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PHY3101
Spring 2008
Wave Packets
A.
Trigonometry
Euler’s identity,
eiθ = cos θ + i sin θ,
may be new to you, but it provides convenient easy ways to derive some of the trig identities.
For example
ei(α+β) = eiα eiβ
cos(α + β) + i sin(α + β) = (cos α + i sin α) × (cos β + i sin β)
cos(α + β) + i sin(α + β) = cos α cos β − sin α sin β + i(cos α sin β + sin α cos β)
The real and the imaginary parts of this give the well known trig identities:
cos(α ± β) = cos α cos β ∓ sin α sin β
and
sin(α ± β) = sin α cos β ± cos α sin β.
Use the Euler identity to show that
sin2 θ + cos2 θ = 1.
Hint: start with eiα e−iα = 1 and use the Euler Identity.
B.
Wave packets
One application of the previous trig identities involves wave interference. We want to use
trig identities to rewrite the algebraic sum of two waves, A cos(k1 x − ω1 t) + A cos(k2 x − ω2 t)
Start with
cos(α + β) + cos(α − β) = 2 cos α cos β.
Let α = ᾱ, and β = ∆α/2, for reasons that will be clear below, so that
cos(ᾱ + ∆α/2) + cos(ᾱ − ∆α/2) = 2 cos(∆α/2) cos(ᾱ).
Also, let
ᾱ =
k1 + k2
ω1 + ω2
x−
t and
2
2
∆α
k1 − k2
ω1 − ω2
∆k
∆ω
=
x−
t≡
x−
t.
2
2
2
2
2
The bar on ᾱ refers to the average. We now have
ᾱ + ∆α/2 = k1 x − ω1 t and ᾱ − ∆α/2 = k2 x − ω2 t.
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Finally, we have
A cos(k1 x − ω1 t) + A cos(k2 x − ω2 t) = 2A cos
∆ω
∆k
x−
t
2
2
cos(k̄x − ω̄t).
In dealing with wave packets, often the pairs k1 and k2 and also ω1 and ω2 are nearly
equal. So, ∆k k̄ and ∆ω ω̄. In this case the terms on the right hand side of this
last big equation have a specific interpretation. The higher frequency waving all comes
from the cos(k̄x − ω̄t) part, and k̄ ≈ k1 ≈ k2 . And the amplitude of the wave is given by
2A cos(∆kx/2 − ∆ωt/2). Now this “amplitude” changes in space and in time, but it changes
only slowly because ∆k k̄ and ∆ω ω̄. We often say that the cos(∆kx/2 − ∆ωt/2)
modulates the cos(k̄x − ω̄t) wave. Figure 5-15 on page 215 of the textbook does a good job
of demonstrating this.
The relationships of k and ω with λ and f , are k = 2π/λ and ω = 2πf . These are easy to
remember because the units of k are radians/distance while λ has units of cycles/distance,
and 1 cycle = 2π radian. Similarly, the units of ω are radians/time while f has units of
cycles/time.
The previous big equation demonstrates that the sum of two waves (with nearly the same
frequency and wavelength) can be written as the product of two waves. But, in the product
form the first part has a small frequency ∆f = ∆ω/2π and a long wavelength ∆λ = 2π/∆k.
(Here ∆k is small so that 1/∆k is big.) This first part changes slowly in space and in time.
But, the second part has a larger frequency f¯ = ω̄/2π, a shorter wavelength λ̄ = 2π/k̄ and
changes relatively rapidly in space and in time.
The speeds of the high frequency and low frequency parts of the wave need not be the
same. The speed of any sinusoidal wave is
v = λf = ω/k.
For the combined wave above, the speed of the high frequency, second part of the wave is
called the phase velocity,
vphase = ω̄/k̄.
While, the speed of the first part, which creates the modulation (or “packet”), is called the
group velocity,
dω
vgroup = ∆ω/∆k =
,
dk
in the limit that the wavelengths and frequencies of the initial waves are very close to each
other.
For many kinds of waves, the speed of the wave is the same for all frequencies and
wavelengths. This is the case for light in a vacuum and for a wave traveling down a string
under tension. We say that these waves are non-dispersive.
For other waves, such as water waves, light waves through glass and sound waves, when
we look at very high and very low frequencies, the speed of the wave depends upon the
frequency and wavelength. We say that these waves are dispersive.
For a non-dispersive wave, the phase velocity and the group velocity are always the same.
But, for a dispersive wave the two velocities might be very different.
Here is a website
http://physics.usask.ca/~hirose/ep225/animation/dispersion/anim-dispersion.html
that has some nice animations as well as some of the mathematics of creating wave packets.
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C.
The dispersive nature of de Broglie’s electron waves.
De Broglie hypothesized that the position of an electron could be described by a wave
with a frequency f = E/h = 12 mv 2 /h and wavelength λ = h/p. Equivalently, we can use
ω = 2π( 21 mv 2 )/h = πp2 /mh and k = 2πp/h. We can create a wave packet which localizes
an electron’s wave by adding up a combination of different frequencies and wavelengths. In
that case the phase velocity vphase = λf = ωk = v/2. The phase velocities of de Broglie’s
electron wave is half the velocity of the actual electron!
But, use the chain rule for derivatives to evaluate the group velocity, and see that
vgroup = dω/dk =
2πp/mh
p
dω/dp
=
=
=v
dk/dp
2π/h
m
accurately tracks the position of the electron.
You might be wondering why we used the Newtonian kinetic energy in the equation
f = E/h = 12 mv 2 /h. Einstein might have preferred that we use the total, relativistic
p
energy E = p2 c2 + m2 c4 , in f = E/h. These two possibilities clearly give two different
frequencies, but they give the same group velocity! First note that
d 2πp
2π
dk
=
=
,
dp
dp h
h
and also
dω
d 2πE
=
dp
dp
h
p
d 2π p2 c2 + m2 c4
=
dp
h
2
2πpc
= p
2
h p c2 + m2 c4
2πpc2
=
hE
p
2πc2 mv/ 1 − v 2 /c2
p
=
hmc2 / 1 − v 2 /c2
2πv
=
.
h
Now, use these results in the expression for the group velocity
dω/dp
vgroup = dω/dk =
dk/dp
2πv 2π
=
h
h
= v.
Our conclusion is that the group velocity of the electron’s wave packet is, in fact, equal to the
actual velocity of the electron when we use the correct relativistic expressions for the energy
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and the momentum of the electron. When de Broglie first proposed his expressions for the
frequency and wavelength of the electron probability wave he used these correct relativistic
expressions.
Here are some facts about wave packets: You can algebraically add together sinusoidal
waves of different amplitudes, frequencies and wavelengths to obtain a wave packet. If you
try to create a wave packet that is very tightly restricted in position x, then this requires a
broad range of wave numbers k. And, if you desire a very narrow range of wave numbers k,
then the wave packet will necessarily be spread out in position x.
Similarly, if you try to create a wave packet that is very tightly restricted in time t, then
this requires a broad range of wave frequencies ω. And, if you desire a very narrow range of
wave frequencies ω, then the wave packet will necessarily be spread out in time t.
These are elementary consequences of Fourier analysis. One way to express these ideas
is through the expressions
∆k ∆x ≈ 1 and
∆ω ∆t ≈ 1.