Download CHAPTER 2: Special Theory of Relativity

Preliminary mathematics: PHYS 344
Homework #1
Due in class Wednesday, Sept 9th
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
y (Imaginary)
A
x = A cos(j)
Complex numbers (critical!)
P
Trigonometry
Calculus (integration and
differentiation)
y = A sin(j)
j
x (Real)
2 f
1 2 f
 2 2  0
2
x
v t
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.
f(x)
f(x-2)
f(x-1)
f(x-3)
If we let a = v t, where v is positive
and t is time, then the displacement
will increase with time.
So f(x - v t) represents a rightward,
or forward, propagating wave.
Similarly, f(x + v t) represents a
leftward, or backward, propagating
wave.
v will be the velocity of the wave.
0
1
2
3
x
Plucked guitar strings
The Wave Equation
A guitar string wants to be straight;
the restoring force is proportional
to how much it’s stretched, that is,
its curvature:
2

f
 f
x 2
x
where f(x) is the displacement from straight at the point x.
2

f
But Newton’s 2nd Law (F = ma) says that this equals: m 2
t
Setting them equal and letting: v   / m
2 f
1 2 f
 2 2  0
2
x
v t
Waves are a solution to this equation. And v is the wave’s velocity,
also called its phase velocity.
f (x ± vt) solves the wave equation.
Write f (x ± vt) as f (u), where u = x ± vt. So
f f  u

x u x
Now, use the chain rule:
So
f f

x u
u
 1 and
x
2 f 2 f
 2

2
x
u
and
f
f
v
t
u
f f u

t u t
2
2 f

f
2
 2  v
t
u 2
Substituting into the wave equation:
2 f
1 2 f
 2 2
2
x
v t
2 f
1  2 2 f 

 2 v
 0
2
2 
u
v  u 
f can be any twice-differentiable function.
u
v
t
What does a typical wave look like?
It’s rigid and moves.
x
Some waves (like water waves) change their shape in time, but, for
this to occur, additional terms must occur in the wave equation.
Wavelength, etc.
Wavelength
l
Spatial quantities:
x
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.
k-vector magnitude: k = 2p/l
wave number: k = 1/l
Period
t
Temporal quantities:
t
Temporal quantities:
angular frequency: w = 2p/t
cyclical frequency: n = 1/t
The Wave Equation for Light Waves
where E is the light electric field
(the light magnetic field satisfies
the same equation)
 2E
1  2E
 2
 0
2
2
x
v t
We’ll use a cosine-wave solution:
E ( x, t )  A cos[k ( x  vt )   ]
or
where:
E ( x, t )  A cos(kx  wt   )
w
k
v
For simplicity, we’ll
just use the forwardpropagating wave
for now.
The speed of light in
vacuum, usually called
“c”, is 3 x 1010 cm/s.
Once we know the wavelength (and hence k = 2p/l), we also know
the frequency, w, etc.
The Phase Velocity
How fast is the wave traveling? The phase velocity is
the wavelength / period:
l
x
v = l/t
The wave moves one wavelength, l,
in one period, t.
Since n = 1/t:
v = lv
In terms of the k-vector, k = 2p/l, and
the angular frequency, w = 2p/t, this is:
v =w/k
The Phase of a Wave
The phase is everything inside the cosine.
E (x,t) = A cos(j), where j = k x – wt – 
j = j(x,t)
and is not a constant, like !
When the wave is at a maximum,
we say it has phase = 2mp.
Phase = 0
Phase =
5p/2
At a minimum, it’s (2m+1)p.
At zeroes, the phase is
2mp + p/2 or 2mp + 3p/2.
x
Phase = p/2
Phase = p
Taylor Series and Approximations
x
x2
x3
f ( x)  f (0)  f (0) 
f (0) 
f (0)  ...
1!
2!
3!
Taylor Series:
Example:
x x 2 x3 x 4
exp( x)  1      ...
1! 2! 3! 4!
because all derivatives are
also exp(x) and exp(0) = 1
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:
exp( x)  1  x
Other functions it’ll be convenient to approximate in this manner:
exp( x )  1  x
2
2
1
 1 x
1 x
1
 1  12 x
1 x
1
1  x2
 1  12 x 2
Complex Numbers
y (Imaginary)
x = A cos(j)
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.
P
A
y = A sin(j)
j
x (Real)
So, instead of using an ordered pair, (x,y), we can write:
The tilde under the P
means that P is complex.
P
~ = x+iy
= A cos(j) + i A sin(j)
where i  1
Complex-Number Theorems
Any complex number, z, can be written:
z = Re{ z } + i Im{ z }
Omitting the tildes
for simplicity.
So
Re{ z } = 1/2 ( z + z* )
and
Im{ z } = 1/2i ( z – z* )
where z* is the complex
conjugate of z ( i  –i )
The magnitude, | z | = A, of a complex number
is:
| z |2 = Re{ z }2 + Im{ z }2 = z z*
y (Imaginary)
x = |z| cos(j)
P
To convert z into polar form, A exp(ij):
tan(j) = Im{ z } / Re{ z }
|z|
y = |z| sin(j)
j
x (Real)
Euler's Formula
exp(ij) = cos(j) + i sin(j)
so the point, P
= A cos(j) + i A sin(j), can be written:
~
P = A exp(ij)
~
where:
A = Amplitude
j
= Phase
Properties of exp(ij):
Re{exp(ij)} = cos(j)
Re{-i exp(ij)} = sin(j)
Euler’s Formula Theorems
If exp(ij )  cos(j )  i sin(j )
exp(ip )  1
exp(  ij )  cos(j )  i sin(j )
exp(ij )  exp(ij ) exp(-ij )  1
2
1
cos(j )   exp(ij )  exp(ij ) 
2
1
sin(j )   exp(ij )  exp(ij ) 
2i
A1exp(ij1 )  A2 exp(ij2 )  A1 A2 exp i (j1  j2 ) 
A1exp(ij1 ) / A2 exp(ij2 )  A1 / A2 exp i (j1  j2 )
We can also differentiate exp(ikx) as if
the argument were real.
d
exp(ikx)  ik exp(ikx)
dx
1 = i × i
Proof:
d
cos(kx)  i sin(kx)  k sin(kx)  ik cos(kx)
dx
 ik i sin(kx)  cos(kx)
And we can integrate it, too:

1
exp(ikx) dx  exp(ikx)
ik
Waves Using Complex Numbers
The electric field of a wave can be written:
E (x,t) = A cos(kx – wt – )
Since exp(ij) = cos(j) + i sin(j), E(x,t) can also be written:
E (x,t) = Re { A exp[i(kx – wt – )] }
or
E (x,t) = 1/2 A exp[i(kx – wt – )] + c.c.
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."
Waves Using Complex Amplitudes
Define E(x,t)
to be the complex field—without the Re:
~
E  x, t   A exp i  kx  w t    


E  x, t    A exp(i ) exp i  kx  w t  
where we've separated the constants from the rapidly changing stuff.
The resulting complex amplitude is:
E0  A exp(i )
So:
E  x, t   E0 exp i  kx  w t 
As written, this field is
complex!
How do you know if E0 is real or complex?
Sometimes people use the “~” under the E’s, but not always.
So always assume it's complex.
Complex numbers simplify waves!
Adding waves of the same frequency, but different amplitude
and/or absolute phase, yields a wave of the same frequency.
This is hard using trigonometric functions, but it's easy with
complex exponentials:
Etot ( x, t )  E1 exp i(kx  w t )  E2 exp i (kx  w t )  E3 exp i (kx  wt )
 ( E1  E2  E3 ) exp i(kx  w t )
where all absolute phases are lumped into E1, E2, and E3.
~
E2
E3
~
~
E1 E2 E3
E1
x or t
The 3D Wave Equation for the Electric
Field and Its Solution
A wave can propagate in any
direction in space. So we must allow
the space derivative to be 3D:
or
 2E  2E  2E
1  2E
 2  2  2 2 0
2
x
y
z
v t
which has the solution:
where
and
2
1

E
2
E  2
0
2
v t
E( x, y, z, t )  E0 exp[i(k  r  w t )]
k   kx , k y , kz 
r   x, y, z 
k  r  kx x  k y y  kz z
k 2  kx2  k y2  kz2
E0 exp[i(k  r  wt )] is called a plane wave.
A plane wave’s contours of maximum field, called wave-fronts or
phase-fronts, are planes. They extend over all space.
Wave-fronts
are helpful
for drawing
pictures of
interfering
waves.
A wave's wavefronts sweep
along at the
speed of light.
A plane wave's wave-fronts are equally
spaced, a wavelength apart.
They're perpendicular to the propagation
direction.
Usually, we just
draw lines; it’s
easier.