1 Wave Theory of Light Line Spectra of AtomsQuantum Hypothesis Photoelectric Effect: Particles of Light Bohr’s Explanation of Line Spectra de Broglie’s Matter Waves Schrodinger Equation: A Wave Equation of Particles Bohr-de Broglie Model of Hydrogen Atom 

2 Chapter 2. Quantum Revolution: Failure of Everyday Notions to Apply to Atoms 2.1 Wave Theory of Light wave –traveling or standing wave wave length λ – distance between successive crests or troughs wave number : = # of waves that fit into a 1-cm length Wave amplitude A - distance from the horizontal axis to the crest frequency: = # of crests passing a point within 1 s 2.1 Wave Theory of Light wave –traveling or standing wave wave length λ – distance between successive crests or troughs wave number : = # of waves that fit into a 1-cm length Wave amplitude A - distance from the horizontal axis to the crest frequency: = # of crests passing a point within 1 s * c – speed of light ( x cm s-1)

3 Light wave consist of traveling electromagnetic waves with the same speed c, but different λ oscillating electric field E and magnetic field B traveling wave Spectroscopy - study of the interaction of light and matter

Department of Chemistry, KAIST 5 Diffraction and Interference of Light Waves Interference pattern:- shining λ through a pair of space slits, diffraction interference pattern (alternate bright and dark bands) Huygens’ wave construction: crests and troughs of the waves coming from one slit alternately reinforce or cancel those from the other. Cancellation – “out of phase” and call destructive interference. Reinforcement- “in phase” and call constructive interference

6 Spectrum of Electromagnetic Radiation X-ray (λ=1Å) ~ order of the spacing between atoms in a crystal X-ray shines on a crystal → X-ray diffraction

7 2.2 Line Spectra of Atoms Emission and absorption spectra

8 Fraunhofer’s solar spectrum with dark (i.e., absent) lines Na-D line (Flame spectra of sodium salt) gas-discharge spectrum of Hydrogen Electrical discharge emission

9 Hydrogen Atom Spectrum (difficult to explain with classical theory) Hydrogen Atom Spectrum (difficult to explain with classical theory) H-atom spectra → exhibit several distinct groups (series) of spectral lines Each series can be expressed by Rydberg Formula Rydberg constant (109, cm -1 ) n 1 =1 Lyman, n 1 = 2 Balmer, n 1 = 3 Paschen series H-atom

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11 Black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. Despite the name, black bodies are not actually black as they radiate energy as well. The amount and type of electromagnetic radiation they emit is directly related to their temperature. Black bodies below around 700 K (430 °C) produce very little radiation at visible wavelengths and appear black (hence the name). Black bodies above this temperature, however, begin to produce radiation at visible wavelengths starting at red, going through orange, yellow, and white before ending up at blue as the temperature increases. 2.3 Ultraviolet Catastrophe and Planck’s Quantum Hypothesis

12 Precise measurement by Lummer & Pringsheim → a continuous spectrum with a peak wavelength that becomes shorter and shorter as the temperature increases. However, classical electromagnetic wave theory until then (1860s) → the spectral intensity increases as λ decreases, going to ∞ at very short λ (i.e., violet). This disagreement is called “ ultraviolet catastrophe ”. Wave theory Lummer & Pringsheim (point) Planck’s quantum theory (blue curve) Blackbody radiation Blackbody radiation of heated metal (failure of classical theory)

Quantum Hypothesis (M. Nature makes a jump !! Quantum Hypothesis (M. Planck, 1900): Nature makes a jump !! Radiant energy of the light → the energy of waves could be described as consisting of small packets, bundles (calledquanta) Light energy, Planck’s constant ( x J s) (Appendix A: detailed formula) Classical energy Quantized energy where, I: spectral radiance or per unit time per unit surface area per unit solid angle per unit frequency or wavelength where, I: spectral radiance or energy per unit time per unit surface area per unit solid angle per unit frequency or wavelengthtimesurface areasolid angleenergytimesurface areasolid angle The Quantum of Energy

15 Max Planck was told in 1890s that there was nothing new to be discovered in physics !!

Photoelectric Effect: Particles of Light Shining the light on metal surface (photocathode) → photoelectrons are ejected. i) of the incident light > “threshold” of photoelectron Increase of the incident light → sudden increase of photoelectrons. The amount of photoelectrons → proportional to the brightness of the light. ii) The kinetic energy of electrons → depends only on the but not its intensity. (another clue to quantization of energy)

18 (i) Frequency and intensity dependence

(ii) The kinetic energy of electrons vs frequency of light

20 The result was a conflict with the classical electromagnetic wave Classical wave theory:  Energy of the light is proportional to its Intensity only (intense red light > dim blue light)  Waves can have any amounts of energy  Kinetic energy of electrons should increase with the light intensity

21 Einstein’s Hypothesis i) In order to overcome the conflict between the photoelectric effect (Hertz’s data) and the electromagnetic wave theory → The energy quanta is composed of packets called photons and the energy of each photon is given by, Electron emission process→ a collision between a photon and an electron embedded in the metal.

22 Einstein’s Hypothesis ii ) A photoelectric equation was proposed, The light behaves as a wave (in diffraction) and as a particle (in blackbody radiation and photoelectric emission). - wave-particle duality K= final kinetic energy of ejected electron ν o = threshold frequency below which no electrons can be ejected Work Function - Energy to free an electron from the metal

23 Millikan’s experimental data in Fig. 2.7 (10 years later)

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Nuclear Atom and the Quantum: Bohr’s Explanation of Line Spectra N. Bohr’s Stationary State Hypothesis i) An atom could only exist in certain allowed state of specific total energy, stationary state ii) An atom could make a upward or a downward jump, transition, by an absorption or an emission of a photon, respectively.

Department of Chemistry, KAIST 26 Bohr’s Model for Hydrogen Atom

27 The stationary state for the simplest atom corresponds to circular orbits of various fixed radii of the single electron. (ex. H, He +, Li 2+, Be 3+ ) The model postulates the quantization of angular momentum L as a criterion for fixing the radii of the orbits. On the basis of the model, the Rydberg constant in can be derived as where μ is the “reduced mass” of the electron-proton pair = m e m p /( m e + m p ) = m e /(1 + m e /m p ) = ( m e ) 1/ μ = 1/ m e + 1/ m p

28 However, Bohr’s Theory … Couldn’t be extended into atoms with more than one electron Bohr's theory fits a lot of experimental results, but it can’t explain why orbits are quantized and how atoms behave the way they do. If Newtonian mechanics governs the workings of an atom, electrons would rapidly travel towards and collide with the nucleus.

de Broglie’s Matter Waves: Beginning of a New Mechanics de Broglie Waves i) The integers in Rydberg formula → only possible in a constrained wave motion, as in a vibrating violin string. ii) A string tied down at both ends → vibrate with certain wavelength λ, and, L - length of string n =1 : fundamental tone n = 2, or over: overtones

30 de Broglie Wavelength From Planck’s formula E=hc/λ and Einstein E= mc 2 wavelength for a photon: Confirmed Experimentally by Davisson and Germer (US) and G.P. Thomson (England) Similarly, wavelength for a particle: X-ray scattering by Al foil Electron scattering by Al foil

31 Louis de Broglie (French) B.S. History and Science WW I, army service in radio communications Ph.D. Thesis (1924): matter wave Creator of the Wave Mechanics  Big Impact on the Modern Electron microscopes !!

32 The wave nature of the electron must be invoked to explain the behavior of electrons when they are confined to dimensions on the order of the size of an atom.

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34 Bohr-de Broglie Model of the Hydrogen Atom de Broglie de Broglie Bohr’s orbit → becomes circular standing-wave vibration of the electron wave Newton’s second law of motion for a circular orbit in cgs-e & using the Coulomb potential energy. Radii r n : Total energy: → λ e = 2πr/n = h/mv → v = nh/2πmr → v 2 = e 2 /mr quantum number

Details of derivation

37 Energy Levels for Bound States E = 0 r → ∞ free state E = -ve r = na 0 bound state allowed energy E n Bohr radius a 0

39 Hydrogen-Like Atoms In terms of a 0 and taking into account of the interaction between the electron and the nucleus charge Ze ( ex. He +, Li 2+ ) Multiply both sides of Rydberg formula by hc left hand side - △ E = hv = hc/λ right hand side – difference of two energy levels (Term values) Bohr’s assignment of the Balmer series of atomic hydrogen

40 Hydrogen Atom Spectrum H-atom spectrums → exhibit several distinct groups (series) of spectral lines Each series can be expressed by Rydberg Formula Rydberg constant (109, cm -1 ) n 1 =1 Lyman, n 1 = 2 Balmer, n 1 = 3 Paschen series H-atom

Hydrogen-Like Atoms

The Bohr explanation of the three series of spectral lines.

Schrödinger Equation: Wave Equation for Particles Schrödinger Equation: standing-wave motion of a particle of mass m under the influence of a potential V(x,y,z). The wave function Ψ(x,y,z) should take the form of sin, cos or exp. Its second derivative would take the same form as the original form. Ψ – wave function → orbital E – total energy V – potential energy

ö The Schrödinger Equation  De Broglie’s work attributes wave-like properties to electrons in atoms, and the uncertainty principle shows that detailed trajectories of electrons cannot be defined.  Consequently, we must deal in terms of the probability of electrons having certain positions and momenta.  These ideas are combined in the fundamental equation of quantum mechanics, the Schrodinger equation.  He reasoned that an electron (or any other particle) with wave- like properties should be described by a wave function that has a value at each position in space.  This wave function [Ψ(x,y,z)] is the “height” of the wave at the point in space defined by the set of Cartesian coordinates (x, y, z).  Schrodinger wrote down the equation satisfied by y for a given set of interactions between particles.

The meaning of wave function: probability density  The square of the wave function Ψ 2 for a particle as a probability density for that particle.  In other words, [Ψ 2 (x, y, z)ΔxΔyΔz] is the probability that the particle will be found in a small volume ΔxΔdyΔz about the point (x, y, z).

Department of Chemistry, KAIST 46 Schrödinger Equation: wave equation of a particle It plays a role analogous in quantum mechanics to Newton's second law in classical mechanics. The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wave function to generate the evolution of the wave function in time and space

47 An illustration of both states, a dead and living cat. According to quantum theory, after an hour the cat is in a quantum superposition of coexisting alive and dead states. Yet when we look in the box we expect to only see one of the states, not a mixture of them Erwin Rudolf Josef Alexander Schrödinger ( , Austria) 1926, Publication on the Equation 1933, Nobel Prize Schrödinger Cat

48 A. PiccardA. Piccard, E. Henriot, P. Ehrenfest, Ed. Herzen, Th. De Donder, E. Schrödinger, J.E. Verschaffelt, W. Pauli, W. Heisenberg, R.H. Fowler, L. Brillouin, P. Debye, M. Knudsen, W.L. Bragg, H.A. Kramers, P.A.M. Dirac, A.H. Compton, L. de Broglie, M. Born, N. Bohr, I. Langmuir, M. Planck, Mme. Curie, H.A. Lorentz, A. Einstein, P. Langevin, Ch. E. Guye, C.T.R. Wilson, O.W. RichardsonE. HenriotP. EhrenfestEd. HerzenTh. De DonderE. SchrödingerJ.E. Verschaffelt W. PauliW. HeisenbergR.H. FowlerL. BrillouinP. DebyeM. KnudsenW.L. BraggH.A. KramersP.A.M. DiracA.H. ComptonL. de BroglieM. BornN. BohrI. LangmuirM. PlanckMme. CurieH.A. LorentzA. EinsteinP. LangevinCh. E. GuyeC.T.R. Wilson O.W. Richardson

A Particle in a Box (Appendix B)

50 One-dimensional particle in a box with infinite energy barriers or

51 General solution or With boundary conditions therefore, B=0

52 or

53 With boundary condition x=L therefore  (L)=0

54  A ≠ 0 for ψ to be meaningful. Therefore, inside [ ] = n . Define E n to be value of E for solutions for allowed values of n Correct h to in the lecture note.

55 Quantization of energy states results from the imposition of boundary conditions.

56 Normalized wavefunctions for particle in a box (see p.104 for normalization procedure) Plots of  and  2 for the first four energy levels Which can absorb shorter λ wave between ethene and butadiene?