ch0 introduction
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bnbvnTRANSCRIPT
Instructor: Prof. Hao Zeng (Email: [email protected])Office: 225 Fronczak Hall Class: Mon, Wed and Fri, 1:00-1:50 PM,
219 Fronczak HallOffice hours: Tuesday 2-3 PM or by appointment
Textbook:
Griffiths, “Introduction to Quantum Mechanics” (2nd edition)
Bransden & Joachain, “Quantum Mechanics” (2nd edition)
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Instructor Coordinates
TOPIC UNITS
LEARNING OUTCOMES
Students are expected to master the following
subjects:
OUTCOME ASSESSMENT
Learning on topics is assessed as follows:
the origins of quantum
theory
blackbody radiation, photoelectric effect, Bohr model
of hydrogen atom, [1,2,5]HW, quizzes, Midterm Exam
the wave function and
Schrodinger equation
de Broglie’s hypothesis, wave-particle duality,
interpretation of wave function, the Schrodinger
Equation, Born’s Statistical Interpretation, Probability
Normalization, Momentum, Heisenberg uncertainty
principle [1,2,3]
HW, quizzes, Midterm Exam
Time-independent
Schrodinger equation
Stationary States, The Infinite Square Well,
The Harmonic Oscillator, The Free Particle
The Delta Function Potential,
The Finite Square Well [1,2,3]
HW, quizzes, Midterm Exam, Final Exam
formalism of quantum
mechanics
Hilbert Space, Observables,
Eigenfunctions of a Hermitian Operator,
Generalized Statistical Interpretation,
The Uncertainty Principle, Dirac Notation [3]
HW, quizzes, Final Exam
Quantum Mechanics in three
dimensions and angular
momentum
Schrodinger equation in spherical coordinates, the
hydrogen atom, orbital angular momentum,
eigenvalues and eigenfunctions of L2 and Lz, general
angular momentum, spin angular momentum, spin
one-half, total angular momentum, addition of angular
momentum [1,2,3]
HW, quizzes, Final Exam
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Homework (15%): Approximately one problem set per two weeks of lectures will be assigned. The homework is due in class on the due date (one week after it is assigned). You must show both your work and correct answers to earn full credit.
In class quizzes (15%): Mostly conceptual questions concentrating on materials to be covered in class.
Mid-term Exam (30%): An open-book mid-term exam will be held out of class, time and location to be announced.
Final Exam (40%): TBA
A final letter grade will be assigned based on your accumulative score (> 60% for C and > 90% for A).
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Grades
QM is difficult
“I think I can safely say that nobody understands quantum mechanics.”
Max Planck
"One should not hold against him too much that in his speculations he might have occasionally overshot the goal, as for example in his hypothesis of the quanta of light."
Richard Feynman
Albert Einstein
“GOD does not play dice with the universe!”
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No matter how much it has been questioned or objected, quantum theory has never failed an observational test and has beaten off innumerable challenges.
Then, why should we care
no quantum mechanics, no modern technology• All electronic devices, e.g., computers, mobile phones• Lasers• Superconductors• Nano materials
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Some Tips
A pragmatic attitude or approach to quantum mechanics:
First: Accept it or assume it is true.
Accept its principles and the related results, no matter how
peculiar they look.
Never get stuck for too long. Just move on and come back later.
Second: Practice, in class and after class
In the preface of many quantum mechanics textbooks, a common
advice is to do exercises. No practice, no understanding
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What are we going to learn? A language that describes
atoms, electrons and photons alike.
What are the expectations?
• Read ahead (and gain basic understanding) before coming to class
• Do every single HW. Collaboration allowed, but you should produce your own work
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Basic knowledge of classical physics (classical
mechanics, statistical mechanics,
electromagnetism)
Linear algebra, eigen functions and eigenvalues,
matrix presentation, inner products, etc
Ordinary differential equation
Integration
Required background
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Do not skimp on math!
Some History
ma F
Newton’s second Law Kinetic energy
21
2T mv
E T V
Mechanical energy of the systemdpF
dt
Until early 20th century: Classical Newtonian Mechanics…
Deterministic view: All the parameters of one particle can be determined exactly at any given time
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“All fundamental discoveries in physics have already
been made, and subsequent developments will be in
the sixth place of decimals.” (Michelson, 1894)
“There is nothing new to be discovered in physics
now. All that remains is more and more precise
measurement.”
(Lord Kelvin)
15
1. Michelson’s experiments (1887):
light speed is a constant,
regardless of the movement of the light source
Special theory of relativity
2. Black body radiation (late 19th century):
energy emitted discontinuously,
Planck constant
the beginning of quantum mechanics
Lord Kelvin
“…two small, puzzling clouds remained on the horizon.”
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In the beginning of the 20th century, there emerged more
and more experiments that could not be reconciled with the
classical physics.
These challenges were fundamental rather than technical
and led to a revolution in physics
Blackbody Radiation
Photoelectric Effect
Compton effect
Stern-Gerlach experiment
Spectra of Hydrogen 17
Black body Radiation
Black body: a perfect absorber of light.
A good approximation: Cavity kept at
constant temperature and blackened in
the interior. Once light enters the cavity
through the aperture, it will almost
never come out.
A good light absorber is also a good light
emitter (not the same incident light!)
𝑅 𝑇 = 𝜎𝑇4
R: total emissive power (J Stefan, 1879) : Stefan’s constant 5.6710-8 Wm-2K-4
Stefan-Boltzmann Law
(1)
Spectral Distribution of Black Body RadiationLet’s look at the spectral distribution of black body radiation
𝑅 𝜆, 𝑇 emissive power (spectral emittance)
𝑅 𝜆, 𝑇 𝑑𝜆: power emitted per unit area at T, corresponding to radiation with wavelength between and +d
O. Lummer and E. Pringsheim (1899)
• For fixed , 𝑅 𝜆, 𝑇 increases with increasing T
• At each T, there is a max for which 𝑅 𝜆, 𝑇 is maximum
• max varies inversely with T𝑅𝜆,𝑇
𝜆 𝑚𝑎𝑥𝑇 = 𝑏
Wien’s displacement law
(2)
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𝑅 𝑇 = 0
∞
𝑅 𝜆, 𝑇 𝑑𝜆
Spectral Distribution of Black Body Radiation
Wien’s Law (1893)
Based on thermodynamics
𝜆, 𝑇 (wavelength) spectral distribution function (monochromatic energy density) 𝜆, 𝑇 𝑑𝜆: energy density (energy per unit volume) in wavelength interval (, +d) at T
𝜆, 𝑇 = 𝜆−5𝑓(𝜆𝑇)
𝑓(𝜆𝑇) is a function of variable (𝜆𝑇), which can not be determined by thermodynamics (or classical physics)
To determine 𝜆, 𝑇 , we need to find 𝑓(𝜆𝑇)
(3)
(show: 𝜆, 𝑇 =4
𝑐𝑅 𝜆, 𝑇 )
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21
“n” Space
ny
nz
nx
• Each intersection point represents a
distinct combination of (n1, n2, n3);
• Each mode occupies a volume of 1.
𝑛 = 𝑛𝑥 𝑖 + 𝑛𝑦 𝑗 + 𝑛𝑧 𝑘
Spectral distribution of blackbody radiationRayleigh and Jeans (1905) (classical electromagnetic theory and equipartition of energy):
Thermal radiation within a cavity exists in the form of standing EM waves; the
number of modes per unit volume per unit wavelength 𝑛 𝜆 =8𝜋
𝜆4 (show)
𝜌 𝜆, 𝑇 =8𝜋
𝜆4 𝜀 (1)
If 𝜀 is the average energy of the mode, then
𝜀 = 𝑘𝑇 2The average energy of a classical oscillator is
Average energy per degree of freedom of a dynamical system in equilibrium is 𝑘𝑇/2 (classical law of equipartition of energy).
For a linear harmonic oscillator, 𝑘𝑇/2 kinetic and 𝑘𝑇/2 potential energy.
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𝜌 𝜆, 𝑇 = 8𝜋
𝜆4 𝑘𝑇
Rayleigh-Jeans Law
Spot the problem!
Ultraviolet Catastrophe
Rayleigh-Jeans model blows
up at high frequencies!!
24
𝜆, 𝑇 Exp.𝜌 𝜆, 𝑇 =
8𝜋
𝜆4 𝑘𝑇
Q: What is wrong with the classical model?
Planck’s quantum theory(1) Treat blackbody as a large number of atomic oscillators ( simple
harmonic oscillator), each of which emits and absorbs electromagnetic waves
(2) Each atomic oscillator can have only discrete values of energy that must be
multiples of h
= 𝑛ℎ = 𝑛ℎ𝑐
𝜆, n = 0, 1, 2, …
h = 6.63 x 10-34 J•s ( Planck’s constant, obtained by fitting the exp.)
Spectrum of the atomic Oscillators
Classical Planck’s model
Energy is quantized!25
kTT4
8),(
Recall Rayleigh-Jeans Law kT is the average energy
Replace kT in Rayleigh-Jeans law with
𝜌 𝜆, 𝑇 = 8𝜋
𝜆4 𝐸 =
8𝜋ℎ𝑐
𝜆5
1
𝑒ℎ𝑐𝜆𝑘𝑇−1
Quantization of energy is totally at
variance with classical physics.
At large
Planck’s law 1 1h hc
kT kT
hc
hE
e e
kTT4
8),(
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According to Planck’s theory: E = nh,
1h
kT
hE
e
Why?1 1
1hc
kT
kT
hc hce
kT
(show)
(𝜈 =𝑐
𝜆)
Photoelectric Effect
monochromatic light ,
Classical Picture:
1. IP is proportional to the intensity of the incident light.
2. If the incident light is strong enough, there should always be IPproduced, regardless of the frequency of the light.
Photocurrent IP: The electrons in the
cathode absorb the electromagnetic
energy of light and escape into the
vacuum, forming photocurrent.
Classical wave theory:
the energy of a light wave is given
by its intensity.
𝐴 = 𝐴0sin(𝑡 − 𝑘𝑥), 𝐸1
2𝐴0
2
by Philipp Lenard, 1900
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Electrons ejected from metallic surfaces irradiated by high frequency EM waves.
vacuum tube
Experimental Findings1. When frequency is above a threshold 0,
no matter how weak the light is, there is IP.
2. When is below a threshold 0, no matter
how strong the light is, there is no IP.
(a) The maximum kinetic energy
of any single emitted electron
increases linearly with frequency
above some threshold value and is
independent of the light intensity.
(b) The number of electrons
emitted per second (i.e. IP) is
independent of frequency and
increases linearly with the light
intensity
Change
Fix light intensity
Fix
Change light intensity
Fix
Change light intensity
v029
Classical picture Experiment
Photoelectric effect should
occur for any frequency as
long as the intensity is high
enough to give enough
energy to eject electrons
There is a threshold frequency
Maximum kinetic energy of
electrons should increase
with intensity of light
Maximum kinetic energy of
electrons should be
independent of frequency
Maximum kinetic energy of
electrons is independent of
intensity of light
Maximum kinetic energy of
electrons is linearly proportional
to frequency of light
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Photon, the particle (quanta) of light
Einstein (1905, Nobel price 1921):
light is discrete rather than continuous. In a beam of light, there are
many massless particles, photons. Each photon has an energy of:
Ephoton =h =ħω (ħ =h/2π, h: Planck’s constant)
The difficulty of the classical picture: light as continuous wave
suggests that “the energy of light” is related to frequency, in
addition to intensity. But how?
Only when ħω > EW (EW: work function of the metal),
electrons can be knocked off.
1. Maximum kinetic energy of photoelectrons:
Ekin,max = ħω - EW= ħ(ω - ω0)
Below this frequency limit ω0, no electrons can
leave the metal. Agrees with the experiments.
2. The intensity of photocurrent (the number of photoelectrons):
Increasing the intensity of the light beam increases the
number of photons, and hence increasing the photocurrent.
Elight=N*ħω 31
Potential well
The duality of light
E = ħω =h (ħ=h/2π, ω=2π)
Light is a particle (with discrete energy)
But it also as a frequency (diffraction, interference).
It is both a particle and a wave.
This duality is incompatible with classical physics.
𝑉0 =ℎ
𝑒𝜈 −
𝑊
𝑒
ℎ
𝑒Slope is
R.A. Millikan, 1916
e: 1.602× 10-19 Ch is determined to be 6.5610-34 Js, agrees well with value determined from black body radiation
The wave-particle duality is a general character of ALL physical quantities!
V0: Stopping potential
𝐾𝑚𝑎𝑥 = 𝑒𝑉0
Sto
pp
ing
po
ten
tial
(V
)
Bohr model for hydrogenRutherford’s atomic model (1911)
Vast majority of -particles passed
straight through the foil.
Approximately 1 in 8,000 were
deflected.
most of the atom was made up of
'empty space'.
Planetary model
Electrons circling
the central nucleus
The orbits and energy are
continuous in this model.
The atom would collapse within
10-10 second if it collapses.
A clear contradiction to reality.
Planetary model: unstable
A circling electrons radiates energy.
electron
nucleus
Line spectrum had been known for more
than a century. No one had thought very
deeply about what their relationship might
be with atoms.
hydrogen spectrum
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In 1913 Bohr, by accident, stumbled across Balmer's
numerology for the hydrogen spectrum, and came up with a
workable model of the atom.
Balmer's formula
Balmer series: four lines of visible light
Why is the atomic spectrum discrete instead of continuous?
Cannot be explained by Rutherford model (continuous orbits).
Indicating that the electron stays at some discrete orbits
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Bohr’s theory
1. The planetary model is valid.
2. The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies.
3. When an electron is in an “allowed” orbit it does not radiate. Thus the model simply throws out classical electromagnetic theory. (A hypothesis without any explanation)
4. Electrons can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck relation
5. The angular momentum of the allowed orbits is quantized (discrete)
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∆𝐸 = 𝐸2−𝐸1= ℎ𝜈
𝐿 = 𝑛ℎ
2𝜋= 𝑛ℏ, 𝑛 = 1,2,…∞
Quantization of the orbits
Quantization of the energy
Z=1 for
hydrogen
centripetal
forceCoulomb
force
2 2 2 221
2 2 2
e e e ee
k e k e k e k eE m v
r r r r
22 e
e
k em v
r
222 2 2
2
2 2
2
( ) ( ) ( ) =e
e e
e
e n
e e
k enm vr n m v
m r r
nL m vr n r
k e m
Q: What is the energy required to ionize a hydrogen atom?37
2 2
2
e em v k e
r r
2
22
2 2 2
13.6
2 2
e ee
n
k e mk e eVE
r n n
Δ𝐸 =𝑚𝑒𝑒
4
2(40)2ℏ2
1
𝑛𝑎2−
1
𝑛𝑏2
= ℎ
=ℎ𝑐
𝜆(na < nb)
1
𝜆=
Δ𝐸
ℎ𝑐=
𝑚𝑒𝑒4
ℎ𝑐2(40)2ℏ2
(1
𝑛𝑎2−
1
𝑛𝑏2)
= 𝑅∞ (1
𝑛𝑎2 −
1
𝑛𝑏2)
𝑅∞ =𝑚𝑒𝑒
4
802ℎ3𝑐
=10 973 731.6 m−1
Rydberg constant
Lyman series Paschen series
Balmer series
hydrogen spectrum explained
Rydberg formula
n=6
Rydberg unit of energy (atomic physics):
𝐸 = −𝑚𝑒𝑒
4
2(40)2ℏ2
1
𝑛2
“If this nonsense of Bohr should in the end prove to be right,
we will quit physics!"
Otto Stern: Stern-Gerlach experiment, Spin quantization (p181-183)
Student of Einstein
Max von Laue: discovery of the diffraction of X-rays in crystals
Student of Max Planck
Other experiments: Compton effect, Stern-Gerlach experiment
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-- Otto Stern and Max von Laue