Lectures on Quantum Mechanics Leon A. Takhtajan Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651, USA E-mail address:
[email protected]
Contents Chapter 1. Classical Mechanics 1. Lagrangian Mechanics 1.1. Generalized coordinates 1.1.1. Notations 1.2. The principle of the least action 1.2.1. Examples 1.2.2. Exercises 1.3. Symmetries and Noether theorem 1.4. Integration of equations of motion 1.4.1. One-dimensional motion 1.4.2. The motion in a central field 1.4.3. The Kepler problem 1.5. Legendre transformation 1.6. The action functional in the phase space 2. Hamiltonian Mechanics 2.1. Canonical Hamilton’s equations 2.2. The action as a function of coordinates 2.3. Classical observables and Poisson bracket 2.4. Symplectic and Poisson manifolds 2.4.1. Symplectic manifolds 2.4.2. Poisson manifolds 2.5. Hamilton and Liouville representations 2.5.1. Hamilton’s description of dynamics 2.5.2. Liouville’s description of dynamics
7 7 7 7 8 11 13 13 17 17 17 17 17 21 21 21 23 25 26 27 31 35 38 38
Chapter 2. Foundations of Quantum Mechanics 1. Observables and States 1.1. Physical principles 1.1.1. Notations 1.2. Basic axioms 1.3. Heisenberg’s uncertainty principle 1.4. Dynamics 2. Heisenberg’s commutation relations 2.1. Free particle 2.1.1. Coordinate (Schr¨ odinger) representation. 2.1.2. Momentum representation.
39 39 39 40 41 45 47 49 49 50 53
3
4
CONTENTS
2.1.3. Motion of free quantum particle. 2.1.4. Several degrees of freedom. 2.2. Quantization of Newtonian particle 3. Harmonic oscillator and holomorphic representation 3.1. Harmonic oscillator 3.2. Holomorphic representation 3.2.1. Wick symbols of operators 4. Stone-von Neumann theorem 4.1. Weyl commutation relations 4.2. Stone-von Neumann theorem 4.3. Invariant formulation of Stone-von Neumann theorem 5. Quantization 5.1. Weyl quantization 5.2. The -product 5.3. Deformation quantization Chapter 3. Schr¨ odinger Equation 1. General properties 1.1. Self-adjointness 1.2. Discreteness of the spectrum 2. One-dimensional Schr¨ odinger equation 3. Angular momentum and SO(3) 4. Two-body problem 5. Hydrogen atom and S0(4) 6. Semi-classical asymptotics Chapter 4. Path Integral Formulation of Quantum Mechanics 1. Feynman path integral for the evolution operator 1.1. Free particle 1.2. Path integral in the phase space 1.3. Path integral in the configuration space 1.4. Harmonic oscillator 1.5. Several degrees of freedom 1.6. Path integral in the holomorphic representation 2. Gaussian path integrals and determinants 2.1. Free particle and harmonic osicllator revisited 2.1.1. Gaussian integral for the free particle 2.1.2. Gaussian integral for harmonic oscillator 2.2. Determinants 2.2.1. Dirichlet boundary conditions 2.2.2. Periodic boundary conditions 2.2.3. First order differential operators 2.3. Semi-classical asymptotics 2.3.1. Using Feynman path integral 2.3.2. Rigorous derivation
53 55 57 58 58 62 64 64 64 66 71 74 74 79 82 89 89 89 90 90 90 90 90 90 91 91 92 93 95 96 98 99 99 99 99 102 104 105 109 112 114 114 116
CONTENTS
5
Chapter 5. Integration in Functional Spaces 1. Gaussian measures 1.1. Finite-dimensional case 1.2. Infinite-dimensional case 2. Wiener measure and Wiener integral 2.1. Definition of the Wiener measure 2.2. Conditional Wiener measure and Feynman-Kac formula 2.3. Relation between Wiener and Feynman integrals 2.4. Gaussian Wiener integrals 2.4.1. Dirichlet boundary conditions 2.4.2. Periodic boundary conditions 2.4.3. Traces
123 123 123 124 126 126 129 131 131 132 133 134
Chapter 6. Spin and Identical Particles 1. Spin 2. Charged spin particle in the magnetic field 2.1. Pauli Hamiltonian 2.2. Electron in a magnetic field 3.