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