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## Quantum physics quantum theory, quantum mechanics

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**1. **Quantum physics(quantum theory, quantum mechanics) Part 2

**2. **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

**3. **Problems from 1st lecture estimate Sun’s 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
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) ??

**4. **Outline Introduction
cathode rays …. electrons
photoelectric effect
observation
studies
Einstein’s explanation
models of the atom
Summary

**5. **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

**6. **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

**7. **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”

**8. **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)

**9. **Hallwachs’ experiments “photoelectric discharge”
“photoelectric excitation”

**10. **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

**11. **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.“

**12. **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)

**13. **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

**16. **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 = Planck’s 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

**17. **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

**18. **WHY CAN'T WE SEE ATOMS? “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

**19. **J.J. Thomson’s 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! Models of Atom

**20. **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)

**22. **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

**23. **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.

**24. **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

**25. **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.

**26. **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
values of constants
k = 1/(4pe0) = 8.98· 109 N m2 /c2
m e = 0.511 MeV/c2
h =
e = elementary charge = 1.602 · 10-19 C
Z = nuclear charge = 1 for hydrogen, 2 for ?, 79 for Au
(needed for solution of problems)

**27. **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

**29. **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 )

**30. **Excitation:
states of electron in hydrogen atom:

**31. **Energy levels and emission Spectra En = E1 · Z2/n2
Hydrogen
En = - 13.6 eV/n2

**32. ** 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;

**33. **Momentum of a photon Relativistic relationship between a particle’s 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: E=mc2 ? m = E/c2 m = h?/c2

**34. **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

**35. **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)
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!
Heisenberg’s “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 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 E. Schrödinger (1926)

**36. **Uncertainty principle Uncertainty principle: (Werner Heisenberg, 1925)
it is impossible to simultaneously know a particle's exact position and momentum ?p? ?x ? h = h/(2?) h = 6.63 x 10-34 J ? s = 4.14 x 10-15 eV·s h = 1.055 x 10-34 J ? s = 6.582 x 10-16 eV·s
(?p means “uncertainty” in our knowledge of the momentum p)
also corresponding relation for energy and time: ?E? ?t ? h = h/(2?)
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”

**37. ** 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."

**38. **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(l+1)
Magnetic quantum number - l ? m ? l, (there are 2 l + 1 possible values of m)
Spin quantum number: ms= ?½

**39. **Comparison with Bohr model

**40. **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.

**41. **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

**42. **Problems The solar irradiation density at the earth's distance from the sun amounts to 1.3 kW/m2 ; assuming all photons to have the wavelength at the maximum of the spectrum (?max = 490nm), calculate the number of photons per m2 per second.
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)

**43. **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??