Using Calculus
Terry A. Ring
University of Utah

Mathematical Tools
• 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

How do we get Equations
• Balances
– 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

Maxwell’s Equations

Light
Maxwell’s Equations + Schrodinger’s Equation

Derivatives
• What is a Derivative?

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!
– J
A
= D
AB dC
A
/dx + v x
C
– How a cloud disperses.
A

• Geometry
– Slope of hill/trail
• Flow of Water in the bath or sink
– Area dh/dt = Q in
– Q out
• Pressure Drives Flow in a Pipe
– - dP/dx =  d 2 v x
/dy 2

Integrals
• What is an Integral?
0 x

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  v z
• Geometry r dr
– Area of a Lake

Quantum Mechanics
Terry A. Ring
University of Utah

Movement of the Electron around the Nucleus

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"

Time Dependent
Wave Equation!

Time Independent

Spherical Coordinates r f q

Three Part Solution

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.

Wave
Functions
Probability to find an electron

Energy of the electron

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

Red/Blue Clouds in Space

Zeeman Effect
Light Emission in Magnetic Field

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

Periodic Table of the Elements
Dmitri Mendelyeev, 1869.

Light Emission from Elements
Predicted

Shape of Molecules
B
2
H
6

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