Quantum physics (quantum theory, quantum mechanics)

Quantum physics(quantum theory, quantum mechanics) Part 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 facitsaltus” violated • Opens path to further developments

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) 

Outline • Introduction • cathode rays …. electrons • photoelectric effect • observation • studies • Einstein’s 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) • Johann Hittorf (1869): discharge mediated by rays emitted from negative electrode (“cathode”) -- “Kathodenstrahlen” “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: • Vstopdoes not depend on light intensity • energy of electrons does depend on color (frequency) of light

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(thatfor which quantum of energy is equal to work function), below which no electron emission happens • 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 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

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10-7m (violet) and 7 10-7 m (red); • diameter of atoms: 10-10m • 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. 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!

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)

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

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.

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 ·ħ, 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 • Potential energy in Coulomb field • U = kq1 q2 /R • 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 = n2ħ2/mekZe2 = n2 a0 /Z • ao = ħ2/meke2 = 0.53 x 10-10 m = “Bohr radius” • Energy for stationary orbit n • En = - k2Z2me2e4/2n2ħ2 = - 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 (neededfor solution of problems)

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 10-6eV m Ground state and excited states

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:

En = E1 · Z2/n2 Hydrogen En = - 13.6 eV/n2 Balmer Series m = 2 Visible UV Energy levels and emission Spectra

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;

Momentum of a photon • Relativistic relationship between a particle’s momentum and energy: E2 = p2c2 + m02c4 • 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)

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. • 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!

QM : Heisenberg • 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 “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  ħ/2 h = 6.63 x 10-34 J  s = 4.14 x 10-15 eV·s ħ = 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) • 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] • [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) ·√ l(l+1) • 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 Planck’s constant: Angular momentum (about any axis) shown 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 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 • 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

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)

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”

Quantum physics (quantum theory, quantum mechanics)