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PHY401 Quantum Mechanics 1

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PHY401 Quantum Mechanics

1

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)

2

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

3

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).

4

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!”

6

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

7

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

8

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

9

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

12

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

14

“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.”

16

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)

19

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 )

20

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.

22

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),(

26

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

28

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

30

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

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

34

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

35

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)

36

∆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

39

-- Otto Stern and Max von Laue