the cause of earth’s magnetism.
VI. Conclusions
On the whole it can be seen that there are many interesting unexplored territories in the
field of quantum gyroscopes. Interest in this area will enable mankind to search for vital
truths about the nature.
References
1. [Peniaz] 1978
2. Phys Rev Lett. 1986 Nov 17;57(20):2500-2503
3. Morton, Jr., S. Harold, 1993, Hamiltonian and Lagrangian Formulations of
Rigid Body Rotational Dynamics Based on Euler Parameters, J. Astronaut.
Sci., 41, pp. 561–5991.
4. S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon: Oxford, 1966).
5. S. Chikazumi, Physics of Ferromagnetism (Oxford: New York, 1996).
6. http://nobelprize.org/nobel_prizes/physics/laureates/2006/phyadv06.pdf