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Physical Review

On the Process of Space Quantization

Isidor Isaac Rabi

Published Feb 15, 1936• 663 citations
Publisher

Abstract

The effect of a rapidly varying weak magnetic field on an oriented atom possessing nuclear spin is discussed. The results are applied to an experiment by Frisch and Segr\`e. It is shown that by methods embodying these principles one can measure nuclear spin even when the h.f.s. separation is very small. A possible further application is discussed whereby it is possible to measure the sign of nuclear magnetic moment vector with respect to the spin vector.

DOI
10.1103/physrev.49.324

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The effect of a rapidly varying weak magnetic field on an oriented atom possessing nuclear spin is discussed. The results are applied to an experiment by Frisch and Segr`e. It is shown that by methods embodying these principles one can measure nuclear spin even when the h.f.s. separation is very small. A possible further application is discussed whereby it is possible to measure the sign of nuclear magnetic moment vector with respect to the spin vector.
The nonadiabatic transitions which a system with angular momentum JJJ makes in a magnetic field which is rotating about an axis inclined with respect to the field are calculated. It is shown that the effects depend on the sign of the magnetic moment of the system. We therefore have an absolute method for measuring the sign and magnitude of the moment of any system. Applications to the magnetic moment of the neutron, the rotational moment of molecules, and the nuclear moment of atoms with no extra-nuclear angular momentum are discussed.
The transitions between states of different space quantization which take place when an atom is passed through a weak inhomogeneous magnetic field are calculated by approximate methods taking into account the coupling between the nucleus and the orbital electrons. Two types of field are considered, namely, the field introduced by Majorana and the one introduced by G\"uttinger. For the Majorana field a discussion of the cases III arbitrary J=12J=\frac{1}{2}J=21​, JJJ arbitrary I=12I=\frac{1}{2}I=21​, and III and JJJ arbitrary with explicit application to the nitrogen atom is given. For the G\"uttinger field the general equations are set up for a field of any strength and are then applied to the case I=J=12I=J=\frac{1}{2}I=J=21​.
The present paper shows that the nature of the polarization of charged spin-1/2 particles moving in a uniform magnetic field changes dramatically in a relatively weak transverse axisymmetric magnetic field. The direction along which the spin projection is quantized has a fixed orientation with respect to the axes of a cylindrical coordinate system and can form a substantial angle with the direction of the uniform magnetic field. The presence of spin quantization is proved both by the fact that the commutator of the Hamiltonian operator and the projection of the polarization operator in the direction of quantization is zero and by analyzing the Bargmann-Michel-Telegdi equation for this given case. Finally, the possibilities of detecting this effect and utilizing it are discussed.
Collision experiments employing a preparation and/or an analysis of spin polarization provide a powerful tool to study individual interaction mechanisms. For atoms or ions with nonvanishing nuclear moments the hyperfine interaction before and after the collision may affect the spin polarization and any collision-induced orientation and alignment, thereby substantially reducing experimentally observable effects. This disturbance can be avoided by applying an external magnetic field, as is demonstrated here for the inelastic collision process He+{\mathrm{He}}^{+}He++Na(\ensuremath{\uparrow}) \ensuremath{\rightarrow}He+{\mathrm{He}}^{+}He++Na∗{\mathrm{Na}}^{\mathrm{*}}Na∗(3p). Easily attainable magnetic field strengths are sufficient to decouple electronic and nuclear spins whereby the figure of merit for both, orientation and alignment, is increased by more than one order of magnitude.
This chapter provides the formalism of quantum mechanics to the description of measurements. It describes how an atomic system behaves in a slowly varying applied field. In the Stern-Gerlach (S-G) experiment an atomic or molecular beam is passed through an inhomogeneous magnetic field. The field causes a spatial separation of systems having different values of the magnetic moment and allows one to determine the moment by measuring the deflection of the emergent beams. The S-G magnets and the recombining fields merely force the wave function to undergo a sequence of contortions, which are then carefully undone; the experiment is vacuous, and is not a measurement. The mathematical formalism of quantum mechanics also allows to compute the beam trajectory. If one accepts J. von Neumann's formulation of the quantum mechanics, one is led to far reaching and Dot entirely palatable conclusions.
Quantization of spin and orbital angular momenta in standard quantum mechanics is rooted in the fundamental rotational symmetry of space in conjunction with the results of experiments on atomic and subatomic particles conducted with the Stern-Gerlach apparatus. We explain some of the most important features of spin and orbital angular momenta as inescapable consequences of elementary Stern-Gerlach experiments viewed in light of the rotational symmetry of space. The rotation operator is first derived for spin-½ particles, then extended to spin-1 particles by treating the case of a hydrogen atom in its ground state, where the spin of the proton combines with that of the electron to create a spin-1 triplet state. Along the way, we examine the evolution of a particle’s spin state under an applied magnetic field, where the interaction energy arises from the existence of a magnetic dipole moment for each particle in proportion to its spin angular momentum. Whereas a constant magnetic field B0 causes the particle’s magnetic moment to gyrate around the direction of B0, it will be seen that the presence of a small oscillating magnetic field perpendicular to B0 brings about the so-called Rabi oscillations, which involve periodic flippings of the particle’s spin orientation. In the case of a hydrogen atom in its ground state, the interaction between the magnetic dipole moments of the proton and the electron splits the ground state between a lower energy singlet having a net angular momentum of zero (spin-0 state), and a higher energy triplet having a total angular momentum of ħ (spin-1 state). This hyperfine splitting is the source of the well-known 21-cm line observed in the spectrum of the hydrogen atom, which arises from transitions between the atom’s singlet and triplet states. In the final section, we put forward an argument (originally made by Richard Feynman) that ties the rotational symmetry of space to the conservation of angular momentum in quantum mechanics. The argument is, in fact, quite general, enabling one to deduce other conservation laws from various inherent symmetries of quantum mechanical systems.