Book contents
- Frontmatter
- Contents
- Preface
- Physical constants
- 1 Basic formalism
- 2 Fundamental commutator and time evolution of state vectors and operators
- 3 Dynamical equations
- 4 Free particles
- 5 Particles with spin ½
- 6 Gauge invariance, angular momentum, and spin
- 7 Stern–Gerlach experiments
- 8 Some exactly solvable bound-state problems
- 9 Harmonic oscillator
- 10 Coherent states
- 11 Two-dimensional isotropic harmonic oscillator
- 12 Landau levels and quantum Hall effect
- 13 Two-level problems
- 14 Spin ½ systems in the presence of magnetic fields
- 15 Oscillation and regeneration in neutrinos and neutral K-mesons as two-level systems
- 16 Time-independent perturbation for bound states
- 17 Time-dependent perturbation
- 18 Interaction of charged particles and radiation in perturbation theory
- 19 Scattering in one dimension
- 20 Scattering in three dimensions – a formal theory
- 21 Partial wave amplitudes and phase shifts
- 22 Analytic structure of the S-matrix
- 23 Poles of the Green's function and composite systems
- 24 Approximation methods for bound states and scattering
- 25 Lagrangian method and Feynman path integrals
- 26 Rotations and angular momentum
- 27 Symmetry in quantum mechanics and symmetry groups
- 28 Addition of angular momenta
- 29 Irreducible tensors and Wigner–Eckart theorem
- 30 Entangled states
- 31 Special theory of relativity: Klein–Gordon and Maxwell's equations
- 32 Klein–Gordon and Maxwell's equations
- 33 The Dirac equation
- 34 Dirac equation in the presence of spherically symmetric potentials
- 35 Dirac equation in a relativistically invariant form
- 36 Interaction of a Dirac particle with an electromagnetic field
- 37 Multiparticle systems and second quantization
- 38 Interactions of electrons and phonons in condensed matter
- 39 Superconductivity
- 40 Bose–Einstein condensation and superfluidity
- 41 Lagrangian formulation of classical fields
- 42 Spontaneous symmetry breaking
- 43 Basic quantum electrodynamics and Feynman diagrams
- 44 Radiative corrections
- 45 Anomalous magnetic moment and Lamb shift
- Bibliography
- Index
1 - Basic formalism
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Physical constants
- 1 Basic formalism
- 2 Fundamental commutator and time evolution of state vectors and operators
- 3 Dynamical equations
- 4 Free particles
- 5 Particles with spin ½
- 6 Gauge invariance, angular momentum, and spin
- 7 Stern–Gerlach experiments
- 8 Some exactly solvable bound-state problems
- 9 Harmonic oscillator
- 10 Coherent states
- 11 Two-dimensional isotropic harmonic oscillator
- 12 Landau levels and quantum Hall effect
- 13 Two-level problems
- 14 Spin ½ systems in the presence of magnetic fields
- 15 Oscillation and regeneration in neutrinos and neutral K-mesons as two-level systems
- 16 Time-independent perturbation for bound states
- 17 Time-dependent perturbation
- 18 Interaction of charged particles and radiation in perturbation theory
- 19 Scattering in one dimension
- 20 Scattering in three dimensions – a formal theory
- 21 Partial wave amplitudes and phase shifts
- 22 Analytic structure of the S-matrix
- 23 Poles of the Green's function and composite systems
- 24 Approximation methods for bound states and scattering
- 25 Lagrangian method and Feynman path integrals
- 26 Rotations and angular momentum
- 27 Symmetry in quantum mechanics and symmetry groups
- 28 Addition of angular momenta
- 29 Irreducible tensors and Wigner–Eckart theorem
- 30 Entangled states
- 31 Special theory of relativity: Klein–Gordon and Maxwell's equations
- 32 Klein–Gordon and Maxwell's equations
- 33 The Dirac equation
- 34 Dirac equation in the presence of spherically symmetric potentials
- 35 Dirac equation in a relativistically invariant form
- 36 Interaction of a Dirac particle with an electromagnetic field
- 37 Multiparticle systems and second quantization
- 38 Interactions of electrons and phonons in condensed matter
- 39 Superconductivity
- 40 Bose–Einstein condensation and superfluidity
- 41 Lagrangian formulation of classical fields
- 42 Spontaneous symmetry breaking
- 43 Basic quantum electrodynamics and Feynman diagrams
- 44 Radiative corrections
- 45 Anomalous magnetic moment and Lamb shift
- Bibliography
- Index
Summary
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed.
State vectors
It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following.
The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac.
A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉*, are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
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- Quantum Mechanics with Basic Field Theory , pp. 1 - 23Publisher: Cambridge University PressPrint publication year: 2009