ε π ε λ - La Salle University by linfengfengfz

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									                            CHEM 301: QUANTUM MECHANICS
                      Chapter 1: From Classical to Quantum Mechanics

  LIGHT
   Waves                           c = λν        (ν = frequency)
    Energy of photons              E = hν
    Mass of photon                 m = E/c2 = hν/c2
    Momentum                       p = mc = hν/c = h/λ
    Blackbody (Planck)                             8π hc
                                    ρ (λ ) =
                                               λ (e hc / λkT − 1)
                                                5


  MATTER
    Classical Mechanics
       Momentum                    p = mv  (v = velocity)
       Kinetic Energy              KE = ½ mv2 = p2/2m
    Wave Mechanics
                                   λ = h/p
       deBroglie Relationship
  BOHR MODEL OF ATOM
    Energy of Hydrogen nth                 e 4 me
                                   En = − 2 2 2
       orbital                           8ε o h n
                 th
                                      ε on2h2
    Radius of H n orbital          r=          = 52.9 pm for n = 1
                                      π me e 2
                                    1       1   1          
                                      = RH  2 − 2
                                           n               
                                                                   n1, n2 =1,2,3, … n1 < n2
    Rydberg-Ritz Equation           λ       1 n2           
       (Hydrogen emission)
                                             e 4 me
                                        RH =      2
                                                    = 1.097 x 107        m-1
                                              2
  CONSTANTS (SI UNITS)

    Planck’s Constant              h          = 6.626 x 10-34 J·s
                                   ħ = h/2π = 1.055 x 10-34 J·s
    Boltzmann Constant             k          = 1.380 x 10-23 J/K
    Bohr Radius                    ao         = 5.292 x 10-11 m-1
    Speed of Light                 c          = 3.000 x 108 m/s
    Electron Mass                  me         = 9.109 x 10-31 kg
    Proton Mass                    mp         = 1.673 x 10-27 kg
    Electron Charge                e          = 1.602 x 10-19 C
    Energy Conversions             1 eV       = 1.602 x 10-19 J
                                        -1
                                   1 cm       ~ 1.986 x 10-23 J
                                      = energy corresponding to light of 1/λ (λ in units of cm)



Gentry, 2012
               CHAPTER 1: FROM CLASSICAL TO QUANTUM MECHANICS
                                 Lecture Outline


DUAL NATURE OF LIGHT
    Light sometimes behaves like waves - Maxwell’s Equations
         - Light can be described as electro-magnetic waves
                Gives rise to interference patterns
         - Wavelength (λ) and frequency (ν) depend on speed of light (c)
                c = λν , where c is independent of λ and ν
    BUT… Light sometimes behaves like particles (photons ) - Photoelectric Effect
         Einstein introduced concept of photons (light particles) in 1905
            Discrete particles with defined energy: E = hν
         Photoelectric Effect – light striking a metal electrode in a vacuum tube knocks an
                        electron off the metal, and that ejected electron has kinetic energy.
            1) Energy from light is used to ionize the electron.
            2) Any excess energy becomes kinetic energy (KE) of the electron
                                        V
                                                   hν




                                        e–
                                   electrode
                                               vacuum tube


         Dilemmas of Photoelectric Effect
            - There is minimum frequency for the light, below which there is no ejection of
                   electrons – this is due to quantum nature of atomic orbitals
            - Kinetic energy of ejected electrons depends only on frequency, not intensity of light
                   Particle model says KE of each ejected electron is due to incoming energy of a
                            single light particle striking the electrode and kicking off an electron.
                       Higher light intensity means more photons, but each of those photons still
                            have only the one energy and so ejected electrons have only one KE.
                   Wave model would allow range of kinetic energies.
            - Ejection is instantaneous with light hitting the electrode
                   Particle model says electron ejected as soon as particle hits
                   Wave model requires several wavelengths to hit electrode -not instantaneous
         Mass of photon particle
           Photons have no “rest mass”, but do have relativistic mass due to velocity
               From relativity: E = mc2      Applied to photon: E = hν therefore m = hν/c2
    Wave model is not wrong, it is just incomplete – need both models
      Particle model does not explain interference patterns
      Wave model does not explain momentum of light striking matter
      “Light is considered to act as a photon when interacting with matter, but to propagate
          through space as a wave.” (LaPaglia)
      Can never measure both types of attributes at same time – only one or the other


Gentry, 2012
Energy of Light is Quantized - Blackbody Radiation
    Blackbody radiation = light given off from a cavity inside a heated block of material
            Radiation depends only on temperature (T), not on the material
    ρ(λ) = radiation density of emitted radiation as function of wavelength, λ
            = radiated energy per unit volume of material and per unit wavelength
                                                                       5000K
                                                                                                                                         Rayleigh-




                                                                                                         Radiation density, ρ
                                            Radiation density, ρ
                               bl. body                                                                                                  Jeans law
                               radiation.
                                                                       4500K


                                                                        4000K

                                                                         3500K                                                  Experimental

                heated block
                 of material                                       0           1000       2000    3000
                                                                               Wavelength (nm)                                      Wavelength




    Rayleigh-Jeans Law - Classical Model
         Assumed
            (a) Blackbody radiation due to collection of independent oscillators operating at
                   different wavelengths (or frequencies)
            (b) Classical electromagnetism: number of oscillator modes per frequency is
                   proportional to square of the frequency
            (c) Each oscillator has average thermal energy, Eave = kT
                                                                        8π                8π kT
               Rayleigh-Jeans: ρ ( λ ) =                                   4
                                                                                E ave =
                                          λ           λ4
                        k = Boltzmann’s Constant = 1.380x10-23 J/K
             Works for long wavelengths
               ... BUT fails for short wavelengths (“Ultraviolet Catastrophe”)

    Planck’s Law (1901) - Quantum Model
       Planck empirically found that get correct result if assume light energy is quantized
          Oscillators at given frequency can only absorb and emit discrete packets of energy
              Average energy may not be surrounding thermal energy if energy steps too big

               Planck’s Law: E = n hν, n=0,1,2,...   h = Planck’s constant = 6.626 x 10-34 J·s
                                                         8π hc
               Planck’s Blackbody Radiation: ρ (λ ) = 5 hc / λkT
                                                     λ (e        − 1)

         Planck’s law becomes equivalent to Rayleigh-Jeans at long λ since
                                 x2 x3
                    ex = 1+ x +     +    + ⋅ ⋅ ⋅ and only (1+x) are important for small x
                                  2! 3!
         Planck’s solution to blackbody problem was totally empirical at the time,
            For years he resisted believing that energy was truly quantized
            He thought the error was in his equation, not in his classical picture of nature



Ch 1: Classical vs. Quantum Mechanics                                             -3-
    Why did Rayleigh-Jeans fail and Planck succeed?
         Classical model
                  Assumes that light can have any specified amount of energy no matter what
                     the wavelength or frequency – so average energy is just available kT
                  Thus, thermal energy of blackbody is able to transfer energy to all the oscillators
                     no matter what their frequency, even if energy transfer is only small
                  Once the oscillators absorb energy, the ultraviolet catastrophe is a result of the
                     1/λ4 mathematical artifact of applying Maxwell’s equations.

         Planck’s quantum model
                  Assumes high frequency (small wavelength) oscillators are limited in their
                     ability to absorb energy from the thermal body
                  Planck’s Law, E = nhν, n=0,1,2,... says that light energy can only be present in
                     discrete multiples.
                  Can’t wait around for more light to strike the oscillator to add up to minimum
                     energy. Not like holding a match for a longer time under a pot of water.
                  Planck’s Law consistent with photon particle model. Single photons have specific
                     energy, E = hν. Additional energy due to multiple photons each with same hν.
                             Classical   Quantum   Allowed Energy Levels                         Quantum Energy Levels
                              Model       Model    (gap depends on freq.)
                                                         En = n·hv
                                                                                     E


                                                                                            kT
                                                         E2 = 2·hv
                                                                                            0
                    Energy




                                                                                     ρ(λ)




                                                                                                    Blackbody Radiation
                                                         E1 = 1·hv

                                                                                                    λ


                  Actually have spread of probability for populating each of these discrete levels.
                         Thermal energy is probabilistic - not a simple yes/no cutoff

                         Boltzmann distribution: e-∆E/kT                    k = Boltzmann’s Constant = 1.38x10-23J/K




WAVENUMBERS, cm-1 [a practical note for spectroscopy]
    Wavenumbers ≡ cm-1 ≡ 1 / λ (with λ measured in units of cm)
    Wavenumbers formally have units of inverse distance, BUT used as equivalent to energy
      It is the energy that corresponds to a photon of wavelength λ.
      Energy of one photon, E = hν = hc/λ
                             1 cm-1 → 1.986 x 10-23 Joules




Ch 1: Classical vs. Quantum Mechanics                      -4-
DUAL NATURE OF MATTER

    Matter, like light, can also
       - be described alternatively as particles or waves,
       - and is quantized.
    Particle Nature of Matter
         This is the classical Newtonian model. Things of mass have substance and moving
                 objects follow classical trajectories.
         Deterministic model      (quantum mechanics is probabilistic model)
    Wave Nature of Matter
         Matter can be bent, diffracted, and have interference patterns in the same way that light
                waves can be manipulated.

         1925 – Davisson and Germer Diffraction Experiment
            The two observed interference pattern of electrons in crystalline nickel, similar to x-
                   ray diffraction patterns. Consistent with waves
         Today – Electron Microscopes
            Electron microscopes use magnetic fields as lenses to bend and focus electron waves
                   similar to optical lenses focusing light waves

         Double-Slit Diffraction
           Pass electron beam through either single or double slit
            Like light waves, an electron beam will radiate in all directions if passes through
                    narrow slit
            In case of double slits:
                Get constructive interference if both waves simultaneously have a peak (in phase)
                Get destructive interference if two waves are 180º out of phase
            Can view diffraction pattern with a TV phosphorescent screen
            NOTE: Only get interference if electron wave simultaneously passes through
                    BOTH slits!
                No interference if electron particle passes through one slit or the other slit
                    Only would see two sets of straight-through particle streams if have particles

             Single Slit                Double Slit                  Double Slit Interference

                                                                                    Constructive
                                                                                    Interference
                                                                                    (both waves peak)
                                                                                     Destructive
                                                                                     Interference
                                                                                     (waves out of phase)




Ch 1: Classical vs. Quantum Mechanics          -5-
    deBroglie Relationship (1924)
         deBroglie reasoned that matter could show same duality as light
            No experiments existed in 1924 to say electrons act like waves.
         Proposed relationship between particle (momentum, p) and wave (wavelength, λ)
            a.k.a. the “Rosetta Stone” of waves vs. particles for matter
         This is a postulate – not explicitly proved – but 80 years later continues to be consistent
                     with experimental observations
                       h                 h = Planck’s constant = 6.626 x 10-34J·s,
                 λ=                      p = classical momentum = m·v
                       p
         Relationship relates classical mechanics (particle momentum) to waves (have no mass)
         deBroglie relationship was postulated from light equations:
                             h
                E = hν =          and Einstein’s theory of special relativity (1905) E=mc2
                            cλ
               Setting the two energies equal to each other and rearranging gives:
                              h
                    λlight =
                             mc
               deBroglie postulated that matter has same equation as light, but with speed of
                    particle (v) rather than speed of light (c)
                                    h
                       λ matter =
                                    mv
         Helpful relationships from classical mechanics
                  p = mv and KE = ½ mv2 = p2/2m,
                      where KE = kinetic energy, v = velocity, m = mass,
         Examples:
             Electron with energy = 1.5eV,            wavelength = 1.0 nm
             H2 molecule with KE = kT at 300K,        wavelength = 120 pm
             Tennis ball (57g) at 80km/h,             wavelength = 5.2x10-34 m
             Hint: when in doubt when doing calculations, convert everything to SI units.




Ch 1: Classical vs. Quantum Mechanics              -6-
Quantized Nature of Hydrogen Emission
    Rydberg-Ritz Equation
      Empirical observation of emission lines from hydrogen atoms

                        1         1   1 
              cm −1 =     = RH ⋅  2 − 2  , n1, n2 = 1,2,3, …, n1 < n2
                                 n      
                        λ         1 n2 
                        RH = Rydberg constant = 1.097x107 m-1 = 109,700 cm-1

    Prior models of the Atom
       Thomson (1904) – atom was solid sphere including positive protons and negative
              electrons all distributed throughout the sphere - “plum pudding model”
       Rutherford (1911) – atom had positive nucleus surrounded by mostly empty space with
              electrons moving anywhere within that space.
       Neither model explains Rydberg emission – Why get discrete lines?
    Bohr model of Atom (1913)
       Electrons occupy well-defined orbits around the hydrogen nucleus
          Attracted to nucleus by classical electrical attraction forces:
                         qq
               E elect = 1 2    q1 = charge on electron, q2 = charge on nucleus, r = separation
                        4πε o r
       But electrons would spiral inwards if not for orbits being quantized – can only be a given
              distances from nucleus
       Accurately predicted Rydberg-Ritz hydrogen spectrum

                  Summary of Bohr Atom
                                      e4 m
                    Energy = E n = − 2 2e 2
                                    8ε o h n
                                        ε on2h2
                        Radius = r =
                                        π me e 2
                        Bohr Radius = ao = radius for lowest H orbital (n=1)
                           = 52.9 pm
                                                   e 4 me
                        Rydberg Constant = R H =        2
                                                          = 1.097*105 cm-1
                                                    2
                        n    =   quantum number for orbital, n = 1,2,3…
                        me   =   mass of electron
                        e    =   electrical charge of electron
                        εo   =   vacuum permittivity


         Problems with Bohr Model
            It only works for atoms having 1 electron
            Has non-zero angular momentum for ground state
            Doesn’t account for wave nature of electrons !



Ch 1: Classical vs. Quantum Mechanics              -7-

								
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