Wave Theory of Light

1. Chapter 2. The Quantum Revolution Wave Theory of Light Line Spectra of Atoms Quantum Hypothesis Photoelectric Effect: Particles of Light de Broglie’s Matter Waves Bohr’s Explanation of Line Spectra H=E Bohr-de Broglie Model of Hydrogen Atom Schrodinger Equation: A Wave Equation of Particles

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 amplitudeA - distance from the horizontal axis to the crest frequency: = # of crests passing a point within 1 s * c – speed of light (2.99792 x 1010 cm s-1)

3. Light wave consist of traveling electromagnetic waveswith 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

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 callconstructive interference Department of Chemistry, KAIST

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) H-atom spectra → exhibit several distinct groups (series) of spectral lines Each series can be expressed by Rydberg Formula Rydberg constant(109,677.58 cm-1) n1 =1 Lyman, n1 = 2 Balmer, n1 = 3 Paschen series H-atom

10. Department of Chemistry, KAIST

11. 2.3 Ultraviolet Catastrophe and Planck’s Quantum Hypothesis Black bodyis 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 whitebefore ending up at blue as the temperature increases.

12. Blackbody radiation of heated metal (failure of classical theory) 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)

13. The Quantum of Energy 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(6.6261 x 10-34J s) (Appendix A: detailed formula) Classical energy where, I: spectral radiance or energy per unit time per unit surface area per unit solid angle per unit frequency or wavelength Quantized energy

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

17. 2.4 Photoelectric Effect: Particles of Light (another clue to quantization of energy) 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.

18. (i) Frequency and intensity dependence

19. (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 K= final kinetic energy of ejected electron νo= threshold frequency below which no electrons can be ejected 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-particleduality Work Function - Energy to free an electron from the metal

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

24. Department of Chemistry, KAIST

25. 2.5 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.

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

27. The stationary statefor the simplest atom corresponds to circular orbits of various fixed radii of the single electron. (ex. H, He+, Li2+, Be3+) 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 = memp/(me+mp) = me/(1 + me/mp) =(0.999456 me) 1/ μ = 1/ me + 1/ mp

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.

29. 2.6 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/λandEinstein E= mc2 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 ScienceWW 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.

33. Department of Chemistry, KAIST

34. Bohr-de Broglie Model of the Hydrogen Atom 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 rn : Total energy: →λe = 2πr/n = h/mv → v = nh/2πmr → v2 = e2/mr quantum number

35. Details of derivation

37. Energy Levels for Bound States E = 0 r → ∞free state E = -ve r = na0 bound state allowed energy En Bohr radiusa0

39. Hydrogen-Like Atoms In terms of a0 and taking into account of the interaction between the electron and the nucleus charge Ze ( ex. He+, Li2+) 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(109,677.58 cm-1) n1 =1 Lyman, n1 = 2 Balmer, n1 = 3 Paschen series H-atom

41. Hydrogen-Like Atoms

42. The Bohr explanation of the three series of spectral lines.

43. 2.7 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

44. 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.

45. The meaning of wave function: probability density • The square of the wave function Ψ2 for a particle as a probability density for that particle. • In other words, [Ψ2(x, y, z)ΔxΔyΔz] is the probability that the particle will be found in a small volume ΔxΔdyΔz about the point (x, y, z).

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 Department of Chemistry, KAIST

47. Schrödinger Cat 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 (1887-1962, Austria) 1926, Publication onthe Equation 1933, Nobel Prize

48. A. 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. Richardson

49. A Particle in a Box (Appendix B)

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