## PHY 304: Quantum Mechanics I

3 credits | Prerequisites: MAT 212, PHY 223, 225 for Major;

MAT 212, PHY 230 for Minor

### Course rationale

Quantum Mechanics is an integral part of modern physics. It has revolutionized our ideas on how nature behaves at a very small scale. The very notion of deterministic nature breaks down, and probability becomes inevitable in the description of the quantum world. The application of Quantum Mechanics is diverse ranging from Elementary Particles to String Theory. This core course for physics major students will build the foundation of Quantum Mechanics.

### Course content

The Schrodinger Equation: some historical experiments, wave-particle duality, Schrodinger equation, interpretation of wave function, the probability current, normalization condition, Ehrenfest’s theorem; Time Independent Schrodinger Equation: separation of variable, time-independent Schrodinger equation, stationary states, infinite potential well, free particle, phase velocity, and group velocity, bound state and scattering state; Potential Barrier and Tunneling: delta function potential, potential step, potential barrier, quantum tunneling; Quantum Harmonic Oscillator: review of the classical harmonic oscillator, algebraic approach to diagonalize harmonic oscillator, creation, and annihilation operator, commutation relation, differential operator approach to solve harmonic oscillator, Hermite polynomials as eigenfunctions; Mathematical Formalism of Quantum

Mechanics: Hilbert space, linear algebra in bra-ket notation, Hermitian operators, properties, determinate state, eigenvalue problem, generalized statistical interpretation; The Uncertainty Principle: proof of generalized uncertainty principle, Heisenberg’s uncertainty principle, energy-time uncertainty, minimum uncertainty state, coherent state; Schrodinger Equation in Three Dimensions: separation of variables for central potentials, radial equation, Hydrogen atom, angular equation, spherical harmonics; Angular Momentum: commutation relations, quadratic Casimir, quantization of angular momentum, spherical harmonics as eigenfunctions of angular momentum operator; Spin: intrinsic angular momentum, commutation relation, SU(2) algebra, quadratic Casimir, quantization of spin angular momentum, bosons and fermions; Electron in Magnetic Field: Hamiltonian, Larmour precision, Stern-Gerlach experiment explained, the addition of angular momenta, singlet, and the triplet, Clebsch-Gordon coefficient; Time Independent Perturbation Theory: non-degenerate perturbation theory, degenerate perturbation theory; Applications of Time Independent Perturbation Theory: fine structure of Hydrogen, the Zeeman effect, hyperfine splitting of Hydrogen.

### Course objectives

- Understand the concepts of Quantum Mechanics.
- Understand the interpretation of wavefunctions and observables.
- Use appropriate mathematical formalism to solve different problems of interests.
- Use Dirac notation in appropriate places.
- Understand the non-commutativity properties of operators and uncertainty relations.
- Use algebraic and differential operator method to obtain quantization of angular momenta.

### References

- Introduction to Quantum Mechanics (3rd edition) by David J. Griffiths, and Darrell F. Schroeter
- Lectures on Quantum Mechanics (2nd edition) by Steven Weinberg
- Principles of Quantum Mechanics (2nd edition) by R. Shankar
- Quantum Mechanics by Albert Messiah
- Quantum Physics (3rd edition) by Stephen Gasiorowicz
- Quantum Mechanics: A Modern Development (2nd edition) by Leslie E. Ballentine
- The Feynman Lectures on Physics, Vol. 3 by Richard Feynman, Robert B. Leighton, and Matthew Sands