Transcription
Quantum Mechanics:Fundamental Principles andApplicationsJohn F. DawsonDepartment of Physics, University of New Hampshire, Durham, NH 03824October 14, 2009, 9:08am ESTc 2007 John F. Dawson, all rights reserved.
c 2009 John F. Dawson, all rights reserved.ii
ContentsPrefaceIxvFundamental Principles11 Linear algebra1.1 Linear vector spaces . . . . . . . . . . . . . .1.2 Linear independence . . . . . . . . . . . . . .1.3 Inner product . . . . . . . . . . . . . . . . . .1.3.1 The dual space . . . . . . . . . . . . .1.3.2 Non-orthogonal basis sets . . . . . . .1.4 Operators . . . . . . . . . . . . . . . . . . . .1.4.1 Eigenvalues and eigenvectors: . . . . .1.4.2 Non-orthogonal basis vectors . . . . .1.4.3 Projection operators: . . . . . . . . . .1.4.4 Spectral representations: . . . . . . . .1.4.5 Basis transformations . . . . . . . . .1.4.6 Commuting operators . . . . . . . . .1.4.7 Maximal sets of commuting operators1.5 Infinite dimensional spaces . . . . . . . . . . .1.5.1 Translation of the coordinate system .1.6 Measurement . . . . . . . . . . . . . . . . . .1.6.1 The uncertainty relation . . . . . . . .1.7 Time in non-relativistic quantum mechanics .33456791113151517182222232425272 Canonical quantization2.1 Classical mechanics review . . . . .2.1.1 Symmetries of the action .2.1.2 Galilean transformations . .2.2 Canonical quantization postulates2.2.1 The Heisenberg picture . .2.2.2 The Schrödinger picture . .2.3 Canonical transformations . . . . .2.4 Schwinger’s transformation theory3 Path integrals3.1 Space-time paths . . . . . . .3.2 Some path integrals . . . . .3.3 Matrix elements of coordinate3.4 Generating functionals . . . .292930313232353742. . . . . . . . . . .operators. . . . . .4343454646.iii
CONTENTS3.53.63.73.83.9Closed time path integrals .Initial value conditions . . .Connected Green functionsClassical expansion . . . . .Some useful integrals . . . .CONTENTS.47515152544 In and Out states554.1 The interaction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 The time development operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3 Forced oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585 Density matrix formalism5.1 Classical theory . . . . . . . . . . . . . . . . . . .5.1.1 Classical time development operator . . .5.1.2 Classical averages . . . . . . . . . . . . . .5.1.3 Classical correlation and Green functions5.1.4 Classical generating functional . . . . . .5.2 Quantum theory . . . . . . . . . . . . . . . . . .636363666768686 Thermal densities6.1 The canonical ensemble . .6.2 Ensemble averages . . . . .6.3 Imaginary time formalism .6.4 Thermal Green functions .6.5 Path integral representation6.6 Thermovariable methods . .71717272757576.7 Green functions8 Identical particles8.1 Coordinate representation . . . . .8.2 Occupation number representation8.3 Particle fields . . . . . . . . . . . .8.3.1 Hamiltonian . . . . . . . . .77.79798081819 Symmetries9.1 Galilean transformations . . . . . . . . . . . . . .9.1.1 The Galilean group . . . . . . . . . . . . .9.1.2 Group structure . . . . . . . . . . . . . .9.2 Galilean transformations . . . . . . . . . . . . . .9.2.1 Phase factors for the Galilean group. . . .9.2.2 Unitary transformations of the generators9.2.3 Commutation relations of the generators .9.2.4 Center of mass operator . . . . . . . . . .9.2.5 Casimir invariants . . . . . . . . . . . . .9.2.6 Extension of the Galilean group . . . . . .9.2.7 Finite dimensional representations . . . .9.2.8 The massless case . . . . . . . . . . . . .9.3 Time translations . . . . . . . . . . . . . . . . . .9.4 Space translations and boosts . . . . . . . . . . .9.5 Rotations . . . . . . . . . . . . . . . . . . . . . .9.5.1 The rotation operator . . . . . . . . . . .838485868788919495969898100101102105105c 2009 John F. Dawson, all rights reserved.iv.
511511611611912112112112111 Supersymmetry11.1 Grassmann variables . . . . . . . . . . . . . . . . . . . . . . .11.2 Superspace and the 1D-N supersymmetry group . . . . . . .11.3 1D-N supersymmetry transformations in quantum mechanics11.4 Supersymmetric generators . . . . . . . . . . . . . . . . . . .11.5 R-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.6 Extension of the supersymmetry group . . . . . . . . . . . . .11.7 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . .1231231241251281311321339.69.79.89.99.5.2 Rotations of the basis sets . .General Galilean transformations . .Improper transformations . . . . . .9.7.1 Parity . . . . . . . . . . . . .9.7.2 Time reversal . . . . . . . . .9.7.3 Charge conjugation . . . . . .Scale and conformal transformations9.8.1 Scale transformations . . . .9.8.2 Conformal transformations .The Schrödinger group . . . . . . . .10 Wave equations10.1 Scalars . . . . . . . . . . . .10.2 Spinors . . . . . . . . . . .10.2.1 Spinor particles . . .10.2.2 Spinor antiparticles .10.3 Vectors . . . . . . . . . . .10.4 Massless wave equations . .10.4.1 Massless scalers . . .10.4.2 Massless vectors . .II.Applications12 Finite quantum systems12.1 Diatomic molecules . .12.2 Periodic chains . . . .12.3 Linear chains . . . . .12.4 Impurities . . . . . . .12.4.1 Bound state . .12.4.2 Scattering . . .135.13 One and two dimensional wave mechanics13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .13.2 Schrödinger’s equation in one dimension . . . . . . .13.2.1 Transmission of a barrier . . . . . . . . . . .13.2.2 Wave packet propagation . . . . . . . . . . .13.2.3 Time delays for reflection by a potential step13.3 Schrödinger’s equation in two dimensions . . . . . .c 2009 John F. Dawson, all rights 4156
CONTENTS14 The14.114.214.3WKB approximationIntroduction . . . . . . .Theory . . . . . . . . . .Connection formulas . .14.3.1 Positive slope . .14.3.2 Negative slope .14.4 Examples . . . . . . . .14.4.1 Bound states . .14.4.2 Tunneling . . . .CONTENTS.15915915916016016216416416515 Spin systems15.1 Magnetic moments . . . . . . . . .15.2 Pauli matrices . . . . . . . . . . . .15.2.1 The eigenvalue problem . .15.3 Spin precession in a magnetic field15.4 Driven spin system . . . . . . . . .15.5 Spin decay: T1 and T2 . . . . . . .15.6 The Ising model . . . . . . . . . .15.7 Heisenberg models . . . . . . . . .169169169170171173175175176harmonic oscillatorThe Lagrangian . . . . . . . . . . . .Energy eigenvalue and eigenvectors .Other forms of the Lagrangian . . .Coherent states . . . . . . . . . . . .16.4.1 Completeness relations . . . .16.4.2 Generating function . . . . .Squeezed states . . . . . . . . . . . .The forced oscillator . . . . . . . . .The three-dimensional oscillator . . .The Fermi oscillator . . . . . . . . .16.8.1 Action for a Fermi oscillator .17717717818018218518518618919419519617 Electrons and phonons17.1 Electron-phonon action . . . . . . . . . . .17.2 Equations of motion . . . . . . . . . . . . .17.2.1 Numerical classical results . . . . . .17.3 Electron modes . . . . . . . . . . . . . . . .17.4 Vibrational modes . . . . . . . . . . . . . .17.5 Electron-phonon interaction . . . . . . . . .17.6 The action revisited . . . . . . . . . . . . .17.7 Quantization . . . . . . . . . . . . . . . . .17.8 Block wave functions . . . . . . . . . . . . .17.8.1 A one-dimensional periodic potential17.8.2 A lattice of delta-functions . . . . .17.8.3 Numerical methods . . . . . . . . . .19919920220320320821221321421421421521616 The16.116.216.316.416.516.616.716.8.18 Schrödinger perturbation theory22318.1 Time-independent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22318.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225c 2009 John F. Dawson, all rights reserved.vi
CONTENTS19 Variational methods19.1 Introduction . . . . . . . . . . . . . .19.2 Time dependent variations . . . . . .19.3 The initial value problem . . . . . .19.4 The eigenvalue problem . . . . . . .19.5 Examples . . . . . . . . . . . . . . .19.5.1 The harmonic oscillator . . .19.5.2 The anharmonic oscillator . .19.5.3 Time-dependent Hartree-FockCONTENTS.22722722723023023123123523620 Exactly solvable potential problems23720.1 Supersymmetric quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23720.2 The hierarchy of Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23720.3 Shape invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23721 Angular momentum21.1 Eigenvectors of angular momentum . . . . . . . . . . . . . . . .21.1.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.1.2 Orbital angular momentum . . . . . . . . . . . . . . . .21.1.3 Kinetic energy operator . . . . . . . . . . . . . . . . . .21.1.4 Parity and Time reversal . . . . . . . . . . . . . . . . .21.2 Rotation of coordinate frames . . . . . . . . . . . . . . . . . . .21.2.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . .21.2.2 Axis and angle parameterization . . . . . . . . . . . . .21.2.3 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . .21.2.4 Cayley-Klein parameters . . . . . . . . . . . . . . . . . .21.3 Rotations in quantum mechanics . . . . . . . . . . . . . . . . .21.3.1 Rotations using Euler angles . . . . . . . . . . . . . . .21.3.2 Properties of D-functions . . . . . . . . . . . . . . . . .21.3.3 Rotation of orbital angular momentum . . . . . . . . . .21.3.4 Sequential rotations . . . . . . . . . . . . . . . . . . . .21.4 Addition of angular momentum . . . . . . . . . . . . . . . . . .21.4.1 Coupling of two angular momenta . . . . . . . . . . . .21.4.2 Coupling of three and four angular momenta . . . . . .21.4.3 Rotation of coupled vectors . . . . . . . . . . . . . . . .21.5 Tensor operators . . . . . . . . . . . . . . . . . . . . . . . . . .21.5.1 Tensor operators and the Wigner-Eckart theorem . . . .21.5.2 Reduced matrix elements . . . . . . . . . . . . . . . . .21.5.3 Angular momentum matrix elements of tensor operators21.6 Selected problems . . . . . . . . . . . . . . . . . . . . . . . . . .21.6.1 Spin-orbit force in hydrogen . . . . . . . . . . . . . . . .21.6.2 Transition rates for photon emission in Hydrogen . . . .21.6.3 Hyperfine splitting in Hydrogen . . . . . . . . . . . . . .21.6.4 The Zeeman effect in hydrogen . . . . . . . . . . . . . .21.6.5 The Stark effect in hydrogen . . . . . . . . . . . . . . .21.6.6 Matrix elements of two-body nucleon-nucleon potentials21.6.7 Density matrix for the Deuteron . . . . . . . . . . . . .c 2009 John F. Dawson, all rights 291292294
CONTENTSCONTENTS22 Electrodynamics22.1 The Lagrangian . . . . . . . . . . . . . . . . . . . . .22.1.1 Probability conservation . . . . . . . . . . . .22.1.2 Gauge transformations . . . . . . . . . . . . .22.2 Constant electric field . . . . . . . . . . . . . . . . .22.3 Hydrogen atom . . . . . . . . . . . . . . . . . . . . .22.3.1 Eigenvalues and eigenvectors . . . . . . . . .22.3.2 Matrix elements of the Runge-Lenz vector . .22.3.3 Symmetry group . . . . . . . . . . . . . . . .22.3.4 Operator factorization . . . . . . . . . . . . .22.3.5 Operators for the principle quantum number22.3.6 SO(4, 2) algebra . . . . . . . . . . . . . . . .22.3.7 The fine structure of hydrogen . . . . . . . .22.3.8 The hyperfine structure of hydrogen . . . . .22.3.9 The Zeeman effect . . . . . . . . . . . . . . .22.3.10 The Stark effect . . . . . . . . . . . . . . . .22.4 Atomic radiation . . . . . . . . . . . . . . . . . . . .22.4.1 Atomic transitions . . . . . . . . . . . . . . .22.4.2 The photoelectric effect . . . . . . . . . . . .22.4.3 Resonance fluorescence . . . . . . . . . . . . .22.5 Flux quantization . . . . . . . . . . . . . . . . . . . .22.5.1 Quantized flux . . . . . . . . . . . . . . . . .22.5.2 The Aharonov-Bohm effect . . . . . . . . . .22.6 Magnetic monopoles . . . . . . . . . . . . . . . . . 2732732732732732732832923 Scattering theory23.1 Propagator theory . . . . . . . . .23.1.1 Free particle Green function23.2 S-matrix theory . . . . . . . . . . .23.3 Scattering from a fixed potential .23.4 Two particle scattering . . . . . . .23.4.1 Resonance and time delays23.5 Proton-Neutron scattering . . . . .333333333334334334339339III. . . .in one. . . . . . . . . . . . . . . . . . . . .dimension. . . . . . . . . . . . . . . . . . . . . . . . . .Appendices343A Table of physical constants345B Operator Relations347B.1 Commutator identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347B.2 Operator functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348B.3 Operator theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348C Binomial coefficients351D Fourier transforms353D.1 Finite Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353D.2 Finite sine and cosine transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354c 2009 John F. Dawson, all rights reserved.viii
CONTENTSCONTENTSE Classical mechanicsE.1 Lagrangian and Hamiltonian dynamicsE.2 Differential geometry . . . . . . . . . .E.3 The calculus of forms . . . . . . . . . .E.3.1 Derivatives of forms . . . . . .E.3.2 Integration of forms . . . . . .E.4 Non-relativistic space-time . . . . . . .E.4.1 Symplectic manifolds . . . . . .E.4.2 Integral invariants . . . . . . .E.4.3 Gauge connections . . . . . . .355355360366366370370370375376F Statistical mechanics reviewF.1 Thermal ensembles . . . . . . . . . .F.2 Grand canonical ensemble . . . . . .F.2.1 The canonical ensemble . . .F.3 Some examples . . . . . . . . . . . .F.4 MSR formalism . . . . . . . . . . . .F.4.1 Classical statistical averages .F.4.2 Generating functions . . . . .F.4.3 Schwinger-Dyson equations .F.5 Anharmonic oscillator . . . . . . . .F.5.1 The partition function for the. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .anharmonic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .oscillator.381381381382383383386388391393394G Boson calculusG.1 Boson calculus . . . . . . . . . . . .G.2 Connection to quantum field theoryG.3 Hyperbolic vectors . . . . . . . . . .G.4 Coherent states . . . . . . . . . . . .G.5 Rotation matrices . . . . . . . . . . .G.6 Addition of angular momentum . . .G.7 Generating function . . . . . . . . .G.8 Bose tensor operators . . . . . . . .397397399400401403406414417.Indexc 2009 John F. Dawson, all rights reserved.419ix
CONTENTSc 2009 John F. Dawson, all rights reserved.CONTENTSx
List of Figures3.1The closed time path contour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489.1The Galilean transformation for Eq. (9.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8411.1 R-symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13112.1 We plot the potential energy for an electron in two atomic sites. We also sketch wave functionsψ1,2 (x) for an electron in the isolated atomic sites and the symmetric and antisymmetriccombinations ψ (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.2 A molecule containing four atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12.3 A molecule containing six atomic sites, arranged in a circular chain. . . . . . . . . . . . . . .12.4 Construction for finding the six eigenvalues for an electron on the six periodic sites of Fig. 12.3,for values of k 0, . . . , 5. Note the degeneracies for values of k 1, 5 and k 2, 4. . . . . .12.5 A molecule containing six atomic sites, arranged in a linear chain. . . . . . . . . . . . . . . .12.6 Six eigenvalues for the six linear sites of Fig. 12.5, for values of k 1, . . . , 6. . . . . . . . . . .12.7 A long chain with an impurity atom at site 0. . . . . . . . . . . . . . . . . . . . . . . . . . . .12.8 Transmission and reflection coefficients for electron scattering from an impurity for the casewhen ( 0 1 )/Γ0 0.2667 and Γ1 /Γ0 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . .12.9 Two long connected chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13813914014114214314314514613.1 A junction with three legs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15214.1 Two turning point situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16014.2 Potential well. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16414.3 Potential barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16615.1 Spin precession in the rotating coordinate system. . . . . . . . . . . . . . . . . . . . . . . . . 17516.1 Retarded and advanced contours for the Green function of Eq. (16.117). . . . . . . . . . . . . 19016.2 Feynman (F ) (red) and anti-Feynman (F ) (green) contours. . . . . . . . . . . . . . . . . . . 19217.1 Plot of V (x) for the first 10 sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.2 Plot of V (x) and V 0 (x) for site n with wave functions for sites n, n 1, showing the overlapintegrals between nearest neighbor sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.3 Plot of xn (t) for the first 10 sites as a function of time for 0 ω0 Γ 1, and K 0.5, for100 sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.4 Plot of yn (t) for the first 10 sites as a function of time for 0 ω0 Γ 1, and K 0.5, for100 sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.5 Plot of φn (t) for the first 10 sites as a function of time for 0 ω0 Γ 1, and K 0.5, for100 sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xi200202204204205
LIST OF FIGURESLIST OF FIGURES17.6 Plot of dφn (t)/dt for the first 10 sites as a function of time for 0 ω0 Γ 1, and K 0.5,for 100 sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.7 Plot of the electron and phonon energy spectra k and ωk on the periodic chain, as a functionof k. Energies and k values have been normalized to unity. Note that near k 0, the electronspectra is quadratic whereas the phonon spectrum is linear. . . . . . . . . . . . . . . . . . . .17.8 Construction for finding the oscillation frequencies for the six periodic sites of Fig. 12.3, forvalues of k 0, 1, 2, 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17.9 Plot of the right-hand side of Eqs. (17.107) and (17.108), for β 0.5. . . . . . . . . . . . . . .17.10Plot of the right-hand side of Eqs. (17.107) and (17.108), for β 1.0. . . . . . . . . . . . . . .17.11Plot of the right-hand side of Eqs. (17.107) and (17.108), for β 1.5. . . . . . . . . . . . . . .17.12Plot of the energy, in units of 2m/ 2 , as a function of Ka/π for β 1.5. . . . . . . . . . . . .20520821121721721821921.1 Euler angles for the rotations Σ Σ0 Σ00 Σ000 . The final axis is labeled (X, Y, Z). . . . . 25321.2 Mapping of points on a unit sphere to points on the equatorial plane, for x3 0 (red lines)and x3 0 (blue lines). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25422.122.222.322.4The fine structure of hydrogen (not to scale). Levels with the same value of j are degenerate. 319The hyperfine structure of the n 1 and n 2 levels of hydrogen (not to scale). . . . . . . . 322Zeeman splitting of the n 1 hyperfine levels of hydrogen as a function of µB B (not to scale). 324Stark splitting of the n 2 fine structure levels of hydrogen as a function of β e a E0 . isthe fine structure splitting energy. (not to scale). . . . . . . . . . . . . . . . . . . . . . . . . . 325c 2009 John F. Dawson, all rights reserved.xii
List of Tables1.1Relation between Nature and Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . .319.1 The first five energies of the anharmonic oscillator computed using the time-dependent variational method compared to the exact results [?] and a SUSY-based variational method [?]. . 23621.1 Table of Clebsch-Gordan coefficients, spherical harmonics, and d-functions. . . . . . . . . . . 26921.2 Algebric formulas for some 3j-symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27121.3 Algebric formulas for some 6j-symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27222.1 The first few radial wave functions for hydrogen. . . . . . . . . . . . . . . . . . . . . . . . . . 310A.1 Table of physical constants from the particle data group. . . . . . . . . . . . . . . . . . . . . . 345A.2 Table of physical constants from the particle data group. . . . . . . . . . . . . . . . . . . . . . 346xiii
LIST OF TABLESc 2009 John F. Dawson, all rights reserved.LIST OF TABLESxiv
PrefaceIn this book, I have tried to bridge the gap between material learned in an undergraduate course in quantummechanics and an advanced relativistic field theory course. The book is a compilation of notes for a first yeargraduate course in non-relativistic quantum mechanics which I taught at the University of New Hampshirefor a number of years. These notes assume an undergraduate knowledge of wave equation based quantummechanics, on the level of Griffiths[1] or Liboff [2], and undergraduate mathematical skills on the level ofBoas [3]. This book places emphasis on learning new theoretical methods applied to old non-relativisticideas, with a eye to what will be required in relativistic field theory and particle physics courses. The resultprovides an introduction to quantum mechanics which is, I believe, unique.The book is divided into two sections: Fundamental Principles and Applications. The fundamentalprinciples section starts out in the usual way by reviewing linear algebra, vector spaces, and notation in thefirst chapter, and then in the second chapter, we discuss canonical quantization of classical systems. In thenext two chapters, Path integrals and in- and out-states are discussed. Next is a chapter on the densitymatrix and Green functions in quantum mechanics where we also discuss thermal density matrices andGreen functions. This is followed by a chapter on identical particles and second quantized non-relativisticfields. Next the Galilean group is discussed in detail and wave equations for massless and massive nonrelativistic particles explored. Finally, the last chapter of the fundamental principles section is devoted tosupersymmetry in non-relativistic quantum mechanics.In the application section, I start by discussing finite quantum systems: the motion of electrons onmolecules and on linear and circular chains. This is followed by chapters one and two dimensional wavemechanics and the WKB app
Quantum Mechanics: Fundamental Principles and Applications John F. Dawson Department of Physics, University of New
