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Transactions of the Kansas Academy of Science

The Whittaker Quantum Mechanism

F. E. Kester

Published Mar 28, 1935• 0 citations

Abstract

peated periodic process, I was led to a discussion of the latter integral for the special case of an isolated magnet pole, the Dirac1 quantum m, carried in a closed linking with a closed of an elementary charge of electricity e. The analogy is the more interesting in that the two functions here involved are useful in expressing, by minimum values of each, the conditions for of mechanical systems in conservative fields of force-the potential energy function at minimum value for static equilibrium, the Maupertuis function, evaluated for the natural path of a system between any two configurations, at minimum value for kinetic equilibrium with the field for this path. Whittaker2 discussed the interchanges of energy occurring in the special case of an electron flying along the axis of a magnetic wheel the spokes of which are magnets with their south poles (say) at the center, the north poles at the rim. The expedient of placing one set of poles at the axis to avoid the magnetic action of the flying electron upon them is, of course, not feasible. Lorentz3 corrected this feature of the Whittaker mechanism by assuming an isolated pole, giving it the form of a plane circular filament, closed on itself and free to move only in rotation about its own axis, and he generalized the problem to cover any form of the electron's which may link with the filament.

DOI: 10.2307/3624842

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In Professor Whittaker's highly interesting and suggestive application of my model the postulated structure is a system of magnets rigidly connected like the spokes of a wheel, having poles of the same name at the circumference, the corresponding poles of contrary name being all at the centre. He shows that when a bombarding electron, moving in the direction of the axis, approaches such a wheel it sets the wheel turning, the rotation of the poles constituting what he calls a “magnetic current.” If the initial velocity of the bombarding electron falls short of a certain limiting value, the encounter is perfectly elastic, and the electron is reflected back without loss of energy; the point at which its direction of motion becomes reversed may be on either side of the plane of the wheel, depending on the electron's initial velocity of approach. If the initial velocity exceeds the stated limit, the electron passes clear through the system and escapes, but with a definite reduction of velocity, corresponding to a quantum of energy which it has given up to the wheel. When the encounter is elastic, the angular velocity which the wheel acquires during the approach of the electron is lost during the return of the electron. When the encounter is not elastic, and the electron has passed clear through, the wheel is left with an angular velocity ω; and the energy which that represents is the source of the consequent radiation.

Proceedings of the Royal Society of Edinburgh•Jan 1, 1923•1 citation

On Whittaker's Quantum Mechanism

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A brief description is given of the mechanism of Prof. E. T. Whittaker which compels all exchanges between the kinetic energy of electrons and radiant energy to conform to the quantum condition. A modified form of the mechanism is suggested, consisting of two magnetons, or two ring electrons, placed near together with their planes parallel. It is pointed out that the quantum mechanism gives the law of force postulated by Langmuir in his model of a static atom, which leads to results identical with those obtained from the circular orbits of Bohr's theory.

Proceedings of the Physical Society of London•Dec 1, 1921•1 citation

XIV.—Note on Professor Whittaker's Quantum Mechanism in the Atom

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It seems desirable to check Professor Whittaker's model numerically; algebraical expressions when evaluated sometimes have the wrong order of magnitude. Let us suppose that initially the magnet system is at rest, and that the electron starts from the origin with velocity u.

Proceedings of the Royal Society of Edinburgh•Jan 1, 1923•0 citations

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peated periodic process, I was led to a discussion of the latter integral for the special case of an isolated magnet pole, the Dirac1 quantum m, carried in a closed linking with a closed of an elementary charge of electricity e. The analogy is the more interesting in that the two functions here involved are useful in expressing, by minimum values of each, the conditions for of mechanical systems in conservative fields of force-the potential energy function at minimum value for static equilibrium, the Maupertuis function, evaluated for the natural path of a system between any two configurations, at minimum value for kinetic equilibrium with the field for this path. Whittaker2 discussed the interchanges of energy occurring in the special case of an electron flying along the axis of a magnetic wheel the spokes of which are magnets with their south poles (say) at the center, the north poles at the rim. The expedient of placing one set of poles at the axis to avoid the magnetic action of the flying electron upon them is, of course, not feasible. Lorentz3 corrected this feature of the Whittaker mechanism by assuming an isolated pole, giving it the form of a plane circular filament, closed on itself and free to move only in rotation about its own axis, and he generalized the problem to cover any form of the electron's which may link with the filament.

In Professor Whittaker's highly interesting and suggestive application of my model the postulated structure is a system of magnets rigidly connected like the spokes of a wheel, having poles of the same name at the circumference, the corresponding poles of contrary name being all at the centre. He shows that when a bombarding electron, moving in the direction of the axis, approaches such a wheel it sets the wheel turning, the rotation of the poles constituting what he calls a “magnetic current.” If the initial velocity of the bombarding electron falls short of a certain limiting value, the encounter is perfectly elastic, and the electron is reflected back without loss of energy; the point at which its direction of motion becomes reversed may be on either side of the plane of the wheel, depending on the electron's initial velocity of approach. If the initial velocity exceeds the stated limit, the electron passes clear through the system and escapes, but with a definite reduction of velocity, corresponding to a quantum of energy which it has given up to the wheel. When the encounter is elastic, the angular velocity which the wheel acquires during the approach of the electron is lost during the return of the electron. When the encounter is not elastic, and the electron has passed clear through, the wheel is left with an angular velocity ω; and the energy which that represents is the source of the consequent radiation.

A brief description is given of the mechanism of Prof. E. T. Whittaker which compels all exchanges between the kinetic energy of electrons and radiant energy to conform to the quantum condition. A modified form of the mechanism is suggested, consisting of two magnetons, or two ring electrons, placed near together with their planes parallel. It is pointed out that the quantum mechanism gives the law of force postulated by Langmuir in his model of a static atom, which leads to results identical with those obtained from the circular orbits of Bohr's theory.

In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic ﬁelds. One of these ﬁelds is static (time-independent) and directed along a speciﬁc axis, while the other is timedependent, has a diﬀerent amplitude, and rotates in the plane perpendicular to the ﬁrst magnetic ﬁeld. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin ﬂip”, i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic ﬁeld along the z-axis and a circularly polarized time-varying (ac) magnetic ﬁeld of diﬀerent amplitude, with a speciﬁc frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli Equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a diﬀerent approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.

In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip,” i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.