Using Calculus
Terry A. Ring
University of Utah
Mathematical Tools
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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
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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!
– JA= DAB dCA/dx + vxCA
– How a cloud disperses.
• 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
Integrals
• What is an Integral?
0
x
What is an Integral Good for?
• Sum of anything
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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
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
f
r
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
B2H6
• Probability of finding an electron in space
around an atom using Schrodinger’s
Equation