Quantum physics quantum theory, quantum mechanics

1
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)
• 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

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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) ??

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Outline

• Introduction
• cathode rays . electrons
• photoelectric effect
• observation
• studies
• Einsteins 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)
• 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

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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

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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

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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)

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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)
• 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

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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 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

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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

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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 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

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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!

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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
  at ?10-10 m

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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.

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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
• 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.

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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

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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

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

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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 earths 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 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

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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

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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.
• 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!

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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)

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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

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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,
• 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

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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
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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

• 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

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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)

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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