Today
Superposition of waves
Heisenberg uncertainty principle
Review of wave equations
Review of sinusoidal waves:
Wave in time: cos(2πt/T) = cos(ωt) = cos(2πf·t)
T = Period = time of one cycle
t
ω = 2π/T = angular frequency = number of radians per second
Wave in space: cos(2πx/λ) = cos(kx)
λ = Wavelength = length of one cycle
x
k = 2π/λ = wave number = number of radians per meter
k is spatial analogue of angular frequency ω.
(We use k because it’s easier to write sin(kx) than sin(2πx/λ).)
Plane Waves
•  Most general kinds of waves are plane waves
(sines, cosines, complex exponentials) – extend
forever in space
•  ψ1(x,t) = exp(i(k1x-ω1t))
•  ψ2(x,t) = exp(i(k2x-ω2t))
•  ψ3(x,t) = exp(i(k3x-ω3t))
•  ψ4(x,t) = exp(i(k4x-ω4t))
•  etc…
Different k’s correspond to different energies
k=2π/λ
Plane Waves
•  ψ1(x,t) = exp(i(k1x-ω1t))
•  ψ2(x,t) = exp(i(k2x-ω2t))
•  ψ3(x,t) = exp(i(k3x-ω3t))
•  ψ4(x,t) = exp(i(k4x-ω4t))
•  etc…
Different k’s correspond to different energies, since
E = ½mv2 = p2/2m = h2/2mλ2 = h2k2/2(2π) 2m = 2k2/2m
p=h/λ
k=2π/λ
=h/2π
Three deBroglie waves are shown for particles of equal mass.
I
A
II
x
2A
III
A
x
x
The highest speed and lowest speed are:
a. II highest, I & III same and lowest
b. I and II same and highest, III is lowest
c. all three have same speed
d. cannot tell from figures above
ans b. shorter wavelength means larger momentum =
larger speed. III is largest wavelength, I and II are same.
amplitude of wave is not related to speed.
A: Amplitude.
Quiz
E=hc/λ…
A.
B.
C.
D.
…is true for both photons and electrons.
…is true for photons but not electrons.
…is true for electrons but not photons.
…is not true for photons or electrons.
c = speed of light!
E = hf is always true but f = c/λ only applies
to light, so E = hf ≠ hc/λ for electrons.
Superposition principle
•  If ψ1(x,t) and ψ2(x,t) are both solutions to
wave equation, so is ψ1(x,t) + ψ2 (x,t). →
Superposition principle
•  E.g. homework (HW8, Q7b) – superposition
of waves one traveling to the left and to the
right create a standing wave:
ψ (x,t) = ΣnAnexp(i(knx-ωnt))
•  We can make a “wave packet” by combining
plane waves of different energies:
à
Superposition
8
Plane Waves vs. Wave
Packets
Plane Wave: ψ(x,t) = Aexp(i(kx-ωt))
Wave Packet: ψ(x,t) = ΣnAnexp(i(knx-ωnt))
Which one looks more like a particle?
•  In real life, matter waves are more like wave packets.
Mathematically, much easier to talk about plane waves, and
we can always just add up solutions to get wave packet.
•  Method of adding up sine waves to get another function (like
wave packet) is called “Fourier Analysis.” You will explore it
with simulation in the homework.
10
Plane Waves vs. Wave Packets
Plane Wave: Ψ(x,t) = Aei(kx-ωt) :
Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt) :
For which type of wave are position x and
momentum p most well-defined?
A.  p most well-defined for
plane wave, x most
well-defined for wave
packet.
B.  x most well-defined for
plane wave, p most
well-defined for wave
packet.
C.  p most well-defined for
plane wave, x equally
well-defined for both.
D.  x most well-defined for
wave packet, p most
well-defined for both.
E.  p and x equally welldefined for both.
Plane Waves vs. Wave Packets
Plane Wave: Ψ(x,t) = Aei(kx-ωt) :
Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt) :
For which type of wave are position x and
momentum p most well-defined?
A.  p most well-defined for
plane wave, x most
well-defined for wave
packet.
B.  x most well-defined for
plane wave, p most
well-defined for wave
packet.
C.  p most well-defined for
plane wave, x equally
well-defined for both.
D.  x most well-defined for
wave packet, p most
well-defined for both.
E.  p and x equally welldefined for both.
Plane Waves vs. Wave Packets
Plane Wave: Ψ(x,t) = Aei(kx-ωt)
–  Wavelength, momentum, energy: well-defined.
–  Position: not defined. Amplitude is equal everywhere,
so particle could be anywhere!
Wave Packet: Ψ(x,t) = ΣnAnei(knx-ωnt)
–  λ, p, E not well-defined: made up of a bunch of different
waves, each with a different λ,p,E
–  x much better defined: amplitude only non-zero in small
region of space, so particle can only be found there.
Heisenberg Uncertainty Principle
•  In math: Δx·Δp ≥ /2 (or better: Δx·Δpx ≥ /2)
•  In words: Position and momentum cannot both
be determined precisely. The more precisely
one is determined, the less precisely the other is
determined.
•  Should really be called “Heisenberg
Indeterminacy Principle.”
•  This is weird if you think about particles. But it’s
very clear if you think about waves.
Heisenberg Uncertainty Principle
Δx
small Δp – only one wavelength
Δx
medium Δp – wave packet made of several waves
Δx
large Δp – wave packet made of lots of waves
A slightly different scenario:
Plane-wave propagating in x-direction.
Δy: very large à Δpy: very small
y
Tight restriction in y:
Small Δy à large Δpy
à wave spreads out
strongly in y direction!
x
Weak restriction in y:
somewhat large Δy
à somewhat small Δpy
à  wave spreads out
weakly in y direction!
ΔyΔpy ≥ /2
Review ideas from matter waves:
Electron and other matter particles have wave properties.
See electron interference
If not looking, then electrons are waves … like wave of fluffy
cloud.
As soon as we look for an electron, they are like hard balls.
Each electron goes through both slits … even though it has mass.
(SEEMS TOTALLY WEIRD! Because different than our
experience. Size scale of things we perceive)
If all you know is fish, how do you describe a moose?
Electrons & other particles described by wave functions (Ψ)
Not deterministic but probabilistic
Physical meaning is in |Ψ|2 = Ψ*Ψ
|Ψ|2 tells us about the probability of finding electron in
various places. |Ψ|2 is always real, |Ψ|2 is what we measure
Up next:
The Schrödinger Equation
KE
Mass of
particle
PE
Potential
space and time
coordinates
Etot
Complex i,
with i2 = -1
Review: classical wave equations
Electromagnetic waves:
Vibrations on a string:
y
E
x
x
c = speed of light
Solutions: E(x,t)
Magnitude is non-spatial:
= Strength of Electric field
v = speed of wave
Solutions: y(x,t)
Magnitude is spatial:
= Vertical displacement of String
What does
mean?
a) Take the second derivative of E w.r.t. x only
b) Take the second derivative of E w.r.t. x,y,z only
c) Take the second derivative of E w.r.t both x and t
d)  Take the second derivative of E w.r.t x,y,z and t
e)  I don’t have a clue….
‘w.r.t’ = “with respect to”