1. 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)

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

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

4. PROBLEM 2

5. 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.

6. 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

7. Example:Infinite Square Well
By Dirichlet Theorem:

8. Hermitean Operators 8

9. Hermitian Operator Example: If is real , Therefore is a Hermitian9

10. PROBLEM 1( 3.30 Griffith)

11. WAVE FUNCTION IN MOMENTUM SPACE11

12. PROBLEM 3 (Problem 6.3 Scherrer)

13. SCROEDINGER EQUATION IN 3-DIMENSION
Kinetic Energy

14. 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

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

16. 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

17. 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 = eV-nm2
E111 = (a-2 + b-2 + c-2) = (2/25 + 1/16) eV
=0.062 eV Degeneracy = 1
E211 = E (4/25 +1/25 1/16) eV
= eV Degeneracy = 2
E112 = (1/25 + 1/25 + 4/16) eV
= eV Degeneracy = 1