Handwritten lecture notes from V. Balakrishnan's quantum mechanics course, transcribed from the original notebook scans. The course starts from the uncertainty principle and the linear-vector-space formalism, builds up the postulates of quantum mechanics, the Schrödinger equation and stationary states, and then works through the standard one-dimensional problems (box, ring, 2-D box, delta-function well, finite well) and the role of symmetry and parity.

This transcription was done with Claude Opus 5, with every diagram redrawn in TikZ.

Video lectures: Quantum Physics by Prof. Balakrishnan (YouTube playlist).

The scans Qm1--Qm7 cover Lec 1 through Lec 19. Two quirks of the notebook: it carries no separate heading for Lec 9 (that material runs on inside the pages grouped under Lec 8), and it writes Lec 17 out before Lec 16. Split into ten pages, two lectures at a time:

1. Lec 1-2 -- introduction, the uncertainty principle, why orbits and orbital theory fail, the state vector, and the axioms of a linear vector space.
2. Lec 3-4 -- linear independence, span, basis and dimensionality, \(l_2\) (square-summable sequences), Gram--Schmidt, projection operators, completeness, \(L_2(a,b)\), Legendre polynomials, Fourier series and the Fourier inversion formula.
3. Lec 5-6 -- Cauchy sequences, Hilbert space and separability, linear operators, the postulates of quantum mechanics, expectation values, the classical/quantum dictionary (Poisson brackets \(\to\) commutators), the Schrödinger equation and the unitary time-evolution operator.
4. Lec 7-8 -- position and momentum bases, wave functions as expansion coefficients, simultaneous eigenstates and the parity operator, \(\langle x|\hat p|x'\rangle\) and \(\hat p \to -i\hbar\partial_x\), \(\langle x|p\rangle \propto e^{ipx/\hbar}\), the position-space Schrödinger equation, stationary states, and the particle in a 1-D box, a ring and a 2-D box.
5. Lec 9-10 -- symmetry implies degeneracy, why the H-atom ground state has \(L = 0\), the finite well, the attractive delta-function potential and its single bound state, and the two-well (symmetric/antisymmetric) splitting.
6. Lec 11 -- the node theorem, \(\langle K.E\rangle \geq 0\), metastable states and tunnelling, parity of the eigenfunctions of a symmetric potential, the semiclassical estimate \(E_n \sim n^{2r/(r+2)}\) for \(V = \lambda|x|^r\), and then the linear harmonic oscillator by Dirac's ladder-operator method.
7. Lec 12-13 -- oscillator wavefunctions and Hermite polynomials, coherent states as eigenstates of \(a\), the factorization / intertwining method, matrix representation of \(a, a^{\dagger}, x, p\) in the Fock basis, the general uncertainty principle, and the Schrödinger vs Heisenberg pictures.
8. Lec 14-15 -- the probability current, spreading of a wave packet and Ehrenfest's theorem, scattering off a 1-D potential barrier, a particle in a constant force field (Airy equation), and a charged particle in a magnetic field.
9. Lec 16-17 -- angular momentum in Q.M, radial momentum, gauge transformations, the angular-momentum algebra and Schwinger's two-oscillator construction, then Landau levels and the cyclotron frequency.
10. Lec 18-19 -- angular-momentum states from two oscillators, direct products of Hilbert spaces, rotation of states and operators, the \(j = \tfrac{1}{2}\) case and Schur's lemma, electron spin, spin in a constant magnetic field, entanglement, addition of angular momenta, and the Pauli exclusion principle.
11. Lec 23-24

The same material, typeset in LaTeX as a single document with all diagrams redrawn:

• Quantum Mechanics -- PDF (Lec 1-19)

• Classical Mechanics and Dynamical Systems
• Analytic Functions of a Complex Variable

Acknowledgements:

All these html and tex files are created by use of Anthropic's Claude AI using opus 5 with high usage.

For any discrepancies, please refer to the original lectures on youtube or my handwritten notes.