Abstract The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.
Abstract The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.
The separation of the dynamics of the quantum many-body systems into the overall rotation and internal motions leads to their quantization in rotating frames. Whereas the rotating body frame is unique for a rigid body, there is a great degree of freedom for choosing a body frame for a general rotating and vibrating N-particle system. This, in turn, implies the need to study many-body systems for a class of body frames.The concept of vibrational coordinates and of rotational coordinates is particularly clear for the case of a triatomic molecule: three particles form a single plane, in which the molecule vibrates, and at the same time the molecular plane rotates in the space-fixed frame of reference. The overall molecular rotation is made explicit by means of an orthogonal transformation, which brings the space-fixed into the body-fixed axes. The direction perpendicular to the molecular plane uniquely determines one of the three axes of the frame. Rotation about this direction will change the position of the axes in the molecular plane, in fact the axis system. It is already intuitively clear that this axis transformation will not affect the vibrational portion of the kinetic energy operator. The other parts of the kinetic energy, the rotational part and the rotation-vibration coupling part, will then need to adjust themselves to the new definition of the body frame. In this fashion, we may address the problem of the interaction of rotations and vibrations, which is one of the most important problems in molecular studies.The variation of the rovibrational kinetic energy operator following a change of orientation of the axis system is investigated in the present work from the physical and mathematical points of view. Various types of axis embedding are analyzed in a systematic fashion for triatomic molecules of arbitrary structure, including the bisector, principal axis, and Eckart frames. We also consider the bisector axis embedding for tetratomic molecules and the local axis embedding for pentatomic molecules. Different axis conventions lead sometimes to a complex entanglement of rotational and vibrational motions, which may pose serious obstacles to numerically exact approaches.
Classical dynamics of rigid bodies The difficulties in formulating the theory of spin ½ systems outlined at the end of the last chapter may be avoided if we start from a classical theory of rotation in which the translational and orientational coordinates enter on a symmetrical footing, and pass to the quantum theory via the standard canonical quantization procedure. The result is not simply the Pauli theory written in a different matrix representation but a more general physical theory which explicitly incorporates in the wave equation the extended character of the bodies considered, through moment-of-inertia-dependent terms. As expected, the spin ½ character of the wave is manifested through the dependence of the body motion on the total quantum state via quantum potential and torque terms. Spin is not usually approached from this angle in basic quantum mechanics texts so it is worthwhile to build up the theory from scratch (for a more rigorous treatment see Normand (1980) and for further details: Bopp & Haag (1950), Rosen (1951, 1984), van Winter (1954), Dankel (1970), Holland (1988a, b)). We begin by reviewing some basic results in the classical canonical theory of rigid body motion. Our treatment is confined to a rigid homogeneous sphere with moment of inertia I about any axis passing through its centre, where the centre of mass is located. The sphere carries a rigid charge and a magnetic moment. By Chasle's theorem an arbitrary displacement of a rigid body is composed of a translation together with a rotation about a line fixed in the body.
This chapter focuses on the concept of rotational motion. It explains that energy and angular momentum for a particle rotating in two- or three-dimensions are quantized and satisfied cyclic boundary conditions, which imply the quantization of angular momentum's magnitude and z-component. The electronic structure of atoms showcases rotational motion because electrons orbit around nuclei and spin on their axis. The chapter details the solutions for the Schrödinger equation and the vector model of angular momentum. It also discusses the concept of space quantization revolving around the quantum mechanical result of a rotating body that may not take up an arbitrary orientation concerning some specified axis.
In the 1920s, the old quantum theory was gradually being superseded by the new quantum theory. The cornerstone of the old theory was Bohr’s model of the hydrogen atom, which predicted that an electron cannot orbit the proton in the hydrogen atom in any arbitrary fashion. Orbits are “quantized,” meaning that only certain sizes, shapes, and magnetic properties are allowed. The principal quantum number n determined the allowed radii of the orbits, the orbital quantum number l determined the allowed shapes, and the magnetic quantum number m determined the magnetic behavior. Additionally, there is a fourth quantum number s which denotes the fact that the electron has an additional angular momentum, loosely associated with self rotation about its own axis, and that is quantized in units of (1/2)~. The old quantum theory was useful to infer the existence of discrete energy levels in atoms, calculate energy spacings between these levels, and therefore allowed one to interpret atomic spectra. The new quantum theory appeared to be more revolutionary and more powerful. It was triggered by Heisenberg’s discovery of matrix mechanics and Schro¨dinger’s discovery of wave mechanics. These two formalisms would not only predict the quantization of energy and provide a prescription to determine the energy difference between the levels (and thus explain the multiplicity of atomic spectra), but also allow one to calculate easily probabilities of transitions between different quantized energy states. At first, matrix mechanics and wave mechanics looked entirely different in their mathematical appearance and physical meaning. However, Schro¨dinger and Eckart [1] independently showed that the two theories are mathematically equivalent. Toward the end of 1926, Dirac unified the two theories using the concept of state vector and thus established the transformation theory of quantum mechanics. This ultimately had a profound implication for the quantum mechanical (mathematical) recipe to treat spin, as we will show in this chapter. The transformation theory is the mathematical recipe to handle modern quantum mechanics. In Heisenberg’s matrix mechanics, a physical quantity is expressed by a matrix, whereas in Schro¨dinger’s wave mechanics, a physical quantity is expressed by a linear operator. In the unified transformation theory, physical quantities are represented by abstract linear operators called Dirac’s q-numbers, which are linear operators in an infinite-dimensional linear space. Depending upon which types of orthogonal coordinate systems are used in this linear space, either matrix mechanics or wave mechanics emerges.
Chapter 4 deals with Quantum Mechanical (QM) phenomena. The first section describes the first attempts at quantization of energy, momentum, angular momentum, etc. We start with Plank's solution to the black-body problem, introducing the concept of photons: discrete packets of electromagnetic energy. We show how Plank's description of black-body radiation is perfectly fit by the cosmic microwave radiation, providing the strongest evidence for the Big-Bang at the origin of the Universe. We then describe Bohr's solution to the paradoxical stability of atoms, introducing quantization of angular momentum (or action) and correctly predicting the energy levels of the hydrogen atom. We show how Einstein used Plank's idea to explain the photo-electric effect (an effect exploited in CCD cameras and solar cells) and how he used Bohr's understanding of energy levels in atoms to provide an intuitive explanation of Plank's law. Einstein insight opened the way to the field of spectroscopy and the development of the laser. In the second section, following the lead of Dirac, we show that QM is equivalent to Linear Algebra in the space of complex functions (which describe the state of the system). We show that Schrödinger's equation is a natural consequence of that description, once the relation between energy and frequency established by Plank (for photons) and de Broglie (for particles) is taken into account. In the context of the chiral ammonia molecule we discuss the queerness of QM which describes physical systems as superposition of states (such as left and right chirality state of a molecule, dead and alive cat). We show that this queerness is nonetheless supported by experiments and is behind much of our technology: masers and lasers, atomic clocks, semiconductors (diodes, transistors, etc.). In the following section we introduce the representation of operators in QM (the equivalent of matrices in linear algebra): momentum, position, energy, etc. which correspond to measurements on the system. We show that the non-commutability of some operators (common with matrix operations) leads to Heisenberg uncertainty principle: the position and momentum of a particle cannot be determined with absolute precision. We discuss the solution of Schrödinger's equation for free particles demonstrating their equivalence with EM phenomena: particles undergo refraction and diffraction, exemplifying the particle/wave duality inherent to QM systems. We discuss QM tunneling and its use in the Scanning Tunneling Microscope. In the following section we study Dirac's equation: the successful attempt by Dirac to unify QM with Einstein's relativity, with the resulting prediction of anti-particles and spin. We then solve for the energy levels of electrons in the hydrogen atom (from Schrödinger's equation) and show that the existence of spin profoundly affects the behavior of electrons in an atom: they cannot share the same energy level, a rule known as Pauli's exclusion principle. Pauli's principle is central to the understanding of the chemical properties of atoms and fully explains Mendeleev's periodic table of the elements. In the next section we discuss along with two exactly soluble systems (diatomic rotational and vibrational levels), various approximations used to study systems (e.g. molecules) for which there are no analytical solutions. In particular, we introduce Pauling's variational approach, Hückel molecular orbital theory and time independent perturbation theory. We apply those approaches to determine the energy levels in alkenes, in multi-electron atoms and in systems under constant electric or magnetic field (such as in Magnetic Resonance Imaging). In the last section we discuss time dependent perturbation theory. We use it to establish the rules allowing for transitions between energy levels and revisit Einstein description of absorption/emission phenomena.