Advanced Quantum Mechanics

 

Lectures

 

 

Lecture # 1

 

Introduction. Recommended literature. Objectives of the course.

 

Perturbation theory (Part 1). Time-independent perturbations. First and second order corrections

to the energies and wave functions. Conditions for the applicability of perturbation theory.

 

Lecture # 2

 

Perturbation theory (Part 2). 1D harmonic oscillator (refreshment). Matrix mechanics.

How the perturbation theory works: 1D anharmonic oscillator- cubic and quadric anharmonisms.

 

Lecture # 3

 

Perturbation theory (Part 3). Perturbation theory for a degenerate state. Secular equation.

Doubly degenerate level.

 

Lecture # 4

 

Perturbation theory (Part 4). Time-dependent perturbations. Periodic in time perturbations.

Sudden  perturbations. Transitions in the continuous spectrum.  Fermi golden rule. 

Energy-time uncertainty.

 

Lecture # 5

 

Spin (Part 1). Theory of angular momentum (refreshment). Spin algebra. SU(2) group.

Spin operator. Pauli matrices. Properties of Pauli matrices.

 

Lecture # 6

 

Spin (Part 2). Spinors. Transformation of spinors. Covariant and contravariant forms. Metric tensor.

Eigenvalues and eigenfunctions of an arbitrary s=1/2 component. The rotation operator. Representation

of the rotation operator for arbitrary spin. Rotation operator for s=1/2 and s=1.

Time reversal symmetry and Kramers’ theorem.

 

Lecture # 7

 

Symmetries (Part 1). Identical (indistinguishable) particles. Symmetry of the wave function. Slater determinants.

Pauli principle. Fermions and Bosons. u-v transformations for fermions and bosons.

 

Lecture # 8

 

Symmetries (Part 2). Connection between spin and symmetry of the wave functions.  Symmetry ban.

Two particles with s=0 in a box. Two particles with spin s= ½ in a box. Singlet and Triplet. Exchange interaction.

Two particles with spin s=1 in a box. Many-particle states of non-interacting fermions and bosons in a box.

 

Lecture # 9

 

Second quantization. Many-particle wave functions for bosons and fermions.

Commutation relations: commutators and anti-commutators. Psi-operators.

Single-particle operator. Two-body and three-body interactions.

Secondary quantized form of Hamiltonians for fermions and bosons.

 

Lecture # 10

 

Electron in magnetic field. Bohr magneton. Schroedinger equation for the electron

in uniform magnetic field. Landau levels. Gauge invariance.

 

Lecture # 11

 

Complex atoms with large number of electrons. Electrons in a box. Fermi energy. Thomas-Fermi equation.

 

Lecture # 12

 

WKB method (Part 1). One –dimensional Schroedinger equation. Semi-classical wave function.

Conditions for applicability of WKB theory. Boundary conditions for semi-classical wave function.

 

Lecture # 13

 

WKB method (Part 2). Bohr-Sommerfeld’s quantization.  Semi-classical penetration through a potential barrier.

 

Lecture # 14

 

WKB method (Part 3). Semi-classical motion in central-symmetric potential. Semi-classical theory for large orbital moments.

 

Lecture # 15

 

WKB method (Part 4). Semi-classical theory of  hydrogenic atom. Semi-classical approach to molecules (ammonia).

Symmetric double well potential. Rabi frequencies.

 

Lecture # 16

 

Atom in external fields (Part 1). Quantum-mechanical two-body problem (refreshment). Separation of variables.

Central symmetric potentials. Hydrogenic atom.  Atom in electric field. Multipole moments: dipol moment, quadrupol moment.

Stark effect. Linear Stark effect in hydrogenic atom.

 

Lecture # 17

 

Atom in external fields (Part 2). Spin-orbit interaction. Russell-Saunders coupling. Lande’s interval rule. Lande’s factors.

Zeeman effect.

 

Lecture # 18

 

Atom in external fields (Part 3). Theory of Zeeman effect in meso-atom and hydrogenic atom.

Quadratic Zeeman effect.  Paschen-Back effect.

 

Lecture # 19

 

Symmetries in quantum mechanics. Kepler problem in classical mechanics. Planet orbits.

Hidden symmetries of Coulomb problem in classical and quantum mechanics. Runge-Lenz vector and operator.

Fock quantization in hydrogenic atom.

 

Lecture # 20

 

Schroedinger equation in momentum space. Dirac delta-function. Properties of delta-function.

Delta-functional well. Delta-functional barrier.  Transmission through the barrier given by

the sum of two delta-functions.

 

Lecture # 21

 

Periodic potentials in 1D. Dirac comb model. Floquet’s theorem. Bloch’s theorem.

Brillouin zone. Band structure of crystals. Kronig-Penney model.

 

Lecture # 22

 

Elastic collisions (Part 1). Prologue. Classical description of elastic collisions.

Quantum problem of scattering. Scattering amplitude. Differential cross-section. Total cross-section.

Scattering phase shifts. Partial wave description.

 

Lecture # 23

 

Elastic collisions (Part 2). Properties of scattering amplitude. S-matrix. The unitarity condition.

Optical theorem. Reciprocity theorem. Green’s function of free particle.

 

Lecture # 24

 

Elastic collisions (Part 3). Scattering amplitude in momentum representation. Lippmann-Schwinger equation. 

Born approximation. Rutherford’s formula.

 

Lecture # 25

 

Elastic collisions (Part 4). Scattering amplitude for central-symmetric potential in Born approximation.

Properties of the scattering amplitude as a function of energy. Poles of scattering amplitude. Shallow level

in 2D delta-functional well.

 

Lecture # 26

 

Elastic collisions (Part 5). Total cross-section for slow and fast particles. Transport cross-section.

Forward scattering. Backward scattering. Scattering on potentials 1/r^n (general arguments).

 

Lecture # 27

 

Collision theory (Part 6). Epilogue. Elastic collisions in 2D. Unitarity and optical theorem.

Collision of indistinguishable particles. Resonance on a quasi-discrete level.

Inelastic scattering. Breit-Wigner formula.

 

Lecture # 28

 

Modern quantum mechanics (Part 1 - Supersymmetry). Supersymmetric Witten quantum mechanics.

 

Lecture # 29

 

Modern quantum mechanics (Part 2 - Topology). Adiabatic change and geometric phases.

Berry’s phase. Aharonov-Bohm effect. Dirac monopole. Fractional statistics.

 

Lecture # 30

 

Modern quantum mechanics (Part 3 – Nonlocality). The Einstein-Podolsky-Rosen paradox.

Bell inequalities. Quantum computations.

 

 

Tutorials

Tutorial-1: Potential energy as a perturbation.

Tutorial-2:   Time independent perturbation theory - Two identical
spin-1/2 particles interacting through a delta-function potential.

Tutorial-3: Time dependent perturbation - A charged particle subject
to a time-dependent electric field.

Tutorial-4: Addition of angular momentum - 2 and 3 spin problems.

Tutorial-5: Addition of angular momentum (continued).

WKB method-connection formula.

Tutorial-6: WKB (continued).

Tutorial-7: Alpha decay. Equivalence of Coulomb problem and simple
harmonic oscillator in two dimensions.

Tutorial-8: Equivalence of Coulomb problem and simple harmonic oscillator in 2D (continued). 

Mo­tion of an electron in a constant magnetic field - calculation of degen­eracy.

Tutorial-9:             Ammonia molecule.

Tutorial-10:              Ammonia molecule (continued).

Tutorial-11:                Homework problems-1

Tutorial-12:              Homework problems-1

Tutorial-13:              Homework problems-1

Tutorial-14:              Homework problems-2

Tutorial-15:              Homework problems-2

 

 

Recommended literature:

 

L.D.Landau and E.M.Lifshitz. Quantum Mechanics - Non-relativistic theory. Pergamon, 1977

 

J.J. Sakurai. Modern Quantum Mechanics. Addison-Wesley, 1994.

 

C.Cohen-Tannudji, D.Diu and F.Laloe. Quantum Mechanics. Willey NY, 1977