1. Terry A. Ring University of Utah
Using Calculus
Terry A. Ring
University of Utah

2. Mathematical Tools Integral Equations Integral-Differential Equations
Algebra
Geometry
Calculus
Imaginary Numbers
Differential Equations
Series Solutions
Laplace & Fourier Transforms
Partial Differential Equations
Integral Equations
Integral-Differential Equations
Statistics/Probability
Vector Analysis - Linear Algebra
Chaos Theory
String Theory
Fractals

3. How do we get Equations Balances Field Flux Equations Probability
Population
Mass
Energy
Force
Newton’s F = ma
Momentum
Flow
Field Flux Equations
Maxwell’s Equations
Probability
Schrodinger Equation
Heisenburg Uncertainty
Some of these Equations have derivatives and integrals

4. Maxwell’s Equations

5. Light Maxwell’s Equations + Schrodinger’s Equation

6. Derivatives
What is a Derivative?

7. What is the calculation of the slope useful for?
Flux of heat!
Q = k dT/dx
How heat disperses.
You can only feel heat flow!
Flux of mass!
JA= DAB dCA/dx + vxCA
How a cloud disperses.

8. Flow of Water in the bath or sink
Geometry
Slope of hill/trail
Flow of Water in the bath or sink
Area dh/dt = Qin – Qout
Pressure Drives Flow in a Pipe
- dP/dx =  d2vx/dy2

9. Integrals
What is an Integral? x

10. What is an Integral Good for?
Sum of anything
People born during a year
People that die during a year
How much money they make during their life time.
Sum of snow fall during winter.
Path Integrals
Total Flow of water in a pipe
Q = 2 vz r dr
Geometry
Area of a Lake

11. Terry A. Ring University of Utah
Quantum Mechanics
Terry A. Ring
University of Utah

12. Movement of the Electron around the Nucleus

13. Erwin Rudolf Josef Alexander Schrödinger
Born: 12 Aug 1887 in Erdberg, Vienna, Austria Died: 4 Jan 1961 in Vienna, Austria
Nobel Prize in Physics 1933
"for the discovery of new productive forms of atomic theory"

14. Time Dependent
Wave Equation!

15. Time Independent

16. Spherical Coordinatesf r q

17. Three Part Solution

18. Energy of the electron
Energy is related to the Principle Quantum number, n.
This gives 3 of the 4 quantum numbers, the last one is the spin quantum number, s, either +½ or – ½.

19. Wave Functions Probability to find an electron

20. Energy of the electron

21. Electron Transitions give off Energy as Light/Xrays
E=hc/l

23. Red/Blue Clouds in Space

24. Zeeman Effect Light Emission in Magnetic Field

25. Multi-electron Atoms (Wolfgang) Pauli Principle Exclusion Principle
No 2 electrons with same quantum numbers!

26. Periodic Table of the Elements
Dmitri Mendelyeev, 1869.

27. Light Emission from Elements Predicted

28. Shape of Molecules
B2H6

29. Probability of finding an electron in space around an atom using Schrodinger’s Equation