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Quantum physics(quantum theory, quantum

mechanics)Part 2

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Summary of 1st lecture

classical physics explanation of black-body radiation failed (ultraviolet catastrophe)

Planck’s 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

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Problems from 1st lecture

estimate Sun’s surface temperature assume Earth and Sun are black bodies Stefan-Boltzmann law Earth in energetic equilibrium

(i.e. rad. power absorbed = rad. power emitted) , mean

temperature of Earth TE = 290K

Sun’s angular size Sun = 32’ show that for small frequencies, Planck’s average

oscillator energy yields classical equipartition result <Eosc> = kT

show that for standing waves on a string, number of waves in band between and + is n = (2L/2)

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Outline Introduction cathode rays …. electrons photoelectric effect

observation studies Einstein’s explanation

models of the atom Summary

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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)

Johann Hittorf (1869): discharge mediated by rays emitted from negative electrode (“cathode”) -- “Kathodenstrahlen”

“cathode rays” study of cathode rays by many physicists – find

o cathode rays appear to be particles o cast shadow of opaque bodyo deflected by magnetic fieldo negative charge

eventually realizedcathode rays wereparticles – namedthem electrons

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Photoelectric effect1887: 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:oGlass effectively shielded the sparkoQuartz did notoUse of quartz prism to break up light into

wavelength components find that wavelength which makes little spark more powerful was in the UV

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Hertz and p.e. effect

oHertz’ 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”

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Photoelectric effect– further studies

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

by Hertz:o Clean circular plate of Zn mounted on insulating stand;

plate connected by wire to gold leaf electroscopeo Electroscope charged with negative charge – stays

charged for a while; but if Zn plate illuminated with UV light, electroscope loses charge quickly

o 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)

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Hallwachs’ experiments

“photoelectric discharge”

“photoelectric excitation”

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Path to electron

1897: three experiments measuring e/m, all with improved vacuum: Emil Wiechert (1861-1928) (Königsberg)

o Measures e/m – value similar to that obtained by Lorentz

o Assuming value for charge = that of H ion, concludes that “charge carrying entity is about 2000 times smaller than H atom”

o Cathode rays part of atom?o Study was his PhD thesis, published in obscure journal

– largely ignored Walther Kaufmann (1871-1947) (Berlin)

o Obtains similar value for e/m, points out discrepancy, but no explanation

J. J. Thomson

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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.“

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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)

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More studies of p.e. effect

1902: Philipp LenardStudies of photoelectric effect

oMeasured variation of energy of emitted photoelectrons with light intensity

oUse retarding potential to measure energy of ejected electrons: photo-current stops when retarding potential reaches Vstop

oSurprises: 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:

o Assume that incoming radiation consists of “light quanta” of energy h (h = Planck’s constant, =frequency)

o electrons will leave surface of metal with energyE = h – W

W = “work function” = energy necessary to get electron out of the metal

o there is a minimum light frequency for a given metal (that for which quantum of energy is equal to work function), below which no electron emission happens

o When cranking up retarding voltage until current stops, the highest energy electrons must have had energy eVstop on leaving the cathode

o Therefore eVstop = h – W

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Verification of Einstein’s explanation

1906 – 1916: Robert Millikan (1868-1963) (Chicago) Did not accept Einstein’s explanation Tried to disprove it by precise

measurements Result: confirmation of Einstein’s theory,measurement of h with 0.5% precision

1923: Arthur Compton (1892-1962)(St.Louis):Observes scattering of X-rays on

electrons

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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 410-7 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

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J.J. Thomson’s model:“Plum pudding or raisin cake model”

o atom = sphere of positive charge (diameter 10-10 m),

o with electrons embedded in it, evenly distributed (like raisins in cake)

o i.e. electrons are part of atom, can be kicked out of it – atom no longer indivisible!

Models of Atom

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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 < 10-14 m, surrounded by electrons at 10-10 m

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Rutherford model “planetary model of atom”

positive charge concentrated in nucleus (<10-14 m);

negative electrons in orbit around nucleus at distance 10-10 m;

electrons bound to nucleus by electromagnetic force.

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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

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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:

o 1. angular momentum is quantized L = n·(h/2) = n ·ħ, n = 1,2,3,... electron can only be in discrete specific orbits with particular radii discrete energy levels

o 2. electron does not radiate when in one of the allowed levels, or “states” o 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.

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Energies and radii in hydrogen-like atoms

Potential energy in Coulomb field U = kq1 q2 /R

For circular orbit, potential and kinetic energies of an electron are: U = -kZe2/R K = mev

2/2 = kZe2/2R Total energy E = U + K = -kZe2/2R

radius for stationary orbit n Rn = n2ħ2/mekZe2 = n2 a0 /Z ao = ħ2/meke2 = 0.53 x 10-10 m = “Bohr radius”

Energy for stationary orbit n En = - k2Z2me

2e4/2n2ħ2 = - Z2E0 /n2

E0 = k2me2e4/2ħ2 = 13.6 eV (Rydberg energy or Hartree energy)

values of constants k = 1/(4πε0) = 8.98· 109 N m2 /c2

m e = 0.511 MeV/c2

ħ = h/2π = 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

(needed for solution of problems)

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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 )

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Excitation:

states of electron in hydrogen atom:

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Energy levels and emission Spectra

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111mn

R

Lyman n = 1

Balmer n = 2

Paschen n = 3

En = E1 · Z2/n2

Hydrogen En = - 13.6

eV/n2

Balmer Seriesm = 2

UVVisible

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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 earth’s magnetic field, channeled towards poles; ionization/excitation of air caused by charged particles, followed by recombination/de-excitation;

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Momentum of a photon

Relativistic relationship between a particle’s momentum and energy: E2 = p2c2 + m0

2c4

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/, = h/p “(moving) mass” of a photon:

E=mc2 m = E/c2 = h/c2 (photon feels gravity)

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Matter waves Louis de Broglie (1925): any moving particle has

wavelength associated with it: = h/p example:

o electron in atom has 10-10 m; o car (1000 kg) at 60mph has 10-38 m; o 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:o beam of electrons scattered by crystal lattice shows

diffraction pattern (crystal lattice acts like array of slits); experiment done by Davisson and Germer (1927)

o Electron microscope

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QUANTUM MECHANICS new kind of physics based on synthesis of

dual nature of waves and particles; developed in 1920's and 1930's. Schrödinger’s “wave mechanics”

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

for matter waves; basically a formulation of energy conservation.

o its solution called “wave function”, usually denoted by ;

o |(x)|2 gives the probability of finding the particle at x;

o applied to the hydrogen atom, the Schrödinger equation gives the same energy levels as those obtained from the Bohr model;

o the most probable orbits are those predicted by the Bohr model;

o but probability instead of Newtonian certainty!

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QM : Heisenberg Heisenberg’s “matrix mechanics”

(Werner Heisenberg, 1925)oMatrix mechanics consists of an array of

quantities which when appropriately manipulated give the observed frequencies and intensities of spectral lines.

oPhysical observables (e.g. momentum, position,..) are “operators” -- represented by matrices

oThe 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.

oShown to be equivalent to wave mechanics by Erwin Schrödinger (1926)

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Uncertainty principle Uncertainty principle: (Werner Heisenberg, 1925)

o it is impossible to simultaneously know a particle's exact position and momentum

p x ħ/2 h = 6.63 x 10-34 J s = 4.14 x 10-15 eV·sħ = h/(2) = 1.055 x 10-34 J s = 6.582 x 10-16

eV·s (p means “uncertainty” in our

knowledge of the momentum p)o also corresponding relation for energy and time:

E t ħ/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 (represented by “operators”)

in general, P Q ½[P,Q]o [P,Q] = “commutator” of P and Q, = PQ – QPo A denotes “expectation value”

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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."

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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, …oAngular Momentum, L = (h/2) ·√ l(l+1)

Magnetic quantum number - l m l, (there are 2 l + 1 possible values of

m) Spin quantum number: ms= ½

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Comparison with Bohr model

, 1, 2,3,zL n n

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0, Bohr radiusn

n ar a

Z

Angular momentum (about any axis) assumed to be quantized in units of Planck’s constant:

Electron otherwise moves according to classical mechanics and has a single well-defined orbit with radius

Energy quantized and determined solely by angular momentum:

Bohr model Quantum mechanics

2

2, Hartree

2n h h

ZE E E

n

, , ,zL m m l l

020

1 , Bohr radiusZ

ar n a

2

2, Hartree

2n h h

ZE E E

n

Angular momentum (about any axis) shown to be quantized in units of Planck’s constant:

Energy quantized, but is determined solely by principal quantum number, not by angular momentum:

Electron wavefunction spread over all radii; expectation value of the quantity 1/r satisfies

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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.

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Periodic table

Pauli’s 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

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Problems

Calculate from classical considerations the force exerted on a perfectly reflecting mirror by a laser beam of power 1W striking the mirror perpendicular to its surface.

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). Assume the surface temperature of the sun to be 5800K.

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)

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Summary electron was identified as particle emitted in photoelectric

effect Einstein’s explanation of p.e. effect lends further credence to

quantum idea Geiger, Marsden, Rutherford experiment disproves

Thomson’s 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”