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Canonical Quantisation

K. Kong Wan

Published Jul 9, 2019• 0 citations
Publisher

Abstract

Many important quantum observables, e.g., angular momentum, have their origin in corresponding classical observables. Postulate 26.1(OV) does not tell us how to find the operators to describe these observables. We can tackle this problem by formulating a process of quantisation to establish the operators for the corresponding quantum observables. Before we can formulate a quantisation process we must choose an appropriate description of classical observables. This is done in the following section.

DOI
10.1201/b22504-27

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Many important quantum observables, e.g., angular momentum, have their origin in corresponding classical observables. Postulate 26.1(OV) does not tell us how to find the operators to describe these observables. We can tackle this problem by formulating a process of quantisation to establish the operators for the corresponding quantum observables. Before we can formulate a quantisation process we must choose an appropriate description of classical observables. This is done in the following section.
Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.
Vectors and operators in a Hilbert space are independent quantities and there is no obvious link between them. The mathematical description of the states and observables of a quantum system given by Postulates 25.1(PS) and 26.1(OV) in terms of vectors and operators does not tell us how a given state would determine the value of an observable. Physically the relationship between states and observables is about how the measurable values of an observable is related to a state. This relationship should lead to the quantum properties QMP5.3(1) to QMP5.3(3). First we must know how a state can give rise to a probability distribution of measurable values of an observable. This would require a prescription to relate a unit vector to the spectrum of selfadjoint operators. The discussion in Chapter 22 tells us how to obtain such a prescription. Because of the intricacy of the relationship both physically and mathematically we shall set out the relationship for discrete and continuous observables separately.
We suggest a somewhat non-standard view on a set of curious, paradoxical from the standpoint of simple classical physics and everyday experience phenomena. There are the quantisation (discrete set of values) of the observables (e.g., energy, momentum, angular momentum); forbidden simultaneous measurements of the observables in the most cases (e.g., of a coordinate and momentum, of angular momentum projections on difference axis); counter-intuitive relations on the simultaneously measurable quantities (e.g., the famous expression for the square momentum l(l+1)l(l+1)l(l+1) with the maximal projection lll). These and other paradoxes are traditionally related to "purely quantum" phenomenon, i.e., having no analogue in the "classical world" ones. However, there are deep analogies between classical and "quantum" worlds, as soon as the quantum technique is applied to the classical phenomenon. We follow these analogies with the examples of relatively simple and well known models of classical physics, such as a simplified model of light transition through the media, a system of electric charges close to each other and far from the observer; the specific of motion in the Coulomb/Newtonian field. This text can be considered as a mini-course addressed to higher school and undergraduate students who are interested in basics of quantum mechanics, but are not yet ready for systematic study of standard courses. The text may be also useful to those who supervise such students.
Preface. 1. Origins of Quantum Physics. 1.1 Historical Note. 1.2 Particle Aspect of Radiation. 1.3 Wave Aspect of Particles. 1.4 Particles versus Waves. 1.5 Indeterministic Nature of the Microphysical World. 1.6 Atomic Transitions and Spectroscopy. 1.7 Quantization Rules. 1.8 Wave Packets. 1.9 Concluding Remarks. 1.10 Solved Problems. Exercises. 2. Mathematical Tools of Quantum Mechanics. 2.1 Introduction. 2.2 The Hilbert Space and Wave Functions. 2.3 Dirac Notation. 2.4 Operators. 2.5 Representation in Discrete Bases. 2.6 Representation in Continuous Bases. 2.7 Matrix and Wave Mechanics. 2.8 Concluding Remarks. 2.9 Solved Problems. Exercises. 3. Postulates of Quantum Mechanics. 3.1 Introduction. 3.2 The Basic Postulates of Quantum Mechanics. 3.3 The State of a System. 3.4 Observables and Operators. 3.5 Measurement in Quantum Mechanics. 3.6 Time Evolution of the System's State. 3.7 Symmetries and Conservation Laws. 3.8 Connecting Quantum to Classical Mechanics. 3.9 Solved Problems. Exercises. 4. One-Dimensional Problems. 4.1 Introduction. 4.2 Properties of One-Dimensional Motion. 4.3 The Free Particle: Continuous States. 4.4 The Potential Step. 4.5 The Potential Barrier and Well. 4.6 The Infinite Square Well Potential. 4.7 The Finite Square Well Potential. 4.8 The Harmonic Oscillator. 4.9 Numerical Solution of the Schrodinger Equation. 4.10 Solved Problems. Exercises. 5. Angular Momentum. 5.1 Introduction. 5.2 Orbital Angular Momentum. 5.3 General Formalism of Angular Momentum. 5.4 Matrix Representation of Angular Momentum. 5.5 Geometrical Representation of Angular Momentum. 5.6 Spin Angular Momentum. 5.7 Eigen functions of Orbital Angular Momentum. 5.8 Solved Problems. Exercises. 6. Three-Dimensional Problems. 6.1 Introduction. 6.2 3D Problems in Cartesian Coordinates. 6.3 3D Problems in Spherical Coordinates. 6.4 Concluding Remarks. 6.5 Solved Problems. Exercises. 7. Rotations and Addition of Angular Momenta. 7.1 Rotations in Classical Physics. 7.2 Rotations in Quantum Mechanics. 7.3 Addition of Angular Momenta. 7.4 Scalar, Vector and Tensor Operators. 7.5 Solved Problems. Exercises. 8. Identical Particles. 8.1 Many-Particle Systems. 8.2 Systems of Identical Particles. 8.3 The Pauli Exclusion Principle. 8.4 The Exclusion Principle and the Periodic Table. 8.5 Solved Problems. Exercises. 9. Approximation Methods for Stationary States. 9.1 Introduction. 9.2 Time-Independent Perturbation Theory. 9.3 The Variational Method. 9.4 The Wentzel Kramers Brillou in Method. 9.5 Concluding Remarks. 9.6 Solved Problems. Exercises. 10. Time-Dependent Perturbation Theory. 10.1 Introduction. 10.2 The Pictures of Quantum Mechanics. 10.3 Time-Dependent Perturbation Theory. 10.4 Adiabatic and Sudden Approximations. 10.5 Interaction of Atoms with Radiation. 10.6 Solved Problems. Exercises. 11. Scattering Theory. 11.1 Scattering and Cross Section. 11.2 Scattering Amplitude of Spinless Particles. 11.3 The Born Approximation. 11.4 Partial Wave Analysis. 11.5 Scattering of Identical Particles. 11.6 Solved Problems. Exercises. A. The Delta Function. A.1 One-Dimensional Delta Function. A.2 Three-Dimensional Delta Function. B. Angular Momentum in Spherical Coordinates. B.1 Derivation of Some General. B.2 Gradient and Laplacianin Spherical Coordinates. B.3 Angular Momentum in Spherical Coordinates. C. Computer Code for Solving the Schrodinger Equation. Index.