Lecture 7 SMES2201 Semester 1 (2009/2010) Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)

Topics Today Inner Products Hermitean Operators Schroedinger Equations in 3-Dimension: Particle in a Box.

The Hilbert Space Quantum Mechanics Wave Functions Operators State of a System Observables

PROBLEM 2

Inner Product Inner Product is like a dot product of 2 vectors. Define vectors by α and β. Linear Transformation Vectors in Quantum Mechanics are functions in infinite space. Normalization of Wave Function. Wave functions live in Hilbert Space.

Inner Product of Two Functions Schwartz Inequality: Properties of Inner Product: Orthornormal Orthogonal A set of function is complete if If the functions are orthonormal, the coefficients are given by 6

Example:Infinite Square Well By Dirichlet Theorem:

Hermitean Operators 8

Hermitian Operator Example: If is real , Therefore is a Hermitian 9

PROBLEM 1( 3.30 Griffith)

WAVE FUNCTION IN MOMENTUM SPACE 11

PROBLEM 3 (Problem 6.3 Scherrer)

SCROEDINGER EQUATION IN 3-DIMENSION Kinetic Energy

Potential is zero in the box and infinite outside. PARTICLE IN A BOX Potential is zero in the box and infinite outside. x y z a c b when x =0, x = a y = 0, y = b z = 0, z = c

Insert previous equations into Schroedinger Eqn. and divide by XYZ Separate Above Equation by making each component a constant Using Boundary conditions: At

Lowest Energy: No quantum number of a particle in a square box is zero because if so Degeneracy – Number of states that have the same energy. Example: Second level above ground level. and , are degenerates if a = b = c, Degeneracy = 3

Problem on Degeneracy: Find the 3 lowest energy levels and the degeneracy of each electron in the square box when a = b = 5 nm and c = 4 cm. h2/2m = h2c2/8m2c2 = 0.376 eV-nm2 E111 = 0.376 (a-2 + b-2 + c-2) = 0.376 (2/25 + 1/16) eV =0.062 eV Degeneracy = 1 E211 = E 121 0 0.376 (4/25 +1/25 1/16) eV = 0.987 eV Degeneracy = 2 E112 = 0.376 (1/25 + 1/25 + 4/16) eV = 0.1241 eV Degeneracy = 1