QUANTUM MECHANICS

GENERAL RELATIVITY

A GRAND TOUR OF PHYSICS

DR. GEORGE DERISE PROFESSOR EMERITUS, MATHEMATICS THOMAS NELSON COMMUNITY COLLEGE SPRING 2019

(WITH A BIT OF MATHEMATICS)

MAR 22-APR 26, 2019 1:30 – 3:30 TNCC ROOM 328.

A GRAND TOUR OF PHYSICS

DR. GEORGE DERISE PROFESSOR EMERITUS, MATHEMATICS THOMAS NELSON COMMUNITY COLLEGE SPRING 2019

MAR. 22, 2019 1:30 – 3:30 TNCC ROOM 328.

CLASSICAL MECHANICS

LECTURE 1

A GRAND TOUR OF PHYSICS-DR. GEORGE DERISE PROFESSOR EMERITUS, MATHEMATICS THOMAS NELSON COMMUNITY COLLEGE (HISTORIC TRIANGLE) FRIDAYS; MARCH 22 – APRIL 26, 2019: 1:30 – 3:30; ROOM 328.

GOOD WEBSITES: Hyperphysics: http://hyperphysics.phy-astr.gsu.edu/ Khan Academy: https://www.khanacademy.org/ Wikipedia: first paragraphs or introduction usually accessible. YOUTUBE LECTURES: Just search appropriate topic "… youtube"

BOOKS: The Fabric of the Cosmos: Space, Time, and the Texture of Reality: Brian Greene The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory: Brian Greene Fearful Symmetry: A. Zee Particle Physics in the Cosmos (Readings from Scientific American Magazine)

TOPICS: 1. CLASSICAL MECHANICS 2. ELECTRICITY AND MAGNETISM SPECIAL RELATIVITY 3. QUANTUM MECHANICS 4. PARTICLE PHYSICS 5. GENERAL RELATIVITY 6. ADVANCED THEORIES- SUPERSYMMETRY QUANTUM GRAVITY- STRING THEORY CONSTRUCTING THE UNIVERSE ?

CONCEPTS: SYMMETRY BEAUTY IN PHYSICS - IN MATH? SIMPLICITY; ONE ‘THEORY OF EVERYTHING’? GENERALIZATION A BIT OF MATH??

Symmetry is a type of invariance: the property that something does not change under a set of transformations.

“ I am quite convinced; and, believe me, if I were again beginning my studies, I should follow the advice of Plato and start with mathematics, a science which proceeds very cautiously and admits nothing as established until it has been rigidly demonstrated. ”

GALILEO GALILEI LINCEO DIALOGUE ON THE TWO CHIEF WORLD SYSTEMS 1632

If you wish is to become really a man of science, and not merely a petty experimentalist, I should advise you to apply to every branch of natural philosophy, Including mathematics.

THE UNREASONABLE EFFECTIVENESS OF MATHEMATICS IN THE NATURAL SCIENCES EUGENE WIGNER

REAL WORLD: RIGID BODY MOTION

IDEAL WORLD PLATO; THEORY OF FORMS GEOMETRY

The Pythagorean Proposition Elisha S. Loomis

256 proofs!

SECOND EDITION: 370 PROOFS!

PYTHAGOREAN THEOREM WATER DEMO https://www.youtube.com/watch?v=CAkMUdeB06o

a

b c

“MANN MUSS IMMER GENERALIZIEREN” “One should always generalize” Carl Gustav Jacob Jacobi

PYTHAGOREAN THEOREM GENERALIZED TO THREE DIMENSIONS

If an integer n is greater than 2, then the equation has no solutions in non-zero integers a, b, and c.

FERMAT’S LAST THEOREM (NUMBER THEORY)

𝑎3 +  𝑏3  =  𝑐3

𝑎4 +  𝑏4  =  𝑐4

𝑎5 +  𝑏5  =  𝑐5

         . . .𝑎𝑛 +  𝑏𝑛  =  𝑐𝑛

Any relevance to physics at all?

AXIOMATIC METHOD (example: EUCLIDEAN GEOMETRY)

1. Undefined terms ( point, line; plane; between) 2. Axioms ( given two distinct points, there exists one

and only one line connecting them.) 3. Definitions (angle, triangle, polygon) 4. Theorems (the sum of the angles of any triangle is

exactly 180 degrees… the Pythagorean Theorem... …There exists exactly Five Platonic Solids)

The primary criterion for an axiomatic system is...

consistency

REAL NUMBER SYSTEM; THE CONTINUUM

the order of magnitude of the earth-sun distance is 1011 m.

speed of light, c = 3 x 108 m/s; the order of magnitude of c is 108 m/s.

the size of the Milky Way Galaxy is 10 orders of magnitude greater than

the earth-sun distance.

KEPLER’S MYSTERIUM COSMOGRAPHICUM (1596) “Geometry is one and eternal shining in the mind of God”. (letter to Galileo, 1610 )

WILLIAM THOMSON - LORD KELVIN

THE THEORY OF VORTEX ATOMS (1867)

Stability of atoms accounted for by the stability of knots.

Large number of knot types can accommodate all the different elements.

Vibrational oscillations of knots could be the mechanism for atomic spectral lines.

“Geometry, which before the origin of things was coeternal with the divine mind and is God himself.” HARMONICES MUNDI 1619

KEPLER: THREE LAWS OF PLANETARY MOTION (1609-1619)

1. The orbit of every planet is an ellipse

with the Sun at one of the two foci. (1609) 2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time. (1609) 3. The square of the orbital period of a planet is directly proportional to the cube of the semi major axis of its orbit. (1619)

CURRENT VALUES: Orbital Period 29.5 years Semi-Major Axis 9.58 AU D= 890.56 million miles

886.65 million miles

GALILEO SIDEREUS NUNCIUS

1610

s = t2

s = 16t2

s = ½ gt2

GALILEO

s = 16 t 2

400 = 16 t 2

25 = t 2

5 = t

How long would it take to die when leaping off a 40 story building?

SIR ISAAC NEWTON (1642-1726)

THREE LAWS OF MOTION, LAW OF GRAVITATION, INVENTED CALCULUS PHILOSOPHIAE NATURALIS PRINCIPIA MATHEMATICA

NEWTON; “HYPOTHESES NON FINGO”

E IS THE TOTAL ENERGY OF THE PARTICLE

John Couch Adams Urbain Le Verrier

MATHEMATICAL DISCOVERY OF NEPTUNE 1846

LAPLACE'S DEMON the first published articulation of causal or scientific determinism, Pierre-Simon Marquis de Laplace (1814). If the Demon knows the precise location and momentum of every atom in the universe, their past and future values for any given time known by him, the future and the past can be known precisely from the laws of classical mechanics.

Pierre-Simon Laplace, "A Philosophical Essay on Probabilities” 1814

BASIC ASSUMPTIONS OF CLASSICAL MECHANICS

Classical dynamical variables, e.g. position, x momentum, mv kinetic energy, 1/2mv2 take on continuous values.

These dynamical variables satisfy a law (usually a differential equation) GLOBAL initial conditions and / or boundary conditions LOCAL

CLASSICAL MECHANICS: DETERMINISTIC

A state of a system is represented by giving the positions and velocities

of all the particles at a given time t.

This is equivalent to require the uniqueness of the solution of the

Newton's equations once initial conditions are specified.

.

LAW OF CONSERVATION OF ENERGY In a closed system the total amount of energy is conserved i.e. does not change. Energy may change from one form to another, but the total amount of energy in the closed system remains constant.

LAW OF CONSERVATION OF ENERGY H = K + U

LINEAR PENDULUM ~ CIRCULAR TRIGONOMETRIC FUNCTIONS

LINEAR HARMONIC OSCILLATOR (SPRING) ~ UNIFORM CIRCULAR MOTION

UNIFORM ROTATIONAL MOTION ~ LINEAR HARMONIC OSCILLATOR

UNIFORM CIRCULAR MOTION GRAPHED AS A TRIGONOMETRIC (CIRCULAR) FUNCTION

PROPERTIES OF WAVES

2015-04-22 at 2230 GMT,

left to right:

Europa, Io, Ganymede, Callisto. https://astronomy.stackexchange.com

Newtonian Mechanics A.P. French. 1971. p289

TRIGONOMETRIC (CIRCULAR) FUNCTIONS

FOR ONE PARTICLE TRAVELING IN 3 SPACE WE NEED SIX COORDINATES; THREE TO SPECIFY POSITION THREE TO SPECIFY MOMENTUM, THUS WE HAVE A 6 DIMENSIONAL SPACE.

TRUE PERIOD OF THE LINEAR PENDULUM

x

t 0

a

PHYSICS FERMAT’S PRINCIPLE of LEAST TIME Time for light to travel a given distance is minimized

whatever shape a bubble has initially, it will try to become a sphere.

sphere - the shape that minimizes the surface area the shape that requires the least energy

“ACTION PRINCIPLE” IN PHYSICS

“Les loix du movement…” 1746

“I…have discovered a universal principle on which all

laws are based… This is

the principle of least action

(a physical quantity called ACTION

tends to be minimized)

a principle so subtle as to be worthy of a supreme

being.”

Pierre-Louis Moreau de Maupertuis

NEWTON’S MECHANICS CAN BE DERIVED FROM THE PRINCIPLE OF LEAST ACTION

WHY DOES THE BALL FOLLOW A PARABOLIC TRAJECTORY?

THERE IS A MATHEMATICAL EXPRESSION CALLED THE ACTION MINIMIZE THE ACTION: THAT WILL GIVE YOU THE EXACT PARABOLIC TRAJECTORY!

ACTION = SUM (KINETIC-POTENTIAL ENERGY)

RICHARD FEYNMAN: it [the particle] smells all the paths in the neighborhood and chooses the one that has the least action by a method analogous to the one by which light chose the shortest time.

TIME

NOETHER’S THEOREM: CONSERVATION LAWS FOLLOW FROM THE SYMMETRY PROPERTIES OF NATURE.

If a system has a continuous symmetry, then there are corresponding quantities whose values are conserved in time.

SPACETIME SYMMETRIES

INDEPENDENCE OF TIME → ENERGY CONSERVATION.

INDEPENDENCE OF POSITION → MOMENTUM CONSERVATION.

INDEPENDENCE OF DIRECTION → ANGULAR MOMENTUM CONSERVATION.

If the Lagrangian is invariant under a transformation, then there is a conserved quantity.

CALCULUS OF VARIATIONS EASIEST EXAMPLE: FIND THE SHORTEST ARC LENGTH BETWEEN TWO POINTS

GEODESICS

Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point A to another B in the least time.

THE BRACHISTOCHRONE PROBLEM:

SOLUTION TO THE BRACHISTOCHRONE PROBLEM- THE CYCLOID