Today: More Wavefunctions
1. Superposition and uncertainty principle.
2. How is information coded into the wavefunction?
3. Equation for finding the wavefunction!
Exam 2 is coming: Tuesday Nov. 4 7:30-9PM HERE!
• HWK 9 due Friday. 10AM.
• Week 10 online participation available later
today.
• Reading for Fri.: TZ&D Chap. 7. Yup.
Goal for matter waves
Interference pattern
1) Want partial differential equations to
solve for Ψ.
2) Interference pattern from the squared
magnitude,
∝ Ψ (r ,t )
2
Connection to the quanta:
Probability of particle arrival ∝ Ψ
2
Single particles
Connection to photons
LIGHT seems to come in quanta, photons.
Have a good wave theory (Maxwell), but
experiment requires photons, the quanta of
light. The photon properties, energy and
momentum, are quantized:
E ( x, y, z, t ) = E0 exp ⎡i k ir − ωt ⎤
⎣
⎦
(
)
h
E = hf
p=
E= ω
p= k
OR:
λ
deBroglie Idea:
Do the same thing for matter particles
We expect a wave function, Ψ(r,t), to code the
particle momentum, position, and ALL OTHER
PARTICLE PROPERTIES.
E= ω
Encoded into the
time dependence
p= k
Encoded into the
space dependence
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, since
E = mv2/2 = p2/2m = h2/2mλ2 =
2k2/2m
Superposition
Plane Waves vs. Wave Packets
Plane Wave: Ψ (x,t) = Aexp(i(kx-ωt))
Wave Packet: Ψn(x,t) = ΣnAnexp(i(knx-ωnt))
Which one looks more like a particle?
• In real life, matter waves 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.” More in this week’s
homework.
Plane Waves vs. Wave Packets
Plane Wave: Ψ(x,t) = Aexp(i(kx-ωt))
Wave Packet: Ψ(x,t) = ΣnAnexp(i(knx-ωnt))
For which type of wave are position x and
momentum p most well-defined?
A.
B.
p most well-defined for
plane wave, x most
well-defined for wave
packet.
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) = Aexp(i(kx-ωt))
– Wavelength, momentum, energy well-defined.
– Position not well-defined: Amplitude is equal everywhere,
so particle could be anywhere!
Wave Packet: Ψ(x,t) = ΣnAnexp(i(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: ΔxΔp ≥ /2
• In words: If the wavefunction approach is
OK, position and momentum cannot both
be determined completely precisely. The
more precisely one is determined, the less
precisely the other is determined.
• This is weird if you think about particles,
not very weird 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
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.
Review ideas from matter waves:
Electron and other matter particles have wave properties.
See electron interference
If not looking, then electrons are a wave … 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/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
Even heros are abused…
∂Ψ(x,t)
∂2 Ψ(x,t)
+ V (x,t)Ψ(x,t) = i
−
2
2m ∂x
∂t
2
Once at the end of a colloquium I heard Debye saying something like:
“Schrödinger, you are not working right now on very important problems…why
don’t you tell us some time about that thesis of deBroglie, which seems to have
attracted some attention?” So, in one of the next colloquia, Schrödinger gave a
beautifully clear account of how deBroglie associated a wave with a particle,
and how he could obtain the quantization rules…by demanding that an integer
number of waves should be fitted along a stationary orbit. When he had
finished, Debye casually remarked that he thought this way of talking was
rather childish…To deal properly with waves, one had to have a wave equation.
- Felix Bloch
Rest of Today- work towards finding equation that describes
the probability wave for electron in any situation.
Rest of QM -- solving this differential eq. various cases
and using it to understand nature and technology
Look at general aspects of wave equations …
apply to classical and quantum wave equations
a. start by reviewing some classical wave eqns
and solve (violin)
… easier to think about in classical system
b. look at Schrodinger equation
Schrodinger’s starting point:
What do we know about classical waves (radio, violin string)?
What aspects of electron wave eq’n need to be similar and
what different from those wave eqs?
Not going to derive it, because there is no derivation…
Schrodinger just wrote it down.
Instead, give plausibility argument.
What does the PDE
really mean?
∂ E 1 ∂ E
= 2 2
2
c ∂t
∂x
1 2
−k E = − 2 ω E
c
2
2
OR
2
ck = ω
Works for light, why not work for electron?
OR
c k= ω
Energy = pc
simple answer-- not magic but details not useful to you.
Advanced formulation of classical mechanics ⇒ .
Each p, ⇒ partial derivative with respect to x.
Each E, ⇒ partial derivative with respect to time.
∂ E 1 ∂ E
= 2 2
2
∂x
c ∂t
2
2
Works for light, why not work for electron?
simple answer-- not magic but details not useful to you.
Advanced formulation of classical mechanics ⇒ .
Each p, ⇒ partial derivative with respect to x.
Each E, ⇒ partial derivative with respect to time.
light: E=pc, so equal number derivatives x and t.
(KE)
(PE)
electron: E = p2/2m +V, so need 1 time derivative, 2
derivatives with respect to x, plus term for potential
energy V.
(complex answer (?)- need this structure to get same results
under Galilean frame transformation (r,v))