Approximate Methods in Quantum Mechanics

Atomic and molecular physicists have developed extensive and detailed approximate methods for dealing with the relativistic versions of the Hamiltonians appearing in their fields. Nuclear physicists were originally more concerned with non-relativistic problems as the energies they were dealing with were normally small compared with the rest energy of the nucleon. This situation has changed with the appearance of the quark models of nucleons and thus the objective of this paper is to use the standard variational procedures of nuclear physics for problems in relativistic quantum mechanics. The 4 × 4α and β matrices in the Dirac equation are replaced by 2 × 2 matrices, one associated with ordinary spin and the other, which we call sign spin, is mathematically identical to the isospin of nuclear physics. The states on which our Hamiltonians will act will be the usual harmonic oscillator ones with ordinary and sign spin and the frequency ω of the oscillator will be our only variational parameter. The example discussed as an illustration will still be the Coulomb problem as the exact energies of the relativistic bound states are available for comparison. A gap of the order of 2mc2 is observed between states of positive and negative energy, that permits the former to be compared with the exact results.

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Relative intensities of stark components in helium

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1930.0144 ◽

1930 ◽

Vol 129 (809) ◽

pp. 70-91 ◽

Cited By ~ 3

Keyword(s):

Perturbation Theory ◽

Quantum Mechanics ◽

Quantum Number ◽

Experimental Work ◽

Principal Quantum Number ◽

Approximate Methods

Theoretical values for the displacements and relative intensities of Stark components have been calculated by Foster for selected groups of helium lines. These were based on approximate methods in the perturbation theory of quantum mechanics, and on the assumption that the source contains equal numbers of atoms excited to states with a common principal quantum number. Experimental work on this subject has been carried out by Foster, Takamine and Werner, and more extensively by Dewey. The observed displacements agree remarkably well with the theory; but the intensities have shown some variations which cannot be disregarded. This appears surprising at first sight, since the theory reveals close connections between displacements and intensities. It seems, however, well established by Dewey that the P - P (and p - p) combination lines are decidedly too strong for the theory. There appear to be other less striking discrepancies.

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The symmetric spherical oscillator, and the rotational motion of homopolar molecules in crystals

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1931.0024 ◽

1931 ◽

Vol 130 (815) ◽

pp. 551-557 ◽

Cited By ~ 52

Keyword(s):

Quantum Mechanics ◽

Fourier Series ◽

Potential Function ◽

Rotational Motion ◽

Present Author ◽

Perturbation Methods ◽

Quantum States ◽

Approximate Methods ◽

First Approximation ◽

Spherical Oscillator

In a recent paper, L. Pauling has discussed the motion of molecules in crystals. By the use of approximate methods, criteria are deduced in that paper for determining when the motion of molecules in crystals is “rotational,” and when it is “oscillational” about positions of equilibrium. While it served Pauling’s purpose, the investigation did not go deep enough to satisfy the needs of the present author, who desired information about the statistical weights of the lowest quantum states of molecules in crystals, more certain and definite than that provided by Pauling’s analysis. The present more rigorous investigation was accordingly undertaken. The results go somewhat beyond the author’s needs, but they are perhaps interesting in themselves. The investigation involves a problem in quantum mechanics which is capable of exact treatment, without the use of perturbation methods. We imagine a homopolar molecule of type X 2 free to rotate in a field of force of axial symmetry. Symmetry considerations show that if a certain orientation is one of equilibrium, then the orientation obtained by reversing the molecule end for end will also be one of equilibrium; and we see also, if the orientation of the molecule is specified by the co-ordinates θ, ϕ (the axis of θ being the axis of equilibrium), that the potential function V(θ) of the homopolar molecule must be symmetric about the equatorial circle θ = π/2. We suppose that our molecule is tree from axial spin. If V(θ) can be expanded in a Fourier series, then it can be expanded in a Fourier series made up of cosines of even multiples of θ. A good first approximation will be obtained by taking only the first two terms of this series; higher terms can be taken account of later, if desired, by a perturbation method. We therefore consider, with Pauling, the potential function (1) V = V 0 (1 - cos 2θ).

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The application of quantum mechanics to chemical kinetics

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1933.0031 ◽

1933 ◽

Vol 139 (838) ◽

pp. 466-474 ◽

Cited By ~ 70

Keyword(s):

Quantum Mechanics ◽

Potential Energy ◽

Classical Mechanics ◽

Energy Difference ◽

Chemical Processes ◽

Quantum Mechanical ◽

Simple Type ◽

Heat Of Reaction ◽

Initial State ◽

Approximate Methods

Much work has recently been done on the application of quantum mechanics to chemical reactions. In the majority of cases, however, the actual reaction processes have been considered as taking place according to the laws of classical mechanics, quantum-mechanical theory being only employed in calculating the interatomic forces. It has, however, been suggested by various authors that the actual transition processes involved must be treated as non-classical. Some of these authors have claimed that this method of treatment is essential for the true explanation of chemical processes, just as in the case of radioactive disintegration, where it is well established that classical considerations are unable to explain the phenomena observed. It appears, however, to be the general consensus of opinion that for chemical processes the results obtained by a strict quantum-mechanical treatment would differ negligibly from the results of classical mechanics. This opinion appears to be based only on approximate methods of treatment, and no actual figures have been published. The present paper is a contribution to a more exact knowledge of the problem. According to modern views on reaction mechanism, the reacting system passes through a maximum of potential energy in passing adiabatically from the initial to the final state. The energy difference between the initial state and the maximum is the heat of activation for the reaction (E). As a simple type of chemical reaction we may take the system shown in fig. 1. A particle of mass m passes from a to b through a region of varying potential energy V ( x ). Between a and b the potential energy reaches a maximum value E. The energy difference between a and b is Q, the heat of reaction.

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