Using Calculus Terry A. Ring University of Utah. Mathematical Tools Algebra Geometry Calculus...
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Transcript of Using Calculus Terry A. Ring University of Utah. Mathematical Tools Algebra Geometry Calculus...
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! – 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– 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 SchrödingerBorn: 12 Aug 1887 in Erdberg, Vienna, AustriaDied: 4 Jan 1961 in Vienna, AustriaNobel Prize in Physics 1933
"for the discovery of new productive forms of atomic theory"
Time Dependent
Wave Equation!
Time Independent
Spherical Coordinates
r
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/
Red/Blue Clouds in Space
Zeeman EffectLight Emission in Magnetic Field
Multi-electron Atoms(Wolfgang) Pauli Principle Exclusion PrincipleNo 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