1. Modern Physics (II) Chapter 9: Atomic Structure
Chapter 10: Statistical Physics
Chapter 11: Molecular Structure
Chapter 12-1: The Solid State
Chapter 12-2: Superconductivity
Serway, Moses, Moyer: Modern Physics
Tipler, Llewellyn: Modern Physics

2. Modern Physics I Chap 3: The Quantum Theory of Light
Blackbody radiation, photoelectric effect, Compton effect
Chap 4: The Particle Nature of Matter
Rutherford’s model of the nucleus, the Bohr atom
Chap 5: Matter Waves
de Broglie’s matter waves, Heisenberg uncertainty principle
Chap 6: Quantum Mechanics in One Dimension
The Born interpretation, the Schrodinger equation, potential wells
Chap 7: Tunneling Phenomena (potential barriers)
Chap 8: Quantum Mechanics in Three Dimensions
Hydrogen atoms, quantization of angular momentums

3. Chapter 5: Matter Waves
de Broglie’s intriguing idea of “matter wave” (1924)
Extend notation of “wave-particle duality” from light to matter
For photons,
The wavelength is detectable only for microscopic objects
Suggests for matter,
de Broglie wavelength
P: relativistic momentum
de Broglie frequency
E: total relativistic energy

4. Chapter VI Quantum Mechanics in One Dimension
(x,t ) contains within it all the information that can be known about the particle
(x, t ) is an infinite set of numbers corresponding to the wavefunction value at every point x at time t
Properties of wavefunction
Finite, single-valued, and continuous on x and t
The particle can be found somewhere with certainty
Normalization:
must be “smooth” and continuous where U(x) has a finite value

5. Time-independent Schrödinger equation
The one-dimensional Schrödinger wave equation
Time-independent Schrödinger equation
U(x,t ) = U(x), independent of time
E: total energy of the particle
Solution:
Probability density at any given position x (independent of time)
stationary states

6. Time-independent Schrödinger equation
Separation of variables : (x,t ) = (x)·(t )
= E = constant
Independent of t
Independent of x
spatial
temporal

7. A particle in a finite square well
U(x) L U0 I II
III
Region II
Region I, III x
Need to solve Schrodinger wave equation in regions I, II, and III
Consider: E < U0
The wavefunctions look very similar to those for the infinite square well, except the particle has a finite probability of “leaking out” of the well
 > 0
Region I:
Region II:
Region III:

8. Finite square Well Penetration depth n = 2 n = 3
U(x) L U0 I II
III x L U0 I II
III x
n=1
y(x) U0
n = 3
U(x) I II
III x
Penetration depth
No classical analogy !!

9. Example: A particle in an infinite square well of width L
Momentum is quantized. Energy is quantized !
The notion of quantum number: n

10. Chapter VII Tunneling Phenomena
The Square Barrier Potential
U(x) x Uo L I II
III
Resonance transmission at certain energies E > U0
A finite transmission through the barrier at E < U0 if the barrier is made sufficiently thin

11. Expectation values Observables (可觀測量) and Operators (算符)
For a given wavefunction (x,t ), there are two types of measurable quantities: eigenvalues, expectation values
Observables (可觀測量) and Operators (算符)
An “observable” is any particle property that can be measured
Expectation value Q predicts the average value for Q
 (x,t ) is the “eigenfunction” and q is the “eigenvalue”
q = constant
The Schrödinger wave equation:

12. Examples of eigenvalues and eigenfunctions
U = central forces
U = 0, a free particle

13. Chapter VIII Quantum Mechanics in Three Dimensons
Three-dimensional Schrödinger equation
Time-independent Schrödinger equation:

14. Particle in a system with central forces
electron  r
nucleus
a central force !! 
Require use of spherical coordinates

15. ml l n Time-independent Schrödinger equation
principal quantum number orbital quantum number magnetic quantum number ml l n

16. Since |L| and Lz are quantized differently, L cannot orient in the
For any central force U(r ), angular momentum is quantized by the rules
and
Since |L| and Lz are quantized
differently, L cannot orient in the
z-axis direction. |L| > Lz
 = 1, 2, 3, … (n-1)
ml = 0, 1, 2, … 
Degeneracy for a given n
n = 1, 2, 3,…

17. Probability of finding electron of a hydrogen-like atom in the spherical shell between r and r + dr from the nucleus
0.52 Å
l = 0

18. Excited States of Hydrogen-like Atoms
The first excited state: n = 2
fourfold degenerate
2s state, is spherically symmetric
2p states, is not spherically symmetric

20. (2/23/2009, 2h)