Download Quantum Physics Part II Quantum Physics in three units Bright Line

Quantum Physics in three units
• I: How light interacts with particles
• II: Simple systems
Quantum Physics Part II
– a) Atoms ,their structure, interaction with light
– b) Quantum wave functions and quantum
tunneling
• III: More complex systems
– a)Bosons, Fermions, and Bose Condensates
– b)Superconductivity and superfluidity
– c)Quantum entanglement and quantum computing
Review of What We Learned From
Quantum Part I
•
•
•
•
•
•
•
Black Body Radiation
Photoelectric Effect
Einstein’s Interpretation of these two
results
Wave-particle duality
Diffraction and interference
DeBroglie’s Hypothesis of matter waves
The uncertainty principle
Black Body Radiation
• What it is
– The electromagnetic radiation
given off by a hot object
– Does not agree with classical
physics
– Ultraviolet Catastrophe
• Planck’s Solution
Bright Line Spectra
• Spectra of Atoms
– Bright Line, Absorption, From The Sun
– Mathematical Models of Results
Photoelectric Effect
• Shine light on a metal surface
• No electrons emitted unless the frequency
is above a critical threshold value
• Once past the threshold, the amount of
emitted electrons is proportional to the
optical power
– If the energy of light come in
quanta – small packets = hf then
he can explain the results
1
Einstein’s Interpretation
New Discoveries
• Light comes in quantized bundles called
photons
• The energy is give by: E � hf
• The total power of a beam of photons is
E # photons
given by:
P�
t
�
t
• Bright Line Spectra
• Absorption Spectra
• Spectrum from the Sun
hf
• The threshold effect for emitting electrons
is due to the work function for the metal:
KEmax � hf � �
Bright Line Spectra
Prism
Observer
Atomic Spectra: Key to the Structure of the
Atom
A very thin gas heated in a discharge tube emits
light only at characteristic frequencies.
Light Source
1752 – Scottish Physicist
Thomas Melville
• Compares the spectrum for a pure hot gas (not a
flame) to the spectrum of a black body
• Surprising result: Line Spectra
Joseph von Fraunhofer
German Optician
• Takes a spectrum of the sun
• Sees an almost countless
number of dark lines
superimposed on a continuous
background.
Modern- Super High Resolution
Spectrum
2
Gustav Kirchoff
German Physicist
12 March 1824 – 17 October 1887
First to show that the absorption
lines from an emission spectrum
lines up with the absorption lines in
an absorption spectrum.
But…By very careful work, he was
able to determine that there were
some spectral absorption lines on the
sun had no matching lines on earth.
A worldwide search began to find the
elusive element: Result – Discovery
of “Helium”.
How do the spectra relate to the
atoms?
Anders Jonas Ångström
13 August 1814– 21 June 1874 - Swedish
• In 1862 Angstrom studies
spectrum of Hydrogen in
great detail.
Question: How can we make sense
of all of these lines of different
frequencies of light?
Helium comes from Greek word “Helios” (sun)
Johann Jakob Balmer
Swiss Mathematician
Additional Hydrogen
Wavelengths
May 1, 1825 – March 12, 1898
Balmer finds that he can represent
the frequencies of all observed lines
from Hydrogen by a simple formula:
� 1 1 �
1
� R� 2 � 2 �
�2 n �
�
�
�
n1=1
n2=1
n3=1
A constant
An integer > 2
Using this equation, Balmer predicts additional lines
before they are discovered.
Cathode Rays
Late 1800’s: Studies were being conducted in what
happens when electricity is discharged into a rarefied
gas.
All wavelengths
agree with a
generalized model
� 1
1
1�
� R�� 2 � 2 ��
n
n
�
2 �
� 1
1890’s: A result waiting
for an explanation!
J. J. Thomson
British Physicist and Nobel Laureate
18 December 1856 – 30 August 1940
How Thomson Discovers the Electron (11:08)
(This is the same guy who discovered Thomson
Scattering)
Thomson Discovers
Electron (2:53)
Hypothesis: The Plum
Pudding Model of the
Atom:
3
Ernest Rutherford
30 August 1871 – 19 October 1937
New Zealand-born British chemist and physicist
who became known as the father of nuclear
physics.
He is considered the greatest experimentalist
since Michael Faraday.
Lab:
ROLLING WITH RUTHERFORD
27.10 Early Models of the Atom
Rutherford did an experiment that showed that
the positively charged nucleus must be
extremely small compared to the rest of the
atom. He scattered alpha particles – helium
nuclei – from a metal foil and observed the
scattering angle. He found that some of the
angles were far larger than the plum-pudding
model would allow.
27.10 Early Models of the Atom
The only way to account for the large angles
was to assume that all the positive charge was
contained within a tiny volume – now we know
that the radius
of the nucleus
is 1/10000 that
of the atom.
The work was carried out by one
of Hans Geiger’s graduate
students, Earnest Marsden.
Gold Foil Experiment 9:07
Rutherford’s Result Was A Total Surprise
“This is quite the most incredible
event that has ever happened in my
life. It was almost as if you fired a
15” shell at a piece of tissue paper
and it came back and hit you! 1
1Introducing
27.10 Early Models of the Atom
Therefore, Rutherford’s
model of the atom is
mostly empty space:
Quantum Theory, J.P. McEvoy, Oscar Zarate
4
Rutherford’s
Model Had A
Lot of Critics
Bohr Model of Atom
But the biggest problem
with his model was yet to be
explained.
How Do Radio Transmitters Work?
An accelerating charge produces electromagnetic radiation. If
the charge oscillates with a specific frequency, then the radiation
will have the same frequency.
Classical Physics- All accelerating charges
produce electromagnetic energy.
Niels Bohr
(The Great
Dane)
Danish Physicist
Bohr arrives in England in 1911
and initially works with J.J.
Thomson. However, the two do
not get along with each other.
When he arrived he spoke
almost no English. He brought
a dictionary and the complete
works of Charles Dickens to
learn the language.
The Grandfather of Quantum Physics
7 October 1885 – 18 November 1962
Alice and Bob: How Can
Atoms Exist?
However, Bohr
Hits it Off With
Rutherford.
Niels Bohr (The Great Dane)
5
Niels Bohr (The Great Dane)
Physicists Can Learn From Unit Analysis
1 2
mv
2
Energy:
Frequency
f
Planck’s Constant
Angular
Momentum
�kg ��m�2 � �kg ��m�2 1
�s� �s�
�s�2
1
�s�
E � hf
Units for h : �kg �
L � mvr
�kg � �m� �m�
�s �
�m� �m�
�s �
Planck’s Constant has units of Angular Momentum! Is this just a coincidence???
J. J. Nicholson 1912
Angular Momentum
• He attempts to apply a quantum theory to
Thomson’s Plum Pudding model.
• He decides that the thing to quantize in the
atom is angular momentum of the
electron.
• However, he is unable to reconcile these
two ideas.
Bohr’s Great Breakthrough
• In 1913 Bohr combines three ideas together.
– The line spectra formula from Balmer
– The quantizing of angular momentum from Nicholson
– The need to define stable orbits for Rutherford’s
model
Bohr’s
First
Postulate
6
Principle Quantum Number, n
The angular momentum can not
take on any value (as would be
the case for classical physics).
The angular momentum must
be an integer multiple of h/2π
L1 � 1�
Finding the radius of the orbit
This part is done using classical physics. It is very similar to calculating the
orbits of planets around the sun
Planets: Gravity
Atom: Electromagnetic
M �� m
Centripetal Force
Fc �
m planet v 2
r
qelectron � q proton � e qnucleus � Ze
Provided By
FG �
Gmsun m planet
r
2
Solution for planets
L2 � 2�
L � m planet v planet r
...
r�
Ln � n�
2
L
GMm 2
Conservation of Angular Momentum
is Kepler’s 2nd Law
Centripetal Force
Provided By
m
v2
Fc � electron
r
FE �
KZe 2
r2
Solution for atoms
L � melectron velectron r
r�
�2
L2
rn �
n2
2
KZm elece 2
KmZe
Quantinization of Angular Momentum
is Bohr’s 1st Postulate.
Adding Energy to Bohr’s Model
Bohr
defines
radius of
each orbit
rn �
�2
Kmelec Ze
2
n2
• Once the radius and the angular
momentum are known, it is fairly
straightforward to determine the total
energy of the atom depending on which
orbit the electron is in.
• Procedure:
– Determine the Kinetic Energy
– Determine the Potential Energy
– Add them together
Bohr Derives the Balmer Formula
Bohr’s 2nd
Postulate
� 1
1
1 �
� R�� 2 � 2 ��
�
� n1 n2 �
The value for R calculated by
Bohr agrees with the value
calculated by Balmer within a
few percent.
R depends on Planck’s
constant, the speed of light, and
the fundamental constant of
electromagnetic attraction
between charged particles.
The energy of the atom is
quantized.
7
How We Understand the Bohr Atom - 1913
1. The atom is quantized by a single quantum
number “n”, which relates to the angular
momentum of the state that the electron is in.
2. The same number defines the energy of the
atom.
3. Absorption and emission of a photon can only
occur if the energy level between two states is
exactly equal to energy of the photon being
absorbed or emitted.
4. The quantum number, n, defines the “shell” for
the electron.
Bohr’s Formula for Energy
• Overall energy levels:
Z is the number of
protons in the nucleus.
Z2
En � �13.6 2 eV
n
n is a quantum
number. It can be 1,
2, 3, …
Chladni Plate Vibrations
En � �13.6
1
eV
n2
When n=∞, E=0,
electron is ionized
from atom.
What might Chladni patterns look
like in 3D?
The patterns of the hydrogen atom
are complex, but much simpler
than these!
More complicated structure
• Additional spectral lines were observed
• It was proposed by Arnold Sommerfeld that these were
due to the fact that the orbitals were not simply circular in
shape.
• A new quantum number was used. It was called the l
quantum number or the azimuthal quantum number.
• These were called subshells.
• For any given quantum number n, the possible subshells
range from l=0 to l=n-1
• Again, the angular momentum was determined by the
value according to
L2 � � 2l �l � 1�
8
Electron cloud, or probability distribution, for n = 2 states in
hydrogen
Orbital Shapes- Derived Later
Orbitals
How the Periodic Table relates to the
azimuthal quantum number
Chemistry
• You learned about the l quantum numbers
in chemistry.
l number
0
1
2
3
orbital type
S
P
D
F
Overview of Magnetism
How Do They Work 6:25
Magnetic Moment and Orbital Angular Momentum
Orbital Magnetic Moment
�orb � IA
v 2�r
T 2�r
�
� �
1
1
T
v
2�r
T�
v
Charge:
e
�
�
e Lorb
�orb � �
m 2
9
Still more complicated structure
The Zeeman Effect
•
•
•
•
•
The strength of the Zeeman effect depends
on magnetic field
Pieter Zeeman discovers that if you place an
atom in a strong magnetic field, additional
transition lines are observed.
This leads to an understanding that there are
additional energy states.
These are defined by the “magnetic quantum
number”, m.
In the absence of a magnetic field, these
additional states are still present.
For any azimuthal quantum number, l, it was
found that there were possibilities for the m
quantum number according to:
�l � m � l
Measuring the magnetic fields of
stars
• Since the optical splitting depends on the
strength of the magnetic field, observation
of the degree of splitting is a way to
measure the magnetic field strength in
stars.
Example of a selection rule
• When a photon is emitted or absorbed, the
l quantum number must change by ±1.
�l � �1
• The reason for this is that the photon has
angular momentum.
L photon � �1�
Three quantum numbers: n, l, m
• Bohr builds on Sommerfeld’s work and
works out a bunch of details for “selection
rules”.
• These rules showed that certain
transitions between states were not
allowed.
• We will learn more about forbidden
transitions when we get to particle physics.
The fourth quantum number
(The anomalous Zeeman effect)
Wolfgang Pauli – Austrian Theoretical Physicist
25 April 1900 – 15 December 1958
In 1925, additional spectral splitting was
observed that could not be explained.
It was an accepted fact that often theorists
were terrible with experimental equipment.
For some reason, Pauli had the reputation
that by his just stepping into a laboratory he
could make equipment fall apart.
A famous physicist, Otto Stern, would not
allow him into his lab, but would only talk to
him through a closed door.
Other, “forbidden,” transitions also occur but
with much lower probability.
10
Hidden
Rotation
Pauli hypothesized that
the anomalous Zeeman
effect could be
explained by a “hidden
rotation”. This would
result in a fourth
quantum number, “s”,
which would explain the
result.
Intrinsic
spin of
electrons
is either
“up” or
“down”.
Electron Spin
Spin of an electron
• Although it is described as if the electron is
spinning on its axis, that is not how it is
understood.
• Instead, the spin of an electron is said to
be an intrinsic property of the electron (like
its mass).
• We now understand that all fundamental
particles have a property called spin.
Pauli Exclusion Principle
Two electrons can not occupy the same
quantum state. Thus, for each combination
of n, l, m there are at most two electrons one
in the + ½ state and one in the – ½ state.
Why We Can Not Walk
Through Walls?
Alice and Bob
Complex Atoms
Complex atoms contain more than one electron,
so the interaction between electrons must be
accounted for in the energy levels. This means
that the energy depends on both n and l.
A neutral atom has Z electrons, as well as Z
protons in its nucleus. Z is called the atomic
number.
11
The Exclusion Principle
In order to understand the electron
distributions in atoms, another principle is
needed. This is the Pauli exclusion principle:
No two electrons in an atom can occupy the
same quantum state.
The quantum state is specified by the four
quantum numbers; no two electrons can have
the same set.
The Periodic Table of the Elements
We can now understand the organization of the
periodic table.
Electrons with the same n are in the same shell.
Electrons with the same n and l are in the same
subshell.
The exclusion principle limits the maximum
number of electrons in each subshell to 2(2l + 1).
Review of What We Learned From
Quantum Part I
•
•
•
The entire periodic table, all chemical properties, can be
explained by the combined work of the cast of characters that
we have studied so far.
However, this is not the end of the story for quantum theory.
Now the story gets even stranger!
De Broglie’s Hypothesis applied
to atoms
h
h
��
p
�
mv
n� � 2�r For in-phase
h
n
� 2�r
mv
Black Body Radiation
Photoelectric Effect
Einstein’s Interpretation of these two
results
Wave-particle duality
Diffraction and interference
DeBroglie’s Hypothesis of
matter waves
The uncertainty principle
•
•
•
•
A Revolution in Quantum Thought
So Who is
correct?
Bohr?
De Broglie?
Neither?
� mv �
� mv � h
� 2�r �
�
�
�n
� 2� �
� 2� � mv
n
h
� mvr
2�
This is Bohr’s Quantum Condition!
12
Particle-Wave Duality
Bohr-Schrödinger-Heisenberg (6:21)
Werner Heisenberg
• Challenges electron “orbits” as just being an
imaginary tool to visualize the atom.
• He treated atoms as simple oscillators in which
he could define the momentum, p, and the
degree to which the charge, q, was displaced
from equilibrium position.
• He comes up with a very abstract, complicated
algebra.
• It explains the observed quantum results, but
offers no pictures to visualize the atom.
Schrodinger
• Defines a type of wave function that can
be used to solve for many properties of an
atom.
• The Schrodinger equation is a complex
partial differential equation that can be
solved to find this wave function.
• Once the wave function is found, it can be
used to explain all of the observed results.
Imaginary Numbers
i � � �1
Complex Numbers
One of the foundations of
quantum physics
Examples of imaginary numbers:
3i, � 3i, 127.2i, � 15.6i
�3i �2 � 32 i 2 � �9
�� 3i �2 � �� 3�2 i 2 � �9
Imaginary numbers play an important
role in many areas of physics.
13
Complex Numbers
A complex number is one in which part of the number
is real and part of the number is imaginary.
Visualizing Complex Numbers
Imaginary Part
1� 2i
2i
Example of a complex number:
1� 2i
2 � 1i
1i
2
1
Imaginary Part
Real Part
Real Part
Complex numbers are sometimes used in place
of Cartesian coordinates.
Complex Conjugate
Change the sign of the imaginary
part of the complex number.
Visualizing Complex Numbers
Imaginary Part
2i
Example:
A � 1 � 2i
1i
A � 1 � 2i
*
2 � 1i
A
2
1
Real Part
A* is the complex conjugate of A.
Absolute Value of A Complex Number
Equivalent to the length of the vector
described by the complex number
2 � 1i
2
1
1
�1 � 2i ��1 � 2i �
�1 � 2i ��1 � 2i � � 1 � 2i � 2i � 4i 2 � 1 � 4 � 5
Example:
Real Part
2
2 � 1i � 1 � 2 � 5
2
The square of the absolute
value of a complex number
The product of a complex number and its complex
conjugate is equal to the square of the absolute value of a
complex number.
2i
1i
2 � 1i
A*
2
AA* � A
2
14
The phase of a complex number
The phase of a complex number
Changing the phase of a complex number does not
change the magnitude of the complex number
2i
1i
2 � 1i
2i
2 � 1i
1
�
1i
2
1
�
2
Real Part
Real Part
The complex number 2+1i has a magnitude
of 5 and a phase θ.
Origin of the Schrodinger Equation
Solving The Schrodinger Equation
Emily Noether
Principle of
Least Time
Principle of
Least Action
• Very few exact solutions
• Usually done numerically by computer
• The function Ψ that you end up with is the
wave function. It varies at different places
in space.
• The probability of finding the electron at a
particular place in space is given by
Schrodinger
Equation
William Rowan Hamilton
(1805–1865)
Irish Physicist
H � K � PE
The Schrodinger equation
is a complex partial
differential equation that can
be solved to find the wave
function.
P � � *�
Schrodinger’s Wave Function, Ψ
• The Schrodinger wave function is not directly
observable
• Max Born showed that the absolute value
squared of the wave function is equal to the
probability of finding an object at a particular
location.
• No more exact answers, said Born. In
quantum mechanics all we get are
probabilities.
How Physicists Use The Schrodinger Equation (1/2)
Goal: Predict the outcome of a measurement.
•
Max Born (1882–1970)
•
German-British
physicist
Solve Schrodinger’s equation for a given formula for potential
energy, using calculus, and/or using computers to solve the
equation.
You now have the functions, Ψ(x) and Ψ*(x)
•
Pick a special operation that you can apply to the function
Ψ(x) that will give you a new function.
•
Example: For position the operator is xΨ
•
Example: For momentum the operator is
You multiply the function by its location to get a new function..
You use calculus to differentiate the function
and multiply it by some constants.
i�
d
dx
15
How Physicists Use The Schrodinger Equation (2/2)
Wave function for a moving particle
• Now multiply that function new function by Ψ* at every
point in space.
• Carefully add up all of the values for (Ψ* operator Ψ). You
have to consider every valid point is space. Anything
that is non-zero must be included in this sum.
• In reality, this summing process is done by doing an
integral with calculus.
A wave function which
satisfies the non-relativistic
Schrödinger equation with
PE=0. In other words, this
corresponds to a particle
traveling freely through
empty space. The real part
of the wave function is
plotted here.
• The result of this process is a real number that
represents the observable that you will try to measure.
Particle in a Box
One of the few exact solutions to Schrodinger’s Equation.
Solution to the Particle in a Box
All solutions to the equation
have Ψ=0 at x=0 and at x=L.
Some trajectories of a particle in a
box according to Newton's laws of
classical mechanics (A), and
according to the Schrödinger
equation of quantum mechanics (BF). In (B-F), the horizontal axis is
position, and the vertical axis is the
real part (blue) and imaginary part
(red) of the wave function.
Lowest energy state is not E=0.
This is called the zero point
energy. The lowest energy of a
system can never be zero.
� x ��
The states (B,C,D) are energy
eigenstates, but (E,F) are not.
Particle in A Box Visualized 4:17
Particle in a box solution
Relationship between total energy,
KE, and PE. But PE = 0
everywhere inside the box.
E T � K � PE
ET � K
K�
Classical Relationship Between Kinetic
Energy and Momentum. This still holds.
Expectation value for the
“momentum” of the particle = 0
Solution to the Particle in A Box
Uncertainty in
Position
�x � �
�x � �
Uncertainty in
Momentum
�p � �
�x�p � �
p
Expectation value for the
“location” of the particle = the
middle of the box.
� p �� 0
Inside the box
p2
2m
L
2
� n 2� 2
�2
2
3
16
Quantum Tunneling
Minute Physics: What is Quantum Tunneling
Desktop Physics – Quantum Tunneling
Quantum Tunneling and
Radioactive Decay
1:04
2:57
Radioactivity
4:17
Quantum Tunneling through a finite barrier (0:26)
Touch Screens and Quantum Tunneling (6:26)
Simple Harmonic Oscillator
Simple Harmonic Oscillator
Simple Harmonic Oscillator
Simple Harmonic Oscillator
17