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 A –How a cloud disperses.

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 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   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 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 B2H6B2H6

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