1. Introductory Quantum Mechanics/Chemistry

2. Introductory Quantum Mechanics
E = m c2

3. Genealogy of Quantum Mechanics
Wave Theory of Light
(Huygens)
Classical Mechanics
(Newton)
Low
Mass
High
Velocity
Maxwell’s
EM Theory
Electricity and Magnetism
(Faraday, Ampere, et al.)
Relativity
Quantum Theory
Quantum Electrodyamics

4. E = m c2 Energy and Matter Size of Matter Particle Property
Wave Property
Large – macroscopic
Mainly
Unobservable
Intermediate – electron
Some
Small – photon
Few
E = m c2

5. The Wave Nature of Light

6. Continuous Spectrum

7. Line Spectra and the Bohr Model
Colors from excited gases arise because electrons move between energy states in the atom. (Electronic Transition)

8. Line Spectra and the Bohr Model

9. Line Spectra - Bohr Model

10. Bohr Model Experiment
1. Calculate the expected electronic transitions ( λ in nm ) in the Balmer Series. [from higher n’s to n=2]
2. Calculate the series limit transition in the Balmer Series.
3. Using the Spectroscope, record observable lines (color and λ) for the Hydrogen discharge tube and compare them to your Bohr Model calculations.
4. Record observable lines for Oxygen and Water. [Calculate possible lines in the visible region for the Oxygen atom using an appropriate Z.]
5. Compare the observed lines for the three discharge tubes.
6. Using the following sample table (spreadsheet) as a guide, discuss your results.
Tube
Color
Observed λ
Bohr Model λ ( ni →nf )
Hydrogen
…..

12. Line Spectra and the Bohr Model
Limitations of the Bohr Model
Can only explain the line spectrum of hydrogen adequately.
Can only work for (at least) one electron atoms.
Cannot explain multi-lines with each color.
Cannot explain relative intensities.
Electrons are not completely described as small particles.
Electrons can have both wave and particle properties.

13. The Wave Behavior of Matter
The Uncertainty Principle
Wavelength of Matter:
Heisenberg’s Uncertainty Principle: on the mass scale of atomic particles, we cannot determine exactly the position, direction of motion, and speed simultaneously.
If x is the uncertainty in position and mv is the uncertainty in momentum, then
Mathcad Calns

14. E = m c2 Energy and Matter Size of Matter Particle Property
Wave Property
Large – macroscopic
Mainly
Unobservable
Intermediate – electron
Some
Small – photon
Few
E = m c2

15. Quantum Mechanics and Atomic Orbitals
Schrödinger proposed an equation that contains both wave and particle terms.
Solving the equation leads to wave functions.

16. Quantum Chemistry Bohr orbits replaced by Wavefunctions
Three Formulations
Differential Equation Approach by Schrödinger
Matrix Approach by Heisenberg
Operator/Linear Vector-Space approach by Dirac
(Use Scaled-down version here)

17. Operator Algebra Operator Equation Algebraic rules: Commutators
Equality
Addition/Subtraction (linear operators)
Multiplication (order of operation)
Division (reverse operation)
Commutators
Eigenvalue Equation
Compound Operators
Ladder Operators

18. Postulates of Quantum Theory
The state of a system is defined by a function (usually denoted  and called the wavefunction or state function) that contains all the information that can be known about the system.
Every physical observable is represented by a linear operator called the “Hermitian” operator.
Measurement of a physical observable will give a result that is one of the eigenvalues of the corresponding operator for that observable.

19. Postulate I

20. Postulate I …cont…

21. Postulate II Variable Classical Mechanics Quantum Mechanics Position
x , y , z
Momentum
mv (along x-axis)
mv (along y-axis)
mv (along z-axis)
Potential Energy V

22. Postulate II…cont… Variable CM QM Kinetic Energy Total Energy T + V
Starting Point in all Quantum Mechanical Problems.

23. Heisenberg’s Uncertainty Principle
A fundamental incompatibility exists in the measurement of physical variables that are represented by non-commuting operators:
“A measurement of one causes an uncertainty in the other.”

24. Particle in a Box (1D) - 1 ∞ ∞ V V=0 V=∞ V=∞ x aa
Figure Potential energy for the particle in a box. The potential (V) is zero for some finite region (0<x<a) and infinite elsewhere.

25. Particle in a Box (1D) - Interpretations
● Plots of Wave functions
● Plots of Squares of Wave functions
● Check Normalizations
● How fast is the particle moving? Comparison of macroscopic versus microscopic particles.
Calculate v(min) of an electron in a 20-Angstrom box.
Calculate v(min) of a 1 g mass in a 1 cm-box

26. Particle in a Box (3D) – 1 z a y a a xa
Figure The cubic box. For the three-dimensional particle in a box, the potential is zero inside a cube and infinite elsewhere. This could represent the situation of a particle inside a container with perfectly rigid, impenetrable walls. x

27. Particle in a Box (3D) - Solutions

28. Particle in a Box (3D) -Degeneracies
Energy* g
States 3 1
(1,1,1) 6
(2,1,1) (1,2,1) (1,1,2) 9
(2,2,1) (2,1,2) (1,2,2) 11
(3,1,1) (1,3,1) (1,1,3) 12
(2,2,2) 14
(3,2,1) (3,1,2) (2,3,1) (2,1,3) (1,2,3) (1,3,2) 17
(3,2,2) etc 38
(5,3,2) etc; (6,1,1,) etc 54
(5,5,2) etc; (6,3,3) etc; (7,2,1) etc
*Energy given in units of h2/8ma2

29. Quantum Numbers of Wavefuntions
Symbol
Values
Description
Principle n
1,2,3,4,…
Size & Energy of orbital
Azimuthal l
0,1,2,…(n-1)
for each n
Shape of orbital
Magnetic ml
…,0,…+ 
for each 
Relative orientation of orbital's within same
Spin ms
+1/2 or –1/2
Spin up or Spin down
Azimuthal Quantum number
Name of Orbital(CD)
s (sharp) 1
p (principal) 2
d (diffuse 3
f (fundamental 4 g

30. Quantum Mechanics and Atomic Orbitals
Orbitals and Quantum Numbers