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Quantum physics(quantum theory, quantum

mechanics)

- Part 2

Summary of 1st lecture

- classical physics explanation of black-body

radiation failed (ultraviolet catastrophe) - Plancks ad-hoc assumption of energy quanta
- of energy Equantum h?, leads to a

radiation spectrum which agrees with experiment. - old generally accepted principle of natura non

facit saltus violated - Opens path to further developments

Problems from 1st lecture

- estimate Suns temperature
- assume Earth and Sun are black bodies
- Stefan-Boltzmann law
- Earth in thermal equilibrium (i.e. rad.

power absorbed rad. power emitted) ,

mean temperature T 290K - Suns angular size ?Sun 32
- show that for small frequencies, Plancks

average oscillator energy yields classical

equipartition result kT - show that for standing waves on a string, number

of waves in band between ? and ??? is ?n

(2L/?2) ??

Outline

- Introduction
- cathode rays . electrons
- photoelectric effect
- observation
- studies
- Einsteins explanation
- models of the atom
- Summary

Electron

- Cathode rays
- During 2nd half of 19th century, many physicists

do experiments with discharge tubes, i.e.

evacuated glass tubes with electrodes at ends,

electric field between them (HV) - 1869 discharge mediated by rays emitted from

negative electrode (cathode) - rays called cathode rays
- study of cathode rays by many physicists find
- cathode rays appear to be particles
- cast shadow of opaque body
- deflected by magnetic field
- negative charge
- eventually realized
- cathode rays were
- particles named
- them electrons

Photoelectric effect

- 1887 Heinrich Hertz
- In experiments on e.m. waves, unexpected new

observation when receiver spark gap is shielded

from light of transmitter spark, the maximum

spark-length became smaller - Further investigation showed
- Glass effectively shielded the spark
- Quartz did not
- Use of quartz prism to break up light into

wavelength components ? find that wavelength

which makes little spark more powerful was in the

UV

Hertz and p.e. effect

- Hertz conclusion I confine myself at present

to communicating the results obtained, without

attempting any theory respecting the manner in

which the observed phenomena are brought about

Photoelectric effect further studies

- 1888 Wilhelm Hallwachs (1859-1922) (Dresden)
- Performs experiment to elucidate effect observed

by Hertz - Clean circular plate of Zn mounted on insulating

stand plate connected by wire to gold leaf

electroscope - Electroscope charged with negative charge stays

charged for a while but if Zn plate illuminated

with UV light, electroscope loses charge quickly - If electroscope charged with positive charge
- UV light has no influence on speed of charge

leakage. - But still no explanation
- Calls effect lichtelektrische Entladung

(light-electric discharge)

Hallwachs experiments

- photoelectric discharge
- photoelectric excitation

Path to electron

- 1897 three experiments measuring e/m, all with

improved vacuum - Emil Wiechert (1861-1928) (Königsberg)
- Measures e/m value similar to that obtained by

Lorentz - Assuming value for charge that of H ion,

concludes that charge carrying entity is

about 2000 times smaller than H atom - Cathode rays part of atom?
- Study was his PhD thesis, published in obscure

journal largely ignored - Walther Kaufmann (1871-1947) (Berlin)
- Obtains similar value for e/m, points out

discrepancy, but no explanation - J. J. Thomson

1897 Joseph John Thomson (1856-1940) (Cambridge)

- Concludes that cathode rays are negatively

charged corpuscles - Then designs other tube with electric deflection

plates inside tube, for e/m measurement - Result for e/m in agreement with that obtained

by Lorentz, Wiechert, Kaufmann - Bold conclusion we have in the cathode rays

matter in a new state, a state in which the

subdivision of matter is carried very much

further than in the ordinary gaseous state a

state in which all matter... is of one and the

same kind this matter being the substance from

which all the chemical elements are built up.

Identification of particle emitted in

photoelectric effect

- 1899 J.J. Thomson studies of photoelectric

effect - Modifies cathode ray tube make metal surface to

be exposed to light the cathode in a cathode ray

tube - Finds that particles emitted due to light are the

same as cathode rays (same e/m)

More studies of p.e. effect

- 1902 Philipp Lenard
- Studies of photoelectric effect
- Measured variation of energy of emitted

photoelectrons with light intensity - Use retarding potential to measure energy of

ejected electrons photo-current stops when

retarding potential reaches Vstop - Surprises
- Vstop does not depend on light intensity
- energy of electrons does depend on color

(frequency) of light

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Explanation of photoelectric effect

- 1905 Albert Einstein (1879-1955) (Bern)
- Gives explanation of observation relating to

photoelectric effect - Assume that incoming radiation consists of light

quanta of energy h? (h Plancks constant, ?

frequency) - ? electrons will leave surface of metal with

energy - E h ? W W work function

energy necessary to get electron out of the

metal - ? there is a minimum light frequency for a given

metal, that for which quantum of energy is equal

to work function - When cranking up retarding voltage until current

stops, the highest energy electrons must have had

energy eVstop on leaving the cathode - Therefore eVstop h ? W

Verification of Einsteins explanation

- 1906 1916 Robert Millikan (1868-1963)

(Chicago) - Did not accept Einsteins explanation
- Tried to disprove it by precise measurements
- Result confirmation of Einsteins theory,
- measurement of h with 0.5 precision
- 1923 Arthur Compton (1892-1962)(St.Louis)
- Observes scattering of X-rays on electrons

How to see small objects

- seeing an object
- detecting light that has been reflected off the

object's surface - light electromagnetic wave
- visible light those electromagnetic waves that

our eyes can detect - wavelength of e.m. wave (distance between two

successive crests) determines color of light - wave hardly influenced by object if size of

object is much smaller than wavelength - wavelength of visible light between 4?10-7 m

(violet) and 7? 10-7 m (red) - diameter of atoms 10-10 m
- generalize meaning of seeing
- seeing is to detect effect due to the presence of

an object - quantum theory ? particle waves, with

wavelength ?1/p - use accelerated (charged) particles as probe, can

tune wavelength by choosing mass m and

changing velocity v - this method is used in electron microscope, as

well as in scattering experiments in nuclear

and particle physics

Models of Atom

- J.J. Thomsons model
- Plum pudding or raisin cake model
- atom sphere of positive charge
- (diameter ?10-10 m),
- with electrons embedded in it, evenly

distributed (like raisins in cake) - i.e. electrons are part of atom, can be kicked

out of it atom no longer indivisible!

Geiger, Marsden, Rutherford expt.

- (Geiger, Marsden, 1906 - 1911) (interpreted by

Rutherford, 1911) - get ? particles from radioactive source
- make beam of particles using collimators

(lead plates with holes in them, holes

aligned in straight line) - bombard foils of gold, silver, copper with beam

- measure scattering angles of particles with

scintillating screen (ZnS)

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Geiger Marsden experiment result

- most particles only slightly deflected (i.e. by

small angles), but some by large angles - even

backward - measured angular distribution of scattered

particles did not agree with expectations from

Thomson model (only small angles expected), - but did agree with that expected from scattering

on small, dense positively charged nucleus with

diameter

at ?10-10 m

Rutherford model

- planetary model of atom
- positive charge concentrated in nucleus (m)
- negative electrons in orbit around nucleus at

distance ?10-10 m - electrons bound to nucleus by electromagnetic

force.

Rutherford model

- problem with Rutherford atom
- electron in orbit around nucleus is accelerated

(centripetal acceleration to change direction of

velocity) - according to theory of electromagnetism

(Maxwell's equations), accelerated electron emits

electromagnetic radiation (frequency revolution

frequency) - electron loses energy by radiation ? orbit

decays - changing revolution frequency ? continuous

emission spectrum (no line spectra), and atoms

would be unstable (lifetime ? 10-10 s ) - ? we would not exist to think about this!!
- This problem later solved by Quantum Mechanics

Bohr model of hydrogen (Niels Bohr, 1913)

- Bohr model is radical modification of Rutherford

model discrete line spectrum attributed to

quantum effect - electron in orbit around nucleus, but not all

orbits allowed - three basic assumptions
- 1. angular momentum is quantized L n(h/2?)

n h, n 1,2,3,...

?electron can only

be in discrete specific orbits with particular

radii ? discrete energy levels - 2. electron does not radiate when in one of the

allowed levels, or states - 3. radiation is only emitted when electron makes

transition between states, transition also

called quantum jump or quantum leap - from these assumptions, can calculate radii of

allowed orbits and corresponding energy levels - radii of allowed orbits rn a0 n2 n

1,2,3,., a0 0.53 x 10-10 m Bohr

radius n principal quantum number - allowed energy levels En - E0 /n2 , E0

Rydberg energy

- note energy is negative, indicating that

electron is in a potential well energy

is 0 at top of well, i.e. for n ?, at

infinite distance from the nucleus.

Energies and radii in hydrogen-like atoms

- For circular orbit, potential and kinetic

energies of an electron are - U -kZe2/R K mev2/2 kZe2/2R
- Total energy E U K -kZe2/2R
- radius for stationary orbit n
- Rn n2h2/mekZe2 n2 a0 /Z
- ao h2/meke2 0.53 x 10-10 m Bohr radius
- Energy for stationary orbit n
- En - k2Z2me2e4/2n2h2 - Z2E0 /n2
- E0 k2me2e4/2h2 13.6 eV
- values of constants
- k 1/(4pe0) 8.98 109 N m2 /c2
- m e 0.511 MeV/c2
- h h/2p 1.0546 10-34 J s 6.582 10-22

MeV s - e elementary charge 1.602 10-19 C
- Z nuclear charge 1 for hydrogen, 2 for ?, 79

for Au

Ground state and excited states

- ground state lowest energy state, n 1 this

is where electron is under normal

circumstances electron is at bottom of

potential well energy needed to get it out of

the well binding energy - binding energy of ground state electron E1

energy to move electron away from the nucleus (to

infinity), i.e. to

liberate electron this energy also called

ionization energy

- excited states states with n 1
- excitation moving to higher state
- de-excitation moving to lower state
- energy unit eV electron volt energy

acquired by an electron when it is accelerated

through electric potential of 1 Volt - electron volt is energy unit commonly used in

atomic and nuclear physics 1 eV 1.6 x 10-19 J - relation between energy and wavelength E

h? hc/? , hc 1.24 x 10-6 eV m

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Excitation and de-excitation

- PROCESSES FOR EXCITATION
- gain energy by collision with other atoms,

molecules or stray electrons kinetic energy of

collision partners converted into internal energy

of the atom kinetic - energy comes from heating or discharge
- absorb passing photon of appropriate energy.
- DE-EXCITATION
- spontaneous de-excitation with emission of photon

which carries energy difference of the two

energy levels - typically, lifetime of excited

states is ? 10-8 s

(compare to revolution

period ? 10-16 s )

- Excitation
- states of electron in hydrogen atom

Energy levels and emission Spectra

- En E1 Z2/n2
- Hydrogen
- En - 13.6 eV/n2

- IONIZATION
- if energy given to electron binding energy, the

atom is ionized, i.e. electron leaves atom

surplus energy becomes kinetic energy of freed

electron. - this is what happens, e.g. in photoelectric

effect - ionizing effect of charged particles exploited in

particle detectors (e.g. Geiger counter) - aurora borealis, aurora australis cosmic rays

from sun captured in earths magnetic field,

channeled towards poles ionization/excitation of

air caused by charged particles, followed by

recombination/de-excitation

Momentum of a photon

- Relativistic relationship between a particles

momentum and energy E2 p2c2 m2c4 - For massless (i.e. restmass 0) particles

propagating at the speed of light E2 p2c2 - For photon, E h?
- momentum of photon h?/c h/?
- (moving) mass of a photon Emc2 ? m

E/c2 m h?/c2

Matter waves

- Louis de Broglie (1925) any moving particle has

wavelength associated with it ?? h/p - example
- electron in atom has ? ? 10-10 m
- car (1000 kg) at 60mph has ? ? 10-38 m
- wave effects manifest themselves only in

interaction with things of size comparable to

wavelength ? we do not notice wave aspect of us

and our cars. - note Bohr's quantization condition for angular

momentum is identical to requirement that integer

number of electron wavelengths fit into

circumference of orbit. - experimental verification of de Broglie's matter

waves - beam of electrons scattered by crystal lattice

shows diffraction pattern (crystal lattice acts

like array of slits)

experiment done by Davisson and Germer (1927) - Electron microscope

QUANTUM MECHANICS

- new kind of physics based on synthesis of dual

nature of waves and particles developed in

1920's and 1930's. - Schrödingers wave mechanics

(Erwin Schrödinger, 1925) - Schrödinger equation is a differential equation

for matter waves basically a formulation of

energy conservation. - its solution called wave function, usually

denoted by ? - ?(x)2 gives the probability of finding the

particle at x - applied to the hydrogen atom, the Schrödinger

equation gives the same energy levels as those

obtained from the Bohr model - the most probable orbits are those predicted by

the Bohr model - but probability instead of Newtonian certainty!

QM Heisenberg

- Heisenbergs matrix mechanics (Werner

Heisenberg, 1925) - Matrix mechanics consists of an array of

quantities which when appropriately manipulated

give the observed frequencies and intensities of

spectral lines. - Physical observables (e.g. momentum,

position,..) are operators -- represented by

matrices - The set of eigenvalues of the matrix representing

an observable is the set of all possible values

that could arise as outcomes of experiments

conducted on a system to measure the observable. - Shown to be equivalent to wave mechanics by

Erwin Schrödinger (1926)

Uncertainty principle

- Uncertainty principle (Werner Heisenberg, 1925)

- it is impossible to simultaneously know a

particle's exact position and momentum ?p?

?x ? h/2 h 6.63 x 10-34 J ? s 4.14 x

10-15 eVs h h/(2?) 1.055 x 10-34 J ? s

6.582 x 10-16 eVs - (?p means uncertainty in our knowledge

of the momentum p) - also corresponding relation for energy and time

?E? ?t ? h/2 (but meaning here is different) - note that there are many such uncertainty

relations in quantum mechanics, for any pair of

incompatible - (non-commuting) observables.
- in general, ?P? ?Q ? ½??P,Q??
- P,Q commutator of P and Q, PQ QP
- ?A? denotes expectation value

- from The God Particle by Leon Lederman Leaving

his wife at home, Schrödinger booked a villa in

the Swiss Alps for two weeks, taking with him his

notebooks, two pearls, and an old Viennese

girlfriend. Schrödinger's self-appointed mission

was to save the patched-up, creaky quantum theory

of the time. The Viennese physicist placed a

pearl in each ear to screen out any distracting

noises. Then he placed the girlfriend in bed for

inspiration. Schrödinger had his work cut out for

him. He had to create a new theory and keep the

lady happy. Fortunately, he was up to the task. - Heisenberg is out for a drive when he's stopped

by a traffic cop. The cop says, "Do you know how

fast you were going?" Heisenberg says, "No, but

I know where I am."

Quantum Mechanics of the Hydrogen Atom

- En -13.6 eV/n2,
- n 1, 2, 3, (principal quantum number)
- Orbital quantum number
- l 0, 1, 2, n-1,
- Angular Momentum, L (h/2?) v l(l1)
- Magnetic quantum number - l ? m ? l, (there

are 2 l 1 possible values of m) - Spin quantum number ms ?½

Comparison with Bohr model

Quantum mechanics

Bohr model

Angular momentum (about any axis) assumed to be

quantized in units of Plancks constant

Angular momentum (about any axis) shown to be

quantized in units of Plancks constant

Electron otherwise moves according to classical

mechanics and has a single well-defined orbit

with radius

Electron wavefunction spread over all radii

expectation value of the quantity 1/r satisfies

Energy quantized, but is determined solely by

principal quantum number, not by angular momentum

Energy quantized and determined solely by angular

momentum

Multi-electron Atoms

- Similar quantum numbers but energies are

different. - No two electrons can have the same set of

quantum numbers. - These two assumptions can be used to motivate

(partially predict) the periodic table of the

elements.

Periodic table

- Paulis exclusion Principle
- No two electrons in an atom can occupy the same

quantum state. - When there are many electrons in an atom, the

electrons fill the lowest energy states first - lowest n
- lowest l
- lowest ml
- lowest ms
- this determines the electronic structure of

atoms

Problems

- The solar irradiation density at the earth's

distance from the sun amounts to 1.3 kW/m2

calculate the number of photons per m2 per

second, assuming all photons to have the

wavelength at the maximum of the spectrum , i.e.

? ?max). - how close can an ? particle with a kinetic

energy of 6 MeV approach a gold nucleus? (q?

2e, qAu 79e) (assume that the space

inside the atom is empty space)

Summary

- electron was identified as particle emitted in

photoelectric effect - Einsteins explanation of p.e. effect lends

further credence to quantum idea - Geiger, Marsden, Rutherford experiment disproves

Thomsons atom model - Planetary model of Rutherford not stable by

classical electrodynamics - Bohr atom model with de Broglie waves gives

some qualitative understanding of atoms, but - only semiquantitative
- no explanation for missing transition lines
- angular momentum in ground state 0 (1 )
- spin??
- Quantum mechanics
- observables (position, momentum, angular

momentum..) are operators which act on state

vectors