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Quantum physics (quantum theory, quantum mechanics)


classical physics explanation of black-body radiation failed (ultraviolet ... have in the cathode rays matter in a new state, a state in which the subdivision ... – PowerPoint PPT presentation

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Title: Quantum physics (quantum theory, quantum mechanics)

Quantum physics(quantum theory, quantum
  • 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 ltEoscgt kT
  • show that for standing waves on a string, number
    of waves in band between ? and ??? is ?n
    (2L/?2) ??

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

  • 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

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 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
  • 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
  • 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
  • Modifies cathode ray tube make metal surface to
    be exposed to light the cathode in a cathode ray
  • 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, ?
  • ? electrons will leave surface of metal with
  • E h ? W W work function
    energy necessary to get electron out of the
  • ? 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)
  • 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

  • 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
  • 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 lt 10-14 m, surrounded by electrons
    at ?10-10 m

Rutherford model
  • planetary model of atom
  • positive charge concentrated in nucleus (lt10-14
  • negative electrons in orbit around nucleus at
    distance ?10-10 m
  • electrons bound to nucleus by electromagnetic

Rutherford model
  • problem with Rutherford atom
  • electron in orbit around nucleus is accelerated
    (centripetal acceleration to change direction of
  • according to theory of electromagnetism
    (Maxwell's equations), accelerated electron emits
    electromagnetic radiation (frequency revolution
  • electron loses energy by radiation ? orbit
  • 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
  • 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)

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 gt 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
  • 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.
  • 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

Lyman n 1
Balmer n 2
Paschen n 3
  • if energy given to electron gt binding energy, the
    atom is ionized, i.e. electron leaves atom
    surplus energy becomes kinetic energy of freed
  • this is what happens, e.g. in photoelectric
  • 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

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

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
  • 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

  • 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!
  • 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 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)

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 eVs h 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 h/(2?)
  • note that there are many such uncertainty
    relations in quantum mechanics, for any pair of
  • (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
Multi-electron Atoms
  • Similar quantum numbers but energies are
  • 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

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

  • 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
  • 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)

  • 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??
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