Transcript Quantum Mechanics - Bristol

The Realm Of The Bizarre, Freaky and Just Plain Weird
OR
Physics Walks the Planck
Some Famous Quotes
 I think I can safely say that nobody understands
quantum mechanics. (Feynman)
 I don’t like it and I wish I had never had anything to do
with it. (Heisenberg)
 Anyone who is not shocked by quantum theory has
not understood it. (Bohr)
You Must Understand This
 Quantum mechanics, the theory that explains
phenomena in the realm of the very small like atoms,
is right.
 Even so, most physicists find it so conceptually weird
that they just are not comfortable with it.
It All Begins With A Little Problem
 Heat an object it an it glows.
 Make it really hot and it glows orange, or yellow or
even white.
 In the late 1800’s scientists are heating all manner of
things to measure there emission spectra.
 And this guy, Max Planck (1901) finds
a wee problem with the data.
Max And His Hot Body
 Our boy Max wants to
examine not just any hot
hunk of metal like iron, he
wants to use platinum
heated in an insulating
cavity so that the light
emitted comes out of a
small hole.
 Platinum is a “noble” metal
so using it eliminates
impurities and cleans up his
data.
Max And His Hot Body
 Because it absorbs almost
all light shined on it, it’s
called a black body.
 A perfect black body will
absorb 100% of all radiation
hitting it, thus it appears
black.
 There are no perfect black
bodies, but we can get close
enough for government
work.
 Go ahead and giggle now.
Max Collects This Data
Max Has A Problem
 His data doesn’t agree with classical (meaning
Newtonian) theory.
 The issue is one of resonance.
 Pluck one guitar string and others will vibrate slightly
since they are similar in frequency.
 You can try this by opening a piano lid, shouting
loudly, and you’ll hear many strings humming. Each at
a frequency associated with your complex voice
induced sound wave.
A Little About Waves
This wave is a
standing wave.
It is captured by
the cavity its in.
A Little More About Waves
 Heat something and it glows.
Classical theory says it will emit a
little light at all frequencies.
 The problem is energy.
 Eαν
 As frequency goes up so does
energy.
 As one wave induces another
and another, more energy is
released.
 Also, each higher tone is a
multiple of 1/n of the one before.
Resonance occurs in discrete
integer chunks .
 This is HUGE!!!!!
The Ultraviolet Catastrophe
Classical Prediction
All heated objects should
incinerate everything.
Clearly this does not happen,
unless you’re Joan of Arc.
Wanted: Quantum Mechanic
 Our boy Max (in 1900) applies a simple idea to the
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curve, integration by parts.
Maybe each part of the curve can be thought of as a
discrete function unto its self using 1/n.
Like finding the area of a complex shape using a piece
of graph paper.
More importantly, like a piano string maybe the energy
comes in discrete integer based packets of 1/n.
This is also HUGE!!!!!
Max Planck Saves World, Film at 11.
Each red bar represents a discrete
frequency or wavelength of light.
The energy defined by the area of
each bar is related to the frequency.
Wanted: Algebra Student.
 Our boy Max applies the “Guitar Theory”
E α ν where ν (nu) is frequency f
To get the units to work out Max follows a basic premise
of proportionality – create a constant.
E = nhν where h is now known as Planck’s Constan and
n is an integer that describes the “string.”
The implication of n is that strings can’t vibrate in half
measures.
The Quantum Principle Is Born
 In nature certain quantities occur in discrete packets at
discrete intervals. The distance between intervals is
defined by h, Planck’s Constant.
Supermarket Door Openers?
 It had been known for a long time that if you shine
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light on certain metals they will emit electrons.
This is called the photoelectric effect first described by
Heinrich Hertz and Aleksandr Stoletov in 1890.
Nobody knew why it worked.
It was thought to be the reverse of an incandescent
bulb where electricity heats a metal filament which
then emits light in a broad spectrum like a black body.
Except for yet another problem.
Photoelectric Effect
 Light strikes a metal such as
zinc, selenium or silver, and
electrons are ejected from it.
 To determine the energy of
the electrons a voltage or
potential is placed across to
plates which can be changed.
 Electrons that make it across
the gap have same potential
V as the E field across the
gap.
Photoelectric Effect
 Here red light won’t
create electrons with
enough energy to jump
the gap.
Photoelectric Effect
 Here blue light,
which has higher
energy, does kick
out electrons with
enough energy to
jump the gap.
Except For Another Small Problem
 When the voltage was
set to ZERO, meaning
no E field, there was a
threshold below which
light didn’t have enough
umph to kick out
electrons.
 This violated the
classical view.
 Newton loses again.
Observational Summary
 Ordinary white light didn’t create a current no matter
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how bright it was.
Only above a certain frequency, say the UV, are
electrons ejected.
Above this threshold the higher the frequency the
more energetic the electrons.
Increase the brightness (intensity) and you made more
electrons, not more energetic electrons.
There is no time delay between absorption and
emission (This is Huge!!).
Classical Physics Dies, Long Live
Physics.
 None of the observations made sense to classical
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physics.
Conservation of energy and momentum seem to be
violated since intensity means more energy so you
should make more energetic electrons.
Swing a hammer at a bowling ball and it moves.
Swing harder and it moves faster, this makes sense.
The observations said, “swing harder get 2 bowling
balls, real hard and get 3.”
None of it made sense!!!!!!!!!!!
Big Al The Rescue
 All it took was Einstein
to interpret the work of
Planck and Hertz in one
of his 1905 papers.
 The slope of the graph is
h, so nh is no longer
used.
 The units place holder
constant h turns out to
be real. How cools is
that.
Threshold energy
It Suddenly Makes Sense
 E = hν
for black body emission
 Eelectron = hν – Ethreshold
 The threshold energy is also called the stopping
voltage and since it takes energy to stop an electron,
and we know that energy is work, we call Et the WORK
FUNCTION Φ (phi)
 Eelectron = hν – Φ
Einstein’s Nobel Winning 1905 Idea
 To explain the 5 observations that didn’t make sense…
 Light comes in discrete quantized packets.
 We now call them photons (remember QED?).
Einstein Performs Mathematical
Miracle, Makes Massless Thing
Have Momentum.
 To fit the emerging idea, Einstein invented photons to
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pass momentum around. But to meet the idea of
relativity photons had to be massless, otherwise they
can’t move at c.
c = λν
E = hν
And then a miracle happens
p = mv = hν/c = h/λ ……… E = pc
Which is a little ditty called E = mc2 (it’s still 1905)
Hydrogen Found Bohring
The Hydrogen Emission Spectrum
 In 1913 Neils Bohr is trying to explain why the
hydrogen emission spectrum has discrete lines.
 Emission and absorption spectra lines had been
known for many years.
 In 1893 a Swiss school teacher, Johan Jacob Balmer,
explained the pattern in a code like algorithm.
 Building on Planck & Einstein, along with his
Quantum Principle, Bohr speculates that electrons
inhabit discrete energy shells.
Today’s Weather - Balmer
 Balmer cracks the code and finds….
R is the Ryberg
constant and it
makes the math
works out.
Other Emission Series
 Many scientists are heating
atoms and naming
emission spectra series
after themselves.
 Now Enter Bohr.
Ogre’s Have Layers, Atoms Have
Layers – Borh’s Big Idea
 The only way for light to be emitted in discrete
frequencies is that that electrons must move in
discrete ways.
 He said, “the only allowed orbits are those where
angular momentum of electrons mvr is quantized.”
 His quantum leap of insight: The only time an electron
radiates energy is when it jumps between discrete
allowed orbits. The light emitted is then equal to the
energy difference between the two orbits.
The Principle Of Elegance
 Bohr’s model explains the
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photoelectric effect,
emission and absorption,
and the Balmer series.
mvr = nh called h-bar
h = h/2π
It suddenly all makes
sense.
Don’t worry, 2π will make
sense in a minute.
Bohr Drops A Balmer On World
 Bohr expands the R in the Balmer series to account for
the Atomic Number Z and writes it this way. Don’t
forget, E = hν still applies
 ν = 2π2meZ2e4
1 1
h3
k2 n2
{
{
Higher orbital integer
Lower orbital integer
Must be a value more than 2
But less than n.
 And It All Makes Sense And Is Still Elegant.
Except For One Minor Problem
 When shooting x-rays at pure
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crystalline metals we find what we
expect, wave diffraction patterns.
X-rays are generated by shooting
high energy electrons at a metal
target.
Hmm….what if we shoot electrons
at our samples.
OOPS! We get a diffraction pattern!
Hang on, electrons are particles
aren’t they?
Shooting Electrons At Crystals
Causes Physicists Great Diffraction
French Finally Explain Something,
Rest Of World Astounded.
 Since electrons act like waves, perhaps the discrete
orbits they inhabit must have the circumference that
will hold one full wave sequence (remember the UV
catastrophe problem?).
1923 - Louis DeBroglie Does The Wave
 de Broglie's picture of the Bohr model
 The de Broglie relation
 After a lot of arithmetic
 Where k is called the wave number, a convenient way
of reducing a relationship wave amplitude with respect
to time. It merely makes the equation shorter.
 The 2π thing? One rotation of a circle is 2π radians.
 The full cycle of one wave takes places in 2π radians.
 It All Still Makes Sense And Is Still Elegant.
Lambda λ Has An Implication
 λ = h/p implies a another huge idea….
 That all matter behaves like a wave at very small scales.
 When in motion at great speed (mean p is large),
particles exhibit wave like behavior because λ is
insanely small and is related to h which 10-34 scale.
 When p is low, like a bowling ball rolling in this room,
λ is huge an so we perceive the object as physically
solid.
 More about this later.
Wave Particle Duality Causes Cats
To Attempt Suicide, PETA Accuses
Schrödinger Of Feline Cruelty.
Werner Heisenberg Not Certain Of
Who’s Actually Responsible, “just
don’t observe them” he says.
A Tail Of Two Kitties
In A Nutshell
 Bohr’s Principle Of Complementarity: Waves and
particles represent complementary aspects of the same
phenomena.
 Heisenberg’s Uncertainty Principle: It is not possible
to measure the position and momentum of a particle
with arbitrary precision.
 De Broglie Relation: At very small scales matter
behaves like waves.
 Schrodinger develops a way to describe the probability
of a particles location.
Gunfight At The Quantum Corral
 Like water waves, matter can be
manipulated to a create a well in
which waves of matter can be
seen.
 This is a recent advance, like
within the last couple years.
 But what’s the standing wave in
the color image?
To Find Out Dig A Well
 Take any length of wire
and cut it like this picture,
but before gluing it back
together coat the ends
with a super insulator.
 In this way, whatever
electron is placed into
central trap or well will
simply bounce back and
forth since V on the ends
approaches -∞
V  -∞
V  -∞
V=0
L
x=L
x=0
+x
A Very Deep Potential Well
 This diagram represents
the physical artifact we
just made before.
 Because of the infinite
potential outside of the
well the potential inside
can be described as:
U=o for o < x < L
U -∞ for x < 0 & x > L
U(x)
x=0
L
Allow An Electron To Resonate
The electron will now bounce
back and forth like a ball.
L = nλ/2 for n = 1,2,3…
n is the principle quantum
number
The vertical displacement of
the wave, called the
amplitude A, can be
described at any point
along the wave as:
yn(x) = A sin(x nπ/L)
y
x=0
L
Allow An Electron To Resonate
The question now is what
are the odds of finding
the electron at any given
point along x?
Although a standing wave
is drawn imagine the
electron is bouncing back
and forth in wavelike
manner.
y
x=0
L
Coulomb Example
 Near the well wall
acceleration is huge so
velocity is changing
rapidly.
 Near the center of the
well the forces approach
equilibrium and
acceleration is near zero
so the velocity
approaches a constant
value.
e-∞
-∞
Coulomb Example
 The odds of detecting e-
near the well wall is very
low.
 Near the well middle it’s
very high.
 By this example, for n=1,
the probability of
detection is related to
sinx.
e-∞
-∞
The Psi Of Relief
 We can now define the behavior of the e- in a
slightly different way, the wave function ψ.
 ψn(x) = A sin(x nπ/L) for n = 1, 2, 3, ……
 The question now is what is the probability of
detection?
 In our example we can say near the walls it’s 0%
and near the center, at x = L/2 near 100%.
x
Roll The Dice
L
dx
 What are the odds of
finding e- at any point?
 Here we go……………
 (The probability p(x) of
detection in width dx
centered on location x) =
(probability ψn2)(width
dx)
e-∞
-∞
Roll The Dice!
p(x) = ψn2 dx
ψn2 = A2 sin2 ( x nπ/L )
for n = 1, 2, 3 …..
Note, ψn is the wave
function and can have
y(x) be +/-.
Probability must be
positive, hence we square
it to eliminate – values.
Effect Ψ vs.
 As n increase the
probability of
detection
anywhere in the
well increases also.
 Image what would
happen in this
image at n=15.
 As n increases p(x)
approaches 100%.
2
ψ
Quantum Mechanics Reconciles
With Classical Physics. Joy!
 The Correspondence Principle: At large
enough quantum numbers the predictions of quantum
mechanics merges smoothly with classical physics.
Changing the Well
 If we decrease the well end
conditions by decreasing
U(x) < ∞ then the electron
could be found outside the
well.
 ψ2 will follow and the
probability plots will fall
outside the well.
 This makes sense.
y
x=0
L
Enter De Broglie Again
 Since the only valid orbits
match with the De Broglie
wavelength λ, the solutions
for ψ2 follow with n = 1, 2, 3
Central Proton
Enter De Broglie Again
 If we take this 1D plot and
sweep a circle we get a series
of concentric rings.
Central Proton
Hydrogen s Orbital
2
ψ
Plots
Enter The Matrix
 Things become complex mathematically when we
expand from x into y and z.
 We now have a wave function ψ for orthogonal
direction and corresponding ψ2 for each as well.
 We can reduce the complexity – only slightly – using a
polar coordinate system since r corresponds to valid n
values of λ.
Schrodinger’s Mail Box
 To define a particle’s
location you need to
define its position in
space and time.
 Cartesian coordinates
would be x, y, z, t.
 Polar would be r,θ, φ, t
ψ Goes Polar
 Beginning where we did before we define a function
with the appropriate coordinate system.
 n = 1, 2, 3 ….. note 0 is not an option
 l = 0, 1, 2, 3 ….. n-1
 ml = -l, -(l-1),…..+(l-1), +l
The Punch Line Is A Psi of Relief
 The wave function Ψ is a complex vector that defines
the probable location of a particle.
 E is from Planck’s equation and V is related to Φ the
work function.
We Finally Gets Us To Here!
 http://winter.group.shef.ac.uk/orbitron/