**Course Objectives:** Revise and update the mathematical concepts of vectors and tensors to chemical systems by solving eigenvalue and eigenvector problems in matrices and first and second order differential equations that are used for solving the time independent Schrodinger equation. Solve elementary model problems in quantum mechanics, particle in a potential-free box, particle on a ring, harmonic oscillator and particle in a Coulomb potential exactly and demonstrate the solutions for hydrogen atom and molecular rotations and vibrations.

**Learning Outcomes:** At the end of the course, the learners should be able to:

** Use** mathematical techniques in linear algebra for eigenvalues and eigenvectors and first and second order differential equations not only in quantum chemistry but in other areas of physical and theoretical chemistry that will be offered during the whole programme.

**Solve** all the model problems in quantum mechanics for which exact analytical methods and solutions are available and will apply them to analyze the basis behind the postulatory method of quantum mechanics and which forms the foundations for advanced study of the subject.

**Relate** concepts that were originally introduced purely as modern atomic physics to molecular systems through harmonic oscillator, spin and rigid rotator.

**Course Contents:**

Mathematics

• Review of vectors and vector spaces, matrices and determinants, eigenvalues and eigenvectors, similarity transformations, ordinary differential equations- first and second order.

• Solution of differential equations by power series method: solutions of Hermite equation in detail. Orthogonality properties and recurrence relations. Introduction to the solutions of Legendre and Laguerre differential equations, Spherical Harmonics.

**Quantum Mechanics**

• Solution of the Schrodinger equation for exactly solvable problems for bound states such as particle-in-a- box, particle-in-a-ring, harmonic oscillator and rigid rotor.

• Postulates of quantum mechanics, wave functions and probabilities, operators, matrix representations, commutation relationships. Hermitian operators, Commutators and results of measurements in Quantum Mechanics. Eigenfunctions and eigenvalues of operators and superposition principle. States as probability distributions and expectation values. The expansion of arbitrary states in terms of complete set.

• Angular momentum, commutation relationships, basis functions and representation of angular momentum operators, Coupling (addition) of angular momenta

• Solution of the Schrodinger equation for the hydrogen atom, radial and angular probability distributions, atomic orbitals and electron spin, Pauli’s exclusion principle and Aufbau principle.

• The time dependent Schrödinger equation. Co-ordinate and momentum space representation of operators and eigenstates; Role of Fourier transforms and simple examples; Unitary evolution and reversibility. Schrodinger and Heisenberg representations.

**Text Books:**

1. E. Kreyszig, Advanced Engineering Mathematics, 5th edition, Wiley Eastern, 1989.

2. G. Arfken and Hans J. Weber, Mathematical methods for physicists, Prism Indian Edition, 1995.

3. D. A. McQuarrie, Quantum Chemistry, University Science Books, 1983.

4. P. W. Atkins, Molecular Quantum Mechanics, 2nd edition, Oxford University Press, 1983.

5. I. N. Levine, Quantum Chemistry, 3rd edition, Allyn and Bacon, 1983.

6. D. J. Griffiths, Introduction to Quantum Mechanics, Pearson Education, 2005.

7. H. Kuhn, H.-D. Försterling, and D.H. Waldeck, Principles of Physical Chemistry, 2nd edition, Wiley, 2009.

8. J. P. Lowe, Quantum Chemistry, K. A. Peterson, 3rd edition, Academic Press, 2006.