Plane Waves
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 is spatial analogue of angular frequency ω.
(We use k because it’s easier to write sin(kx) than sin(2πx/λ).)
Three deBroglie waves are shown for particles of equal mass.
A, 2f
II
2A, 2f
x
III
A, f
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. f: frequency
• ψ2(x,t) = exp(i(k2x-ω2t))
• ψ3(x,t) = exp(i(k3x-ω3t))
• ψ4(x,t) = exp(i(k4x-ω4t))
k = 2π/λ = wave number = number of radians per meter
I
• Most general kinds of waves are plane waves
(sines, cosines, complex exponentials) – extend
forever in space
• ψ1(x,t) = exp(i(k1x-ω1t))
• etc…
Different k’s correspond to different energies, since
E = ½mv2 = p2/2m = h2/2mλ2 = 2k2/2m
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:

Plane Waves vs. Wave
Packets
Superposition
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.
5
• Method of adding up sine waves to get another function (like
wave packet) is called “Fourier Analysis.” You explored it with
simulation in the homework.
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)
7
– 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.
Superposition
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.
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Heisenberg Uncertainty Principle
A slightly different scenario:
Plane-wave propagating in x-direction.
Δy: very large  Δpy: very small
y
Tight restriction in y:
Small Δy  large Δpy
 wave spreads out
strongly in y direction!
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
x
Weak restriction in y:
somewhat large Δy
 somewhat small Δpy
 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?
Up next:
The Schrödinger Equation
Electrons & other particles described by wave functions (Ψ)
Not deterministic but probabilistic
2
Physical meaning is in |Ψ| = 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
Review: classical wave equations
Electromagnetic waves:
Vibrations on a string:
y
E
c = speed of light
Solutions: E(x,t)
Magnitude is non-spatial:
= Strength of Electric field
How to solve?
Reminder of this class:
only 2 Diff Eq’s:
and
(k & α ~ constants)
x
x
Wave Equation
v = speed of wave
Solutions: y(x,t)
Magnitude is spatial:
= Vertical displacement of String
In DiffEq class, learn lots of algorithms for solving DiffEq’s.
In Physics, only ~8
X differential equations you ever need to solve,
solutions are known, just guess them and plug them in.
How to solve a differential equation in physics:
1) Guess functional form for solution
2) Make sure functional form satisfies Diff EQ
(find any constraints on constants)
1 derivative: need 1 soln  f(x,t)=f1
2 derivatives: need 2 soln  f(x,t) = f1 + f2
3) Apply all boundary conditions
(find any constraints on constants)
(You did this in HW6)
y(x,t)=Asin(kx)cos(ωt) + Bcos(kx)sin(ωt)
y(x,t)=Csin(kx-ωt) + Dsin(kx+ωt)
y
t=0
1) Guess functional form for solution
x
New guess: y(x,t) = Acos(Bx)sin(Ct)
y(x, t) = Asin(Bx)
LHS:
LHS:
RHS:
RHS:
Not OK! x is a
variable. There are
many values of x for
which this is not true!
OK! B and C are constants.
Constrain them so satisfy this.
What is the wavelength of this wave? Ask yourself …
How much does x need to increase to increase kx-ωt by 2π&
sin(k(x+λ)-ωt) = sin(kx + 2π – ωt)
k(x+λ)=kx+2π
k=wave number (radians-m-1)
kλ=2π ◊ k=2π&
&
&
λ&
What is the period of this wave? Ask yourself … How much
does t need to increase to increase kx-ωt by 2π
Speed:
sin(kx-ω(t+Τ)) = sin(kx – ωt + 2π )
ωΤ=2π ◊ ω=2π/Τ &
ω= angular frequency
ω = 2πf
(For your notes: Don’t go over this during class)
ν = speed of wave
What functional form works? Two examples:
y(x,t)=Asin(kx)cos(ωt) + Bcos(kx)sin(ωt)
y(x,t)=Csin(kx-ωt) + Dsin(kx+ωt)
k, ω, A, B, C, D are constants
Satisfies wave eqn if:
(For your notes: Don’t go over this during class)
0
L
Functional form of solution:
y(x,t) = Asin(kx)cos(ωt) + Bcos(kx)sin(ωt)
Boundary conditions?
y(x,t) = 0
at x=0
At x=0: y(x=0,t) = Bsin(ωt) = 0  only works if B=0
y(x,t) = Asin(kx)cos(ωt)
Is that it? Does this eqn. describe the oscillation
of a guitar string? What is k?
With Wave on Violin String:
y(x,t) = Asin(kx)cos(ωt)
Find: Only certain values of k (and thus λ, ω) allowed
 because of boundary conditions for solution
L
0
λ=2π&
&
k&
λ=2L&
& n &
Is there another boundary condition?
y(x,t) = 0 at x=L
n=1
At x=L:
y= Asin(kL)cos(ωt)= 0
n=2
 sin(kL)=0
 kL = nπ (n=1,2,3, … )
 k=nπ/L
n=3
Exactly same for Electrons in atoms:
Find: Quantization of electrons energies (wavelengths) …
 from boundary conditions for solutions
to Schrodinger’s Equation.
y(x,t) = Asin(nπx/L)cos(ωt)
Quantization of k … quantization of λ and ω&
With Wave on Violin String:
Find: Only certain values of k, λ, ω  I.e., the
frequencies of the string are quantized.
Same as for electromagnetic wave in microwave oven:
Exactly same for Electrons in atoms:
Find: Quantization of electrons energies (wavelengths) …
 from boundary conditions for solutions
to Schrodinger’s Equation. (next lecture)
Wow! Wait a minute! We just said that k, λ, and ω
are quantized but yet we can tune a violin by
changing the string tension. What gives?
A) The model we talked about is inconsistent with
the ability to “tune” a violin. But violins are
classical objects anyway, so no problem!
B) You can discuss the violin in terms of quantized
standing waves yet still tune a violin.
 k=nπ/L, and λ= 2L&
n &
Can tune!!
But:
 f=v/λ