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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-