Math and Physics RefresherGeometryCalculusApproximationsDifferential EquationsComplex NumbersPhysics – MechanicsNewton’s Laws:Conservation of Momentum:Conservation of Energy:Circular motion:Physics – ElectromagnetismCoulomb’s LawLorentz Force LawElectromagnetic WavesPhysics – WavesPHY3101 Modern Physics Lecture Notes RefresherMath and Physics RefresherThis course assumes that you have studied Newtonian mechanics and electromagnetism in previous calculus-based physics courses. Listed below are some of the concepts in basic math, calculus, and physics that you are expected to know or to acquire during this course. This is not a complete summary of introductory math and physics. It is only meant to be a refresher of some of the concepts used in this course. Please report any inaccuracies to the professor.Geometry1. Pythagorean Theorem:The square of the hypotenuse of a right triangle is the sum of the squares of the two legs: c a b2 2 2  2. Circumference of a circle: C R23. Volume of a sphere: V R4334. Surface area of sphere:S R42Calculus1. Differentiation:You are expected to be able to take simple derivatives:ddxx nxddxx xddxx xddxe en nx x 1sin coscos sin2. Product Ruleddxf x g x f xdgdxg xdfdxddxx x x x x( ) ( ) ( ) ( )sin cos sin  Example: a fD. Acosta Page 1 1/14/2019 c a bPHY3101 Modern Physics Lecture Notes Refresher3. Chain Ruleddxf g xdfdgdgdxddxx xddxx x xafb gc h c h  Examples: sin cosexp exp2 2 222 24. IntegrationYou are expected to be able to perform simple integrals:x dxnxdx x xdx x xdx e en nx xzzzz111cos sinsin cos A purist would note that a constant should be added to these indefinite integrals.5. Change of VariablesTo use integration tables correctly, you must be able to change variables. For example:In xLun xLdxLnduILndu udx xxxILnn LnFHIK    zzzsinsinsin sin220221422 20LLet Then use 6. Integration By Partsudv uv vdu zzD. Acosta Page 2 1/14/2019PHY3101 Modern Physics Lecture Notes RefresherApproximationsFor small x, the following expansions are useful:1.1112   xx x 2. Binomial Expansion:1 1   x nxna f3. Taylor Expansion:f x f xdfdxx d fdxxxaf af   0202 220!Differential EquationsWe study the Schrodinger Equation in this class, which is a differential equation. Although you are not required to have taken a course in differential equations, we will learn how to solve the simplest ones:1. The exponential function is the only function whose derivative is the function itself:dfdxf xf x Ce Cx is the general solution, where and are constants ( )( )2. Two derivatives of the trigonometric functions give you back the same function with a sign change:d fdxk f xf x A kx B kx A B k222 ( )( ) sin cos , , is the general solution, where and are constantsUsing complex exponentials (see next section), you can also represent this solution as:f x A e B ei k x i k xaf D. Acosta Page 3 1/14/2019PHY3101 Modern Physics Lecture Notes RefresherComplex NumbersThe Schrodinger Equation studied in this class also contains complex-valued functions. Complex (or imaginary) numbers are based on i   1 . A complex number may be represented by z x iy , where x and y are real numbers. It can be represented by a point on a two-dimensional plane:An alternative way to represent a complex number is z re r ii    cos sinafTo find the magnitude of a complex number (the length r), multiply the number by its complex conjugate, then take the square root:| || |**z z z re re rz z z x iy x iy x yi i        or, afaf2 2Note that to take the complex conjugate, replace i with –iIt is possible to represent the sine and cosine functions by complex exponentials:cossin  e ee eii ii i22D. Acosta Page 4 1/14/2019 x iy z = x + iy r PHY3101 Modern Physics Lecture Notes RefresherPhysics – MechanicsNewton’s Laws:1. An object maintains constant velocity unless acted upon by an external force2. The acceleration of an object is proportional to the applied external force divided by the mass of the object (the inertia)F amThis is a vector equation. It can also be written as:Fpp v ddtm where is the momentumThis is your first differential equation! Note that av x ddtddt22 also.3. Force exerted by body 1 on body 2 is equal and opposite to the force that body 2 exerts on body 1. In other words, the force of gravity acting on you is balanced by an opposite force applied by the floor so that you do not fall to the center of theEarth!If the force exerted on a body is constant, then so is the acceleration. In that casev ax v a  tt t122Some examples of forces:Gravitational force near the surface of the Earth: F mg g  where m / s29 8.Newton’s law of Gravity: F Gm mrG  1 22116 6726 10 where N - m kg2 2. /Hooke’s Law: F kx k x   where spring constant, displacementConservation of Momentum:Momentum is defined by p vm. It is a vector quantity. The vector sum of the momentum of all particles before an interaction is the same as it is afterwards. In other words, it is conserved. For two masses which collide (but do not stick):m v m v m v m v1 1 2 2 1 1 2 2 This is an elastic collision. If they do stick, it is inelastic. In that case:m v m v m m v1 1 2 2 1 2  b gD. Acosta Page 5 1/14/2019 M1 M2 r initial finalPHY3101 Modern Physics Lecture Notes RefresherConservation of Energy:In this course, you will learn that energy is always conserved. In Newtonian physics, we learn that the kinetic energy is conserved for elastic collisions (but not inelastic collisions, where some of the energy goes into mass!) The kinetic energy is given by K mv122For the elastic collision of two objects, conservation of energy implies:121212121 122 221 122 22m v m v m v m v Circular motion:In uniform circular motion, the magnitude of the velocity of an object is constant, though its components are not. The magnitude of the centripetal acceleration to achieve uniform circular motion is avrc2The centripetal force responsible for this acceleration is just the mass times this quantity.D. Acosta Page 6 1/14/2019 V a c v dt dv dt r d dv dtrdv dtv