Preliminary mathematics: PHYS 344 Homework #1 Due in class Wednesday, Sept 9 th Read Chapters 1 and 2 of Krane, Modern Physics Problems: Chapter 2: 3, 5, 7, 8, 10, 14, 16, 17, 19, 20

Stuff You Should Absolutely Know Before You Take This Course Complex numbers (critical!) Trigonometry Calculus (integration and differentiation) y (Imaginary) x (Real) P y = A sin(  ) x = A cos(  )  A Some differential equations (I’ll usually solve them for you, but it’s important that you not be afraid of them!)

Please complete this problem by Friday! It will help you determine whether your mathematical background is sufficient for this course. It should be very easy. If it’s hard, you will need to review some mathematics to do well in this course 1st problem

What is a wave? A wave is anything that moves. To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number. If we let a = v t, where v is positive and t is time, then the displacement will increase with time. So represents a rightward, or forward, propagating wave. Similarly, represents a leftward, or backward, propagating wave. v will be the velocity of the wave. f(x)f(x) f(x-3) f(x-2) f(x-1) x f(x - v t) f(x + v t)

Maxwell’s Equations and the Electromagnetic Spectrum Maxwell's main contribution to science was the correction he made to Ampère's circuit law in his 1861 paper On Physical Lines of Force. He added the displacement current term, enabling him to derive the electromagnetic wave equation in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrate the fact that light is an electromagnetic wave. This fact was then later confirmed experimentally by Heinrich Hertz in James Clerk Maxwell (1831–1879)

The Wave Equation Waves are a solution to this equation. And v is the wave’s velocity, also called its phase velocity. A guitar string wants to be straight; the restoring force is proportional to how much it’s stretched, that is, its curvature: where f(x) is the displacement from straight at the point x. x f But Newton’s 2 nd Law ( F = ma ) says that this equals: Setting them equal and letting: Plucked guitar strings

f (x ± vt) solves the wave equation. Write f (x ± vt) as f (u), where u = x ± vt. So and Now, use the chain rule: So  and  Substituting into the wave equation: f can be any twice-differentiable function.

What does a typical wave look like? It’s rigid and moves. Some waves (like water waves) change their shape in time, but, for this to occur, additional terms must occur in the wave equation. x

Wavelength, etc. Spatial quantities: x Wavelength k-vector magnitude: k = 2  / wave number:  = 1/ Temporal quantities: t Period  angular frequency:  = 2  /  cyclical frequency: = 1/  Temporal quantities: Waves look the same, whether we take snapshots of them in space or watch them go by in time, that is, whether we plot them vs. x or t.

The Electromagnetic Spectrum The transition wavelengths are a bit arbitrary. Visible light ranges from ~380 to ~780 nm.

The Wave Equation for Light Waves We’ll use a cosine-wave solution: where E is the light electric field (the light magnetic field satisfies the same equation) The speed of light in vacuum, usually called “ c ”, is 3 x cm/s. For simplicity, we’ll just use the forward- propagating wave for now. where: or Once we know the wavelength (and hence k = 2  / ), we also know the frequency, , etc.

The Phase Velocity How fast is the wave travel- ing? The phase velocity is the wavelength / period: v =  v v =  / k x The wave moves one wavelength,, in one period, . v =  Since = 1/  : In terms of the k-vector, k = 2  and the angular frequency,  = 2  this is:

Amplitude and Absolute Phase E (x,t) = A cos[(k x –  t ) –  ] x Absolute phase = 0 Absolute phase = 2  /3 A A = Amplitude  = Absolute phase (or initial phase)

The Phase of a Wave The phase is everything inside the cosine. E (x,t) = A cos(  ), where  = k x –  t –   =  (x,t) and is not a constant, like  ! When the wave is at a maximum, we say it has phase = 2m . At a minimum, it’s (2m+1) . At zeroes, the phase is 2m  +  /2 or 2m  + 3  /2. x Phase = 0 Phase =  Phase =  /2 Phase = 5  /2

Taylor Series and Approximations Taylor Series: If x is close to zero (i.e., << 1), then we often can approximate the function with the first two terms of its Taylor series: Example: because all derivatives are also exp(x) and exp(0) = 1 Other functions it’ll be convenient to approximate in this manner:

Complex Numbers So, instead of using an ordered pair, ( x, y ), we can write: Consider a point, P = (x,y), on a 2D Cartesian grid Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number. ~ The tilde under the P means that P is complex. where P = x + i y = A cos(  ) + i A sin(  ) y (Imaginary) x (Real) P y = A sin(  ) x = A cos(  )  A

The magnitude, | z | = A, of a complex number is: | z | 2 = Re{ z } 2 + Im{ z } 2 = z z* To convert z into polar form, A exp(i  ) : tan(  ) = Im{ z } / Re{ z } Complex-Number Theorems Any complex number, z, can be written: z = Re{ z } + i Im{ z } So Re{ z } = 1/2 ( z + z* ) and Im{ z } = 1/2i ( z – z* ) Omitting the tildes for simplicity. y (Imaginary) x (Real) P y = |z| sin(  ) x = |z| cos(  )  |z| where z* is the complex conjugate of z ( i  –i )

Euler's Formula exp(i  ) = cos(  ) + i sin(  )  = Phase A = Amplitude where: ~ P = A exp(i  ) ~ so the point, P = A cos(  ) + i A sin(  ), can be written: Re{exp(i  )} = cos(  ) Properties of exp(i  ): Re{-i exp(i  )} = sin(  )

Proof of Euler's Formula: Use Taylor Series: exp(i  ) = cos(  ) + i sin(  ) If we substitute x = i  into exp(x), then:

Euler’s Formula Theorems

We can also differentiate exp(ikx) as if the argument were real. Proof:  1 = i × i And we can integrate it, too:

Waves Using Complex Numbers The electric field of a wave can be written: We often write these expressions without the ½, Re, or +c.c. where " + c.c. " means "plus the complex conjugate of everything before the plus sign." E (x,t) = 1/2 A exp[i(kx –  t –  )] + c.c. E (x,t) = Re { A exp[i(kx –  t –  )] } or Since exp(i  ) = cos(  ) + i sin(  ), E(x,t) can also be written: E (x,t) = A cos(kx –  t –  )

Waves Using Complex Amplitudes Define E(x,t) to be the complex field—without the Re: How do you know if E 0 is real or complex? As written, this field is complex! So: The resulting complex amplitude is: where we've separated the constants from the rapidly changing stuff. Sometimes people use the “~” under the E ’s, but not always. So always assume it's complex. ~

where all absolute phases are lumped into E 1, E 2, and E 3. ~~~ Complex numbers simplify waves! This is hard using trigonometric functions, but it's easy with complex exponentials: Adding waves of the same frequency, but different amplitude and/or absolute phase, yields a wave of the same frequency. x or t

The 3D Wave Equation for the Electric Field and Its Solution or which has the solution: where and A wave can propagate in any direction in space. So we must allow the space derivative to be 3D:

is called a plane wave. A plane wave's wave-fronts are equally spaced, a wavelength apart. They're perpendicular to the propagation direction. Wave-fronts are helpful for drawing pictures of interfering waves. A wave's wave- fronts sweep along at the speed of light. A plane wave’s contours of maximum field, called wave-fronts or phase-fronts, are planes. They extend over all space. Usually, we just draw lines; it’s easier.