PHY622: Mathematical methods for physicsts III

        [Cr:4, Lc:3, Tt:1, Lb:0]

Course Outline

Topics are divided into three groups. First set of topics and one of the other two is to be taught in a given instance.

• Linear Algebra: Vector spaces, Inner product, Linear maps, Vector algebra, Operator algebra, Conjugation of operators, Hermitian operators, Unitary operators, Projection operators, Functions of operators, Matrices, Similarity transformations, Determinant, Trace, Direct sums, Subspaces, Invariant subspaces, Eigenvalues and eigenvectors, Spectral decomposition, Polar decomposition.
• Group Theory: Groups, Subgroups, Classes and Invariant subgroups, Cosets, Factor groups, Homomorphism and isomorphism of groups, Group representations, Reducible and Irreducible representations, Unitary representations, Schur's Lemmas, Lie groups and Lie algebras, Rotation groups $SO(2)$ and $SO(3)$, Special unitary group $SU(2)$, Irreducible representations of $SO(2)$, $SO(3)$ and $SU(2)$ and their applications, Homogeneous Lorentz group, Poincare group, Young diagrams.
• Differential equations: linear and nonlinear differential equations, nonlinear differential equations relevant in physics. Klein-Gordon; Sine-Gordon equation; KdV equations; soliton solutions. Stochastic differential equations, Langevin equation, Fokker Planck equations.

Recommended Reading

• Sadri Hassani, Mathematical Physics, Springer (2013).
• H. J. Weber and G. B. Arfken, Essential Mathematical Methods for Physicists, Academic Press (2004).
• Wu-Ki Tung, Group Theory in Physics, World Scientific (2008).
• M. Hamermesh, Group Theory and Its Application to Physical Problems, Dover Publications (1989).
• Howard Georgi, Lie Algebras in Particle Physics, Levant Books (2009).
• J. V. Jose, and E. J. Saletan, Classical Dynamics: A Contemporary Approach, Cambridge University Press (2002).
• C. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer (2004).