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Journal of Physics Conference Series

Quantisation of angular momenta in atoms

Dzhabrail M Bamatov, Ibragim Musaevich Bamatov, Magomed M. Arsanov

Published Nov 1, 2020• 1 citation
Publisher

Abstract

Abstract The aim of the work was to see if the direction of angular momenta in atoms is quantised. In order to do it, a beam of atoms is split, and deflected because of interactions between atomic dipole moments and an externally applied magnetic field. In which case, the angular momenta in atoms is definitively quantised. Firstly, the detector was checked and then the operation of the detector was characterised; secondly, the beam was detected and characterised with residual magnetisation minimised; thirdly, the beam was detected and characterised with magnetic field present. The sample used to determine the magnetic moment was Potassium. The Bohr’s magneton constant value was found to be (9.22±0.7)*10−24 J/T. The answer is lower than the nominal value but lies within the expected error range. This proved the accuracy of the work done and proved the quantisation of the angular momentum, since it proved the presence of magnetic moment.

DOI
10.1088/1742-6596/1679/2/022046

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Abstract The aim of the work was to see if the direction of angular momenta in atoms is quantised. In order to do it, a beam of atoms is split, and deflected because of interactions between atomic dipole moments and an externally applied magnetic field. In which case, the angular momenta in atoms is definitively quantised. Firstly, the detector was checked and then the operation of the detector was characterised; secondly, the beam was detected and characterised with residual magnetisation minimised; thirdly, the beam was detected and characterised with magnetic field present. The sample used to determine the magnetic moment was Potassium. The Bohr’s magneton constant value was found to be (9.22±0.7)*10−24 J/T. The answer is lower than the nominal value but lies within the expected error range. This proved the accuracy of the work done and proved the quantisation of the angular momentum, since it proved the presence of magnetic moment.
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.
Anisotropy of angular momenta: ideas and methods The fundamental idea forming the basis of early quantum theory was the quantization of the angular momentum of the atom, which, at a later stage, assumed the form of the Bohr–Sommerfeld quantization rule [360], namely ∲ pdq = n ℏ where p and q are the generalized momentum and coordinate. This concept has retained its physical meaning in modern-day quantum mechanics [371, 372]. In the course of the further development of quantum mechanical concepts it became increasingly clear that in the theory of atoms and molecules the angular momentum is a ‘dominant’ notion, the fundamental significance of which is in no small degree connected with its dimension of action, which coincides with that of Planck's constant ℏ. In the interaction between particles and a directed beam of photons the angular momentum of the system is conserved, thus forming the basis for the optical method of producing spatial anisotropy of the distribution of the angular momenta of the ensemble of particles, i.e. its optical polarization . In terms of quantum concepts, spatial anisotropy means that in an ensemble of particles a non-equilibrium population of magnetic sublevels m j of angular momenta is created. We make a distinction between the alignment of momenta, for instance in the absorption of linear polarized light, when there is no difference between the population of ± m j -sublevels, and orientation , when such a difference exists. Anisotropic distribution of angular momenta in an excited state, as produced in absorption, manifests itself directly in the polarization of fluorescence. It is rather likely that understanding of the anisotropic distribution of the angular momenta of atoms was enhanced by Hanle's [184] discovery in 1924.
The choice of an axis of quantization automatically selects those species from a bulk which could become stationary in a field-free environment with related spatial orientations of their coupling angular momenta. But if this axis fails to coincide with the direction of a uniform external magnetic field, a field-imposed transverse torque forces these atoms' total angular momenta into steady nutation that cannot be abolished by coupling-related internal torques. Making any stationary state impossible for these atoms, it also causes the contributions to the energy expectation value to oscillate. This is the result of a rigorous solution to the time-dependent Schrödinger equation describing the coupling dynamics of two spins 1/2 in an external field. Demanding to solve a four-dimensional system of coupled linear differential equations of first order, the bulkiness of even this simplest case already brings the calculations to the verge of becoming infeasible analytically.
Abstract In this chapter we discuss the intrinsic angular momentum called spin. Starting from the magnetic dipole moment related to orbital angular momentum, we focus on the hydrogen atom and consider the effect of an external magnetic field, resulting in Larmor precession and Zeeman splitting. We then turn to the discovery of spin in the Stern-Gerlach experiment. After providing a mathematical description of spin-1/2 systems in terms of two-component spinors and Pauli matrices, we illustrate projective measurements for the example of spin. The relationship between the Lie groups SU(2) and SO(3) is discussed in the context of the transformation of spinors under rotations. We analyse how the presence of spin alters the description of the hydrogen atom. Discussing the addition of angular momenta and the Pauli exclusion principle leads us to the structure of electronic orbitals. Finally, we discuss the occupation of electronic energy levels in heavier nuclei
This chapter analyses the orbital angular momentum to three dimensions and introduces the spin angular momentum of the electron. The description of a localized particle in orbit requires a superposition of eigenfunctions analogous to the packet state that describes motion in one dimension. If the distance between molecules is fixed, all the energy changes are rotational kinetic energy changes, and the quantization of total angular momentum gives rise to discrete rotational energy levels. The analysis of orbital angular momentum is a powerful tool for describing molecular rotation. The Stern-Gerlach experiment and many other lines of evidence show that the measured magnetic moment associated with an electron spin is almost exactly one Bohr magneton. When orbital angular momentum and spin angular momentum both exist in the same atom, the magnetic moments that result from orbital and spin angular momenta interact to cause a splitting in the corresponding energy level.