Download UNIT - STUDY GUIDES - SPH 409 QUANTUM MECHANICS II

COURSE/UNIT STUDY GUIDES
SPH 409: QUANTUM MECHANICS II
Lecturers:
1. Dr S. Mureramanzi (E – mail: [email protected]), Physics Department, Physical
Sciences Building, 2nd Floor, room 203, University of Nairobi.
Introduction to the Course unit
We start the course with a brief review on basic ideas of quantum theory: matter waves, de
Broglie relations, Heisenberg uncertainty principle, and the Schrodinger equation. The following
Chapter deals with approximation methods. This is an important Chapter since the Time
Independent Schrodinger Equation (TISE) cannot be solved exactly for almost all real systems.
In fact, the TISE can be solved exactly for a few problems, such as the one – dimensional box,
the three – dimensional box, the simple harmonic oscillator (SHO) and the hydrogen
atom/hydrogen – like atoms. For other problems, approximation methods have to be used.
Two main approximation methods used are: the perturbation theory and the linear variation
method. The latter is used jointly with other approximations to solve the Schrodinger equation
for simple molecules. As an application of the Time Independent Perturbation Theory, we study
the Weak – Field – Zeeman Effect and the Strong – Field - Zeeman Effect. Other Chapters
include among others, identical particles, the Scattering theory, vibrational and rotational
states of diatomic molecules, the Dirac theory of the free electron and an Introduction to
second quantization.
Aims: The Course aims at introducing some concepts of advanced quantum
mechanics to the students.
Specific objectives: - At the end of the course students should be able to:
1. Solve problems dealing with approximation methods;
2. Make use of the stationary perturbation theory in the study of the Zeeman Effect, the
ground state and the first excited states of the helium atom.
3. Justify the need to introduce the concept of spin into quantum theory and be able to
show how to incorporate its effects into the non – relativistic theory and the Dirac
theory of the electron.
4. Give the main features of the scattering theory and solve problems related to that
theory.
5. State the Born – Oppenheimer approximation and use it to simplify the Schrodinger
equation for H+2 ion and H2 molecule.
6. Solve the Schrodinger equation for H+2 and H2 molecules using the LCAO - MO method
and/or the Heitler – London method.
7. Provide a theoretical explanation for the rotational spectra, vibrational spectra and
rotational – vibration bands of diatomic molecules.
8. Understand the origin and solution of the Dirac equation;
9. Understand the meaning of second quantization for identical particles.
SPH 103: Course outline
1. Recall of Basic Ideas of Quantum Theory: Matter waves, de Broglie relations,
Heisenberg uncertainty principle, the Schrodinger equation;
2. Approximation Methods: Time – Independent Perturbation Theory (non degenerate
and degenerate perturbation theory); application to the Zeeman effect; Variation
Method; Time – Dependent Perturbation Theory (absorption and emission of
electromagnetic radiation; transition probabilities and selection rules).
3. Spin and Pauli matrices: Definitions and properties of the spin operator, Single Spin ½
System and Pauli Matrices, Lowering and raising operators, the spin as a dynamical
variable, rotation in the spin space.
4. Identical Particles: Two – Particle System and the principle of
indistinguishability ; time – independent Schrodinger equation; the
Slater determinant; Pauli Exclusion Principle; Bosons and Fermions; symmetric and
antisymmetric wave functions symmetric and antisymmetric wave functions;
Introduction to the quantum theory of many – electron atoms: Hamiltonian and wave
function, approximation methods.
5. Scattering Theory: Scattering Cross Section, Stationary Scattering States and the
scattering amplitude, Born Approximation, Partial Wave Expansions and Scattering of
identical particles.
6. Quantum Theory of simple molecules: the Hydrogen Molecule Ion, H+2 and the
Hydrogen Molecule: Hamiltonian in the Born – Oppenheimer approximation; the
variation method and the LCAO - MO method; and the Heitler – London method.
7. Vibrational and rotational states: the Rigid Rotor Approximation, Pure Rotational
Spectroscopy, the Harmonic Oscillator and Vibrational Spectroscopy.
8. Dirac Theory of Free electron: relativistic de Broglie relations, the Dirac equation,
electron spin, magnetic moment and spin – orbit coupling.
9. Introduction to Second Quantization: First and Second quantization, Annihilation and
Creation Operators, eigen values of the Simple Harmonic Oscillator Hamiltonian.
Assessment: Assessment of the course shall comprise two CATs given a total of 30 marks and
a Final Exam marked out of 70 marks
References
1. Brandsen, J. H. A. and Joachain, C. J. (2002). Physics of atoms and Molecules (2nd
Ed.) Pearson education in South Asia (India). Published by Dorling and Kindersley.
2. Barchewitz, P. (1970). Spectroscopie Atomique et Moleculaire - Tome I: Mecanique
de l’Atome et de la Molecule – Spectroscopie Atomique. Masson & Cie – Editeurs,
Paris, France.
3. Barchewitz, P. (1970). Spectroscopie Atomique et Moleculaire - Tome II: Mecanique
de l’Atome et de la Molecule – Spectroscopie Moleculaire. Masson & Cie – Editeurs,
Paris, France.
4. Griffiths, D. (2004). Introduction to Quantum mechanics, Prentice - Hall
5. Hanna, M. W. (1969). Quantum Mechanics in Chemistry (Second Edition), Edited by
Walter Kauzmann, Princeton University.
6. Liboff, R. (2003). Introductory Quantum mechanics. Addison - Wesley
7. Peebles, P. J. E.(2003). Quantum Mechanics. Prentice Hall of India Private Limited,
New Delhi, India.
8. Peleg, Y., Pnini, R. and Zaarur, E. (1998). Schaum’s Outlines of Theory and Problems
of Quantum Mechanics. Mc Graw – Hill, N.Y. (USA).
9. Subrahmanyam, N., Lal, B. and Seshan, J. (1984). Atomic and Nuclear Physics. S.
Chand & Company Ltd, New Delhi, India