PHYS 141: Principles of Mechanics

PHYS 141: Principles of Mechanics PART THREE: QUANTUM MECHANICS PART THREE: QUANTUM MECHANICS

I. Blackbody Radiation A. Properties 1. Definition a) A blackbody is a perfect absorber of light at all wavelengths. b) Wien’s Law: The peak wavelength of emission for a bb decreases as temperature increases : lp = 2.898 x 10-3 (m-K)/T, (I.A.1) c) The Stefan-Boltzmann Law: The power = (E/s) = P/m2 of a blackbody is given by: Pbb = sAT4, (I.A.2) where s = 5.67 x 10-8 W/(m2K4).

I. Blackbody Radiation A. Properties c) The Stefan-Boltzmann Law * Further define an energy Flux, F= P/A: F = sT4 (I.A.3) • Hotter blackbodies emit more total power per area. • Properties of blackbody described principally by T. d) Radiance: radiated power as a function of wavelength and temperature (Max Planck). (I.A.4) (I.A.5-6)

I. Blackbody Radiation A. Properties Light bulb off: No light at any l. Radiance UV Visible IR Radio

max= 1000 nm I. Blackbody Radiation A. Properties The combined brightness at each l (color) determines what color a BB appears. Light bulb on: T = 3000 K. = Dim in Violet And Blue Radiance Fairly bright in Red & Orange UV Visible IR Radio

max= 500 nm I. Blackbody Radiation A. Properties Light bulb on: T = 6000 K. = Bright in Y&G Radiance Fairly bright in Red & Violet UV Visible IR Radio

max= 100 nm I. Blackbody Radiation A. Properties Light bulb on: T = 30,000 K. = Very Bright in V,I,B Radiance Bright in G,Y,O,R UV Visible IR Radio

I. Blackbody Radiation A. Properties e) The Ultraviolet Catastrophe! * Classical thermal physics predicted infinite flux and infinite intensity at small wavelengths * The solution: Planck’s hypothesis * “Electric resonators” responsible for bb radiation have discrete (quantized) energies: En = nhf. (I.A.7) where h is Planck’s constant,

I. Blackbody Radiation A. Properties B. The Photoelectric Effect • Einstein proposed the particle model of light: photons. Ephoton = hf = nhc/l. (I.B.1) 2. Extension of Planck’s work on molecular oscillators

I. Blackbody Radiation A. Properties B. The Photoelectric Effect • Einstein proposed the particle model of light: photons. Ephoton = hf = nhc/l. (I.B.1) 2. Extension of Planck’s work on molecular oscillators -e -e -e

I. Blackbody Radiation A. Properties 3. Particle-Theory of Light predictions: RIGHT! a) As the frequency of light increases, the maximum K of electrons also increases: Kmax = hf - f, (B.2) Where f is the “Work Function” that holds the electrons to the plate. b) If f < f0 = “cutoff frequency” = f/h, no electrons ejected. c) Maximum electron energy (as measured by the “stopping potential ,” VS) is independent of intensity.

I. Blackbody Radiation A. Properties

I. Blackbody Radiation A. Properties • Example: Light shines on a substance with f = 1 eV. If  = 500 nm, are electrons ejected from the surface? Step 1: Find the energy of the photons. For  = 500 nm, we have E = hc/ = (6.6 x 10-34 J-s)(3.0 x 108 m/s)/(5 x 10-7 m) = 4.0 x 10-19 J; E = (4.0 x 10-19 J)(1 eV/1.6 x 10-19 J) = 2.5 eV. Step 2: Find Kmax = 2.5eV - 1.0 eV = 1.5 eV > 0. So, yes, electrons are ejected.

II. Light, More Light A. Compton Effect (1923) • Scattered light has lower frequency than incident light. • Recall that relativistic energy is E = pc for a massless particle. Thus, the relativistic momentum for a photon is p = E/c = hf/c = h/. (A.1) ’   e-  e-

II. Light, More Light ’ Conserve momentum & energy for electron initially at rest:   e-  ’=  + (h/mc)(1 - cosq). (A.2) Dl = (h/mc)(1 - cosq). (A.3) e- and (h/mc) = C = Compton Wavelength. (A.4)

II. Light, More Light 3. Example: How much energy is lost by a 1 MeV photon that scatters off an electron with an angle q = 60o? Ei= hf = hc/ = 1.6 x 10-13 J*; = 0.0012 nm. ’=  + (h/mec)(1 - cosq). = 1.2 x 10-12 m + (2.4 x 10-12 m)(1 - .5) = 2.4 x 10-12 m = 0.0024 nm. Since the wavelength doubled, the Ef = hc/’ = 1/2Ei = 0.5 MeV. *1 MeV = 1.602 x10-13 J.

II. Light, More Light B. Light as a Wave 1. And yet: light also behaves as a wave a) Young Double Slit experiment: interference b) Maxwell’s Equations b) Connection between frequency and wavelength lf = c. (B.1) c) “Electromagnetic Spectrum” Low Energy High Energy

II. Light, More Light C. Waves of Matter? • de Broglie (1923): If light is observed to have wavelike properties some times (interference) and particle-like properties other times (photoelectric effect), then what about matter?  = h/mv “de Broglie Wavelength” (I.C.1) • Example: what is  for an electron moving at .01c?  = (6.6 x 10-34 J-s)/(9.11 x 10-31 kg*3.0 x 106 m/s);  = 2.4 x 10-10 m = 0.24 nm.

II. Light, More Light D. Atomic Spectra 1. Spectral Analysis • a) SA: The identification of a chemical substance by its unique spectral lines. • b) Joseph Fraunhofer (1815) : Hundreds of dark lines in the Solar Spectrum • c) The Value of Spectral Analysis • Composition, abundance • Temperature • Motion

II. Light, More Light 2. The visible Hydrogen series (Balmer Series) 1/l = R(1/22 -1/n2), n = 3, 4, 5… (I.D.1) Where R = 1.097 x 107 m-1 is the Rydberg constant. Balmer lines 656 nm (B, n = 3). are transitions 486 nm (B, n = 4). to/from n = 2 434 nm (B, n = 5).

II. Light, More Light E. The Bohr Model 1. General aspects of the model a) Hydrogen Atom as simple example b) Electron orbits are quantized; not all orbits stable. c) To move from one orbit to the next, an electron needs to absorb or emit an exact DE. d) Larger orbital differences mean more energy required to move e) Packet of energy: a photon! f) Model later modified from strict orbits about the nucleus to “energy levels”

II. Light, More Light • 2. The mathematics of quantization • a) Frequency of radiation absorption/emission: Eu - El = hf. (I.E.1) • b) Quantum condition: only specific values of electron orbital angular momentum allowed: L = mevrn = n(h/2p), n = 1, 2, 3, … (I.E.2) Eu hf e- El

II. Light, More Light c) Electrostatic Force: F = kq1q2/r2 (I.E.2a) PE = -kq1q2/r. (I.E.2b) with k = 8.99 x 109 Nm2/C2 is the Coulomb constant and the unit of charge is the Coulomb: e = 1.602 x 10-19 C. q2 r q1

II. Light, More Light d) Total energy of hydrogen atom: PE = -ke2/rn. (Electric Potential energy) (I.E.3) KE = (1/2)mev2. (I.E.4) Thus, E = PE + KE = -ke2/r +(1/2)mev2. Now assume that the electron orbit is circular: F = ke2/rn2 = mev2/rn = 2rn(KE), so (I.E.5) KE = ke2/(2rn), and (I.E.6) E = -ke2/(2rn). (I.E.7) +e rn -e

II. Light, More Light e) Orbital sizes: use the quantum condition (E.1): rn = n(h/2p)/mev, and rn2 = n2(h/2p)2/me2v2. From (E.5), ke2/rn2 = mev2/rn, so v2 = (ke2/me)/rn. Thus, rn = n2(h/2p)2/(ke2me). (I.E.8) Define “the Bohr radius” = r1 = r(n=1) = 0.0529 nm, so rn = n2r1. (I.E.9)

II. Light, More Light • Energy levels of orbits: En = -{mek2e4/(2(h/2p)2}(1/n2), or (I.E.9) En = (-13.6 eV)/n2 = E1/n2. (I.E.10) Energy required for ionization: DEi = E(n = infinity) + (13.6 eV)/n2 = (13.6 eV)/n2. (I.E.11)

II. Light, More Light F. Quantization and atomic spectra (H) • Ground state: n = 1, E1 = -13.6 eV. • First excited state: n = 2, E2 = -3.4 eV. • Energy required to move from E1 to E2: DE = -3.4 eV - -13.6 eV = 10.2 eV. • Hydrogenic atoms: Z = number of protons in nucleus. En = (Z/n)2E1. (F.1) rn = (n2/Z)r1. (F.2)

II. Light, More Light • Photon frequency/wavelength: f = DE/h = [{meke2e4}/{4p(h/2p)3}](1/nf2 - 1/ni2), or f = cR(1/nf2 - 1/ni2), and (I.F.3) 1/ = R(1/nf2 - 1/ni2). (I.F.4)

II. Light, More Light • Quantization and atomic spectra • Spectral types explained! Unique electron orbits => unique energy differences => unique patterns Absorption/Emission spectra understood as upward/ downward transitions of electrons.

II. Light, More Light G. The Wave Nature of Matter (revisited) 1. de Broglie’s wave description of electrons can now be understood in terms of standing waves surrounding a nucleus: only waves that close back on themselves can constructively interfere and “survive”

III. Quantum Mechanics • The Failure of the Bohr Model 1. Couldn’t be applied to more complicated atoms 2. Couldn’t explain “fine structure” 3. Couldn’t explain solids, liquids, molecules 4. Quasi-classical mechanics The Bohr Model, while substantial, was incomplete.

III. Quantum Mechanics B. Wave Mechanics • Define a Wave Function, , that describes a particle. a) Wave function contains information about a particle’s state: position, speed, momentum & energy, spin, etc. b) “Matter Wave” or “Matter Field” c) Depends on position and time: (x,y,z,t). d) Probabilities:  = “Probability density” 2 ~ Probability of finding an electron at position (x,y,z) and time t.

III. Quantum Mechanics B. Wave Mechanics • Calculations: Solution to the “Schroedinger Wave Equation” a) One dimensional case, Cartesian coordinates. Probability of finding a particle between x and x + dx b) Normalization: The particle must be SOMEWHERE. Therefore

III. Quantum Mechanics B. Wave Mechanics • Calculations: Solution to the “Schroedinger Wave Equation” c) Radial Probability distribution. Probability of finding a particle between r and r + dr d) Normalization: The particle must be SOMEWHERE. Therefore (B.1) (B.2)

III. Quantum Mechanics B. Wave Mechanics 3. Example: Ground State of Hydrogen a) Wave function for ground state b) Step 1: Find the constant by normalization (B.3) (B.4a,b) (B.5)

III. Quantum Mechanics B. Wave Mechanics 3. Example: Ground State of Hydrogen c) Now ask: What is the probability of finding an electron inside the Bohr Radius, a0? (see pg. 649 and Appendix A the for integral)

III. Quantum Mechanics C. HUP • Heisenberg Uncertainty (Indeterminancy) Principle • There is an inherent uncertainty in pairs of correlated variables • Momentum-Position Uncertainty Principle xp ≥ (h/2). (III.C.1) The original argument comes from measurement of an electron. The uncertainty in the electron’s position is roughly the wavelength of observation (), and the photon will give some unknown portion of its momentum to the electron (h/) when it strikes the electron. Trying to narrow the position means using smaller , but that imparts more (unknown) momentum to the electron.

III. Quantum Mechanics C. HUP 3. Energy-Time Uncertainty Principle Et ≥ (h/2). (III.C.2) In this UP, the energy of a particle is uncertain (E), over a time scale t ~ (h/2)/E. Warning: Part of the problem is that we tend to view electrons, etc. exclusively as particles. Indeterminancy is a direct result of electrons, protons, etc. having both wave AND particle properties.

III. Quantum Mechanics C. HUP • Example: What is the maximum precision for measuring the position of an electron moving with a speed equal to v = (3.00 ± 0.01) x 106 m/s? xp ≥ (h/2) => xmin = (6.6 x 10-34 J-s/2)/[(9.1 x 10-31 kg)(104 m/s)], = 1.2 x 10-8 m, = 12 nm, or about 100x the size of an atom.

III. Quantum Mechanics C. HUP 5. Example: What is the lifetime of a particle with a spread in energy equal to 1kev? Et ≥ (h/2) => t = (6.6 x 10-34 J-s/2)/[(1000eV)(1.6 x 10-19 J/eV)], = 6.6 x10-19 s.

h h III. Quantum Mechanics D. Quantum Numbers • Principal quantum number: n (e.g., En = E1/n2). a) Related to energy of electron b) n = 1, 2, 3, …  • Orbital Quantum number: l a) related to orbital angular momentum vector of electron L = {l(l +1)}1/2 , where = h/(2), and (D.1) l = 0, 1, 2, …, n-1. (D.2)

h h h III. Quantum Mechanics D. Quantum Numbers 3. Magnetic quantum number: ml. a) Related to direction of electron’s angular momentum - l, - l + 1, - l + 2, …, 0, … l - 2, l - 1, l. (D.3) b) Angular momentum along the “z-axis:” Lz = ml(D.4) Lz ml = 1. Total number of states for a given value of l depends on projection Lz. 0 ml = 0. - ml = -1.

III. Quantum Mechanics D. Quantum Numbers 3. Magnetic quantum number: ml. c) Energy levels are split in the presence of magnetic field. ml = 1. n = 2 l = 1 ml = 0. ml = -1. n = 1 l = 0 ml = 0.

III. Quantum Mechanics D. Quantum Numbers 4. Spin quantum number: ms. a) originally thought of as intrinsic spin of electron b) ms = ±1/2, but s = ½ always (for an electron) c) “fine structure” Electron spin vector is given by (D.5). n = 2 l = 0 ml = 0. n = 1 l = 0 ml = 0.

PHYS 141: Principles of Mechanics