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-30Section 9: f" Partial derivatives: Differentiate with respect to one variable, treating the others as constants: example: if z = x + x2y3 z = 1 + 2x y3 x z = 0 + x2 (3y2) y 3-Dimensional Square Well: I will solve Schrödinger's equation for 0 if 0<x<L, and 0<y<L, and 0<z<L U(x,y,z) = elsewhere Main points to notice in the solution: - The one equation containing x, y and z can be separated into three equations with one variable each. - ψ is the product of the solutions of the three equations. - Separation introduces an arbitrary constant into each equation, the sum of whose squares is related to energy. - Imposing the boundary conditions (ψ = 0 at each wall of the box) restricts these constants, and therefore the energy, to certain values. - The separation constants are proportional to positive integers called the system's quantum numbers. These appear in the system's wave function, so different numbers give you different ψs, which define different states. Ex. 9-1: An electron is confined to a cube 2.00 Å on a side. Find a. the energy of the two lowest energy levels, and b. how many states have each of those energies. Hydrogen: The solution of Schrödinger's equation for a particle in a 3-d box, and the kind of behavior found in the answer, includes all the main features of real hydrogen. I will not show you the details, because they are more complicated than for a particle in a box. To summarize: Put U = into Schrödinger's equation. Separate and solve, something like 3-d square well. Apply boundary conditions (different conditions this time). -31ψ = nasty looking function which includes E (energy) and n, l and ml , constants analogous to nx, ny and nz of square well. Boundary conditions restrict these numbers to values I will put on board. "Subshells" are sometimes lettered: s state means l = 0, p state means l = 1, d state means l = 2, etc. Ex. 9-2: Can a 2d state exist? Ex. 9-3: Neglecting spin, how many states are there with n = 3? What is their energy? Meaning of the quantum numbers: n describes the quantization of energy. Angular momentum, a vector, is also quantized: - l describes the quantization of ⃑ ’s magnitude. - ml describes the quantization of ⃑ ’s direction. Allowed values for L, Lz and θ. Ex. 9-4: Find L and the allowed values of Lz and θ for a p state. Hydrogen’s wave functions: The general expression giving for any n, l and ml is too big and ugly. Just look at a 1s state: a0 = Bohr radius Probability distribution: = probability per unit volume. Probability per unit radius, P(r), is often more useful: 2 Probability it’s in volume dV = dV 2 = (4r 2 dr ) 2 P(r) 1 2a r P(r ) 3 e 0 a 0 P(r)dr = Probability it’s in interval dr wide. 4r 2 Example: 1s state: -321s: More accurate than the Bohr model is to think of the electron as a charged cloud whose density is given by . You could think of a pointlike electron darting around so fast that it looks like a cloud, similar to a fan’s blades looking like a disk when going fast. But unlike a fan, an electron's position at some moment is unpredictable. The reality of something unknowable is highly questionable, so you might as well just think of the electron as spreading out into a cloud when confined to an atom. (And temporarily collapsing to a point if you measure its position.) 2 Ex. 9-5: Find the most probable distance of the electron from the nucleus in a 1s state. Spin: - An electron has angular momentum, as if it was a spinning sphere. Magnitude of ⃑ : Same formula as for L, but quantum number corresponding to l can only be ½: √( ) ( ) = √ Direction of ⃑ : ms, the quantum number corresponding to ms, can only be +1/2 or -1/2. Magnetic moment, ⃑ : (⃑ = I⃑ from PHY 132. Spinning charge amounts to current loops.) It can be shown that . The z component: = +9.27 x 10-24 J/T (lab 9) - Explains why some spectral lines are actually two close lines, and experiments such as Stern & Gerlach’s (See text.) Ex 9-6: Not neglecting spin, how many states are there with n = 3? -33- -34Section 10: Electromagnetic Waves EM waves, such as light, work by induction: A changing electric field induces a changing magnetic field which induces a changing electric field which ... The process is set off by accelerating charges. Example: Oscillating dipole. (The left side of the diagrams has been omitted.) Turn on AC source at t = 0, then: Then, Ultimately: -35Maxwell's Equations: The basic equations of electromagnetism, from which everything else can be derived: By analogy to y = A cos(kx - ωt) from last fall, the fields making up this wave would be given by: (1) E = Emax cos (kx - ωt) k = 2π/λ (2) B = Bmax cos (kx - ωt) ω = 2πf (⃑ and ⃑ are both to the direction of propagation.) Is this model consistent with Maxwell's equations? The two Gauss's laws describe static fields, and so don't apply here. Convert the other two to a differential form: (Er - El)h = - (dB/dt)(h dx) dE dE = – dB dx dt Notation correction: E and B are functions of more than one variable, so these are partial -36derivatives. (Differentiate with respect to one variable, treating others as constants. Notes p.31) so, E = – B x t (3) B = –μ0ε0 E x t Similarly, the Ampere-Maxwell equation (4) (The point of that was to convert the relevant Maxwell equations from the form using intergals into this differential form.) I will show in class how these follow from equations 1 – 4: c= and c= √ Ex. 10-1: The E vector in an electromagnetic wave varies according to E = (600 V/m)cos[(1.2 x 107 m-1)x – (3.6 x 1015 s-1)t]. Find: a. the frequency b. the wavelength c. the expression for B. Electromagnetic Spectrum: low f (or long λ): Radio waves Microwaves Infrared Visible Ultraviolet X - rays high f (short λ): gamma rays Roy G. Biv Energy in EM waves: Wave's energy = energy of the E and B fields it's made of. Notation: U = energy, u = energy/volume Adding the expressions from Phy 132 sec 11, u = ½ε0E2 + B2/(2μ0) Substitute in B = E/c and c2 = 1/(μ0ε0), do some algebra: u = ½ε0E2 + ½ε0E2 -37u = ε0E2 (instantaneous value) As with RMS current & voltage (PHY 132), (E2)av = ½Emax2, so uav = ½ε0Emax2 definition: Intensity: I = EM wave energy density (J/m3) U At (Watts/m2) U = energy, A = area, t = time This suggested (to John Poynting in 1884) a vector whose magnitude is the wave's intensity, and whose direction is the wave's direction of propagation: ⃑ ⃑ ⃑ The Poynting vector Sav = intensity Ex. 10-2: 1.0 watt of power is radiated by the bulb as light. At point P, find the average values of I, S, E and B. -38Momentum and Radiation Pressure: When the wave hits this stationary charge, the electric force, FE, pushes it up and down. Once in motion, it also feels a magnetic force, FM. Over a cycle, cancels , and there is no net electric force. But, with the magnetic force, Thus, the charge gains momentum, which must have come from the wave. If you put ⃑ ⃑ into ⃑ ⃑ (from Phy 131) and make some other substitutions: Momentum of an EM wave: p = U/c A perfect absorber A perfect reflector gains momentum = U/c feels pressure = S/c U = wave's energy S = Poynting vector gains momentum = 2U/c feels pressure = 2S/c Ex. 10-3: If the acceleration due to the Sun's gravity is 5.93 x 10-3 m/s2, and Sav = 1340 W/m2, what is the maximum radius of a dust particle which is repelled by radiation pressure at least as strongly as it is attracted by the Sun's gravity? (Assume the particle's density is 1.0 g/cm2, and that it is very shiny.) -39Section 11: Multi-electron atoms: There are similar states to hydrogen’s, described by the same four quantum numbers. But now, more than one of those states has an electron in it at a given time. Pauli Exclusion Principle: No two electrons in an atom can be in the same state. (Each electron must have a different set of quantum numbers.) So, electrons settle, one per state, into the lowest energies available. A set of values for n, l and ml is called an orbital. Hund's rule: Given a choice of orbitals with equal energy, electrons usually arrange themselves for a maximum number of unpaired spins. For example, nitrogen is not These rules explain the periodic table: similar chemical properties recur as similar outer electron configurations repeat. Ex. 11-1: a. What is the electronic configuration for Carbon (Z = 6)? b. What are n, l, ml and ms for each electron? Molecules: A gas of individual atoms has just translational and electronic energy. Molecules can also rotate and vibrate. Like the electronic energy, rotational and vibrational energies are quantized, which caused classical physics to fail in dealing with them (Equipartition Theorem). Rotational Energy (½Iω2): Diatomic molecule: (simplest example) Reduced mass Moment of inertia Energy levels and the ΔE between them -40Ex. 11-2: The J = 4 to J = 3 rotational transition in HCl produces a line at 120.3 µm. Find the distance between the H and Cl nuclei. (mCl = 5.81 x 10-26kg, mH = 1.67 x 10-27kg) (Atomic mass unit: 1 u = 1/12 mass of a 12C atom = 1.66 x 10-27 kg) Vibrational Energy: ΔE between energy levels Ex. 11-3: An HCl molecule gives off an 8.66 x 1013 Hz photon when it goes from v = 1 to v = 0. Find the maximum speed of the H atom relative to the Cl in the ground state. Solids. (Many atoms in a regular array.) Energy bands: When close, atoms disturb each other's energy levels. The atoms’ slightly different levels, taken together, form bands. (Each band is actually many very close levels.) example, sodium at 0 K: N = number of atoms This partly filled band is what makes metals good electrical conductors. If electrons are going to flow when you turn on an electric field, they have to be able to gain kinetic energy: In a conductor, there are higher energy states they can move into. In an insulator, no states of the right energy are available: (Conductor: Highest band partly full. Insulator: one full, next empty.) Above 0 K, some electrons gain thermal energy and jump above EF: Fermi - Dirac distribution function (Probability that a state of energy E has an electron in it.): -41- f(E) = ⁄ ≈ 1 if E significantly < EF, ≈ 0 if E significantly > EF. EF = Fermi Energy, k = Boltzmann Constant, T = absolute temperature Density of States (No. of States Per Unit Volume with Energy Between E and E + dE): g(E)dE = CE½dE 3 8 2m 2 C h3 = 1.062 x 1056 J-3/2·m-3 = 6.812 x 1027 ev-3/2·m-3 for electrons N(E)dE = number of electrons per unit volume, between E and E+dE = (no. of states per V between E and E+dE)(fraction of states containing e-s) g(E) f(E) N(E)dE = f(E)g(E)dE √ ( )⁄ n = density of charge carriers. (Number of free electrons per unit volume, in a metal.) n= ∫ I will use this to show that EF = ( ) Ex. 11-4: Aluminum’s density is 2.70 g/cm3, and its atomic weight is 26.97. Each atom contributes three free electrons. Find (a) the charge carrier density, (b) the Fermi energy, (c) the Fermi velocity. Section 12: Ex. 12-1: In .05 m3 of aluminum at 300 K, approximately how many free electrons have energies between 11.900 and 11.901 eV? -42Ex. 12-2: In the same piece of aluminum, calculate the number of conduction electrons with energies below 11.0 eV at 300 K. Semiconductors (ex: silicon) Doping (adding impurity atoms): Donor: An atom with too many electrons to fit into lattice gives one off. n-type semiconductor: Most charge carriers are electrons. Acceptor: an atom which takes an electron on. p-type semiconductor: Most charge carriers are holes. p-n junction: Free charges leave junction. Very small I because few free charges are left to flow. Lots of charges flow in, so I is big. The junction is a diode, conducting in only one direction. (It rectifies: converts AC into pulses of DC.) Some holes get stuck on donor atoms in the thin center layer. They repel additional current. The more Ib drains off, the larger Ie and Ic. So small base current controls large e to c current. -43Used as amplifiers, and as "switches" for digital circuitry. Superconductors: Normal resistivity is due to scattering of individual electrons (by lattice defects and phonons). At low temperatures, electrons can bind into Cooper pairs: The pair's total spin is zero ( + ), so Pauli principle doesn't apply. They all go into the same state, and act collectively. Defects, etc, unable to scatter them all at once. There is also a critical magnetic field, above which resistance conducting returns. Magnetic effects, such as Bc, also limit the current density to a critical value, Jc. (J = current/area) Meissner effect: A superconductor expels magnetic fields from its interior. This is done by developing currents on its surface, which cancel the external field. ex: Magnetic levitation: -44The Nucleus: Discovery - Rutherford scattering (1911): Number of alpha particles scattered through large angles by atoms in a gold foil indicated most of the atom's mass is concentrated in a small positive nucleus. r = r0A1/3 Nuclear radius: where r0 = 1.2 fm A = mass number = Z + N Z = atomic number = number of protons N = number of neutrons Isotopes of an element have same Z but different A. A example: Z (ordinary H) (deuterium) (tritium) Ex 12-3: Find the radius and density of an 56Fe nucleus. Binding energy = (c2)(difference between mass of nucleus and particles making it up) Ex: Binding energy of a deuteron - see section 5. Eb (in MeV) = (ZmH + Nmn - matom)(931.5) (masses in atomic mass units: 1 u = 1/12 mass of a 12C atom) mH = mass of 11H = 1.007825 u mn = mass of neutron = 1.008665 u Ex 12-4: Find the average binding energy per nucleon of 168O if the mass of one atom is 15.994915 u. -45Review for Exam 3: 1. What would be the average intensity of an electromagnetic wave in which each cubic meter contained 5.00 J of energy? 2. When a CO molecule changes from the J = 4 to J = 3 rotational state, it emits a 1.91 x 10-3eV photon. What is this molecule's rotational kinetic energy in the J = 3 state? 3. In a metal where the Fermi energy is something greater than 2 eV, how many electrons per unit volume have energies between 0 eV and 2 eV at T = 0 K? 4. The ground state wave function of hydrogen is ψ(r) = (π a03)-1/2e-r/a, where a0 is the Bohr radius. From this wave function, show that the most probable distance of the electron from the nucleus is equal to the Bohr radius, a0. 5. Short answer, 5 points each: a. The energy bands of a certain solid, near absolute zero, are as shown. Is this material an insulator, semi-conductor, or a conductor? b. Consider a free electron in empty space with light falling on it. i. At an instant when the electron is at rest, is the force on it in the direction of ⃑ , ⃑ , or ⃑? ii. Averaged over many cycles of the wave, is the force on the electron in the direction of ⃑ , ⃑ , or ⃑ ? c. Which quantum number (n, l, ml or ms) describes the quantization of the direction of an electron's orbital angular momentum? d. When the electrons in a substance bind into Cooper pairs, what does the substance become? e. Classical physics and the equipartition theorem predict the specific heat of H2 gas (at constant volume) to be larger than what is actually measured at low temperatures. Why is it less than predicted? -46Section 13: Nucleons (protons & neutrons) attract each other by the strong force. Alpha decay: Strong force's range is very short; can't hold back Coulomb repulsion if nucleus is too big. alpha particle Mass numbers add up to same thing on both sides. (conservation of mass) Atomic numbers add up to same thing on both sides. (conservation of charge) Ex 13-1: Write the equation for the alpha decay of Radium 226. Neutrons and protons can decay into each other by the weak force. (Therefore, N Z) Beta decay: antineutrino neutrino ( : Greek "nu") Beta particle is an electron or positron (anti-electron). Neutrino: Does not feel strong force, electromagnetism (no charge), or gravity (little or no rest mass). Therefore, it interacts very weakly with matter. Gamma decay: α or β process, or a collision, excites a nucleus. level by giving off a photon: ( * for excited state.) γ rays are the most penetrating, α's the least. Decay rate, R. Relationships between N (number of nuclei), R, and time. Half-life (T1/2) = time for half of original nuclei to decay. In class, I will show that T1/2 = (ln 2)/λ Then, it drops to a lower energy -47Ex 13-2: Radioactive dating: The ratio of 14C to 12C in all living things is 1.3 X 10-12. The half life of 14 C is 5730 years. If a 100 gram piece of charcoal has an activity of 17 Bq, how old is it? Disintegration energy/ reaction energy: Disintegration: X Y + α, X Y + β + ν, etc. Reaction: Shooting some particle, a, at a nucleus can transmute it into another element: a + X Y + b + ... example: Bombarding uranium, the heaviest natural element, with neutrons can build it up into heavier elements which no one had seen before the 1930's. Q = (total m before - total m after)c2 (c2 = 931.5 MeV/u) (Q = KE of Y & other particles, energy of γ rays, etc.) Ex 13-3: Find the Q value for the α decay of 222.017574 u, 4He: 4.002603 u. 226 Ra. Mass of 226 Ra: 226.025406 u, 222 Rn: If Q is positive, the process can happen spontaneously. If Q is negative, the process can’t happen spontaneously. Nuclear reactions: Fission: Split a nucleus into smaller ones. Fusion: Combine nuclei. Fission: example: In 1939, it was discovered that a bombarding neutron sometimes splits a uranium nucleus, releasing energy: 1 0n + 23592U 236 92U lasts about 10-12s X + Y + neutrons fission fragments (Mass no. & atomic no. must add up to same thing on both sides.) Chain reaction: The neutrons released go on to split other 235Us, releasing even more neutrons, etc. examples: original "A" bomb, nuclear reactors. Critical mass: The minimum mass needed to sustain a chain reaction. (If too few 235Us are around to be split, neutrons escape from sample faster than new ones are released.) -48Fusion: example: Proton - proton chain (The Sun's main process. Requires great temperature and/or pressure.): H + 1H 2H + e+ + ( = neutrino) H + 2H 3He + γ (γ = photon) 1 1 H + 3He 4He + e+ + 1 or He + 3He 4He + 1H + 1H 3 Other examples: H - bomb. Maybe someday, fusion reactors. Effects of radiation: Ionizing radiation damages molecules in cells. If the cell can't repair the damage: - Radiation sickness: Many dead/damaged cells, causing blood abnormalities at a certain dosage; nausea, hair loss, etc at a higher dosage; death at a still higher dose. (It’s kind of like a sunburn that goes more than skin deep.) - Cancer: Damaging a cell's genetic material makes it divide out of control. -Birth defects: Damage to genes in reproductive cells can cause a mutation which is passed down to all future generations. There may be no "safe" level of radiation. Some people claim there is even a small chance of the above from natural background radiation. (Others disagree.) unit: 1 rad = .01 J kg amount of energy absorbed amount of material absorbing it The same number of rads of different kinds of radiation causes different amounts of damage: RBE = Relative Biological Effectiveness = How many rads of x-rays would produce the same damage as 1 rad of the radiation being used. def.: Effective dose in REM = (dose in rad)(RBE) Ex 13-4: A tumor which is ordinarily given a dose of 1000 rad from a Co-60 source (γ rays with RBE = .7) is to be treated with neutrons having an RBE of 3.0. How many rads are needed? -49Section 14: Elementary particles: Starting in the 1930's, hundreds of subatomic particles were discovered, using cosmic rays and particle accelerators. Leptons (ex: e-): Do not feel the strong force. (low mass) Hadrons: Do feel the strong force. Mesons: spin = 0 or 1 (medium mass) Baryons (ex: p+ & n): spin = 1/2 or 3/2 (heaviest) Conservation Laws: Charge, momentum, etc, and: Baryon number: B=1 for baryons, B= –1 for antibaryons, B =0 for other particles Lepton numbers: Le = 1 for e- and , –1 for e+ and , 0 for others. L = 1 for e- and , –1 for + and , 0 for others. L = 1 for e- and , –1 for + and , 0 for others. (Bar over the top means antiparticle.) Ex 14-1: Which reactions can occur, and why? a. p + n p + p + e + b. p + n p + p + ̅ c. p + n p + p + e d. p + n p + p + e + Since hadrons - are very numerous, - are heavy, with measurable diameters, - have patterns in their properties (see text), this suggests they are made of still smaller quarks. So, the most basic building blocks of matter would then be: -50- (And each has an antiparticle: Same mass, but charge and some other properties opposite.) Forces between particles: Due to exchange of field particles. (Analogy: Two kids on skateboards, with boomerangs. To repel, they throw the boomerangs at each other and catch them. To attract, throw them away from each other, they boomerang around, come up behind each kid, and they catch them.) Interaction Property “feeling” interaction Gauge Boson (field particle) Gravitation Mass Graviton (undetected) Electromagnetism Electric charge Photon Strong Color Gluons Weak Weak charge W+, W- & Z (At close range, gravitation is weakest, then Weak, EM, and Strong is strongest.) Example: Electromagnetic repulsion between electrons: (Exists only as much time as uncertainty principle allows "violation" of conservation of energy.) Weak interaction: Affects hadrons and leptons. (Ex: Causes β decay) Strong (or "color") force: Binds quarks into hadrons, and hadrons into nuclei. Color (charge-like property): Red, green & blue. Analogous to negative charge: antired, antigreen & antiblue. -51Whole hadrons are colorless: Meson = a quark & antiquark. (Colors cancel) Baryon = 3 quarks, or 3 antiquarks. (R + B + G = white) Ex 14-2: Give the color and flavor of the other quark in a a. π+ containing a red u quark b. antiproton containing an antiblue u & antired d. Unification of forces: Considerable work is currently being done in trying to develop a single "Theory of Everything." (Something like the unification of electricity with magnetism in the 1800's.) A successful electroweak theory was published in the 1970's. Grand unification theories which include the strong force exist, but await experimental evidence. Including gravity is hardest. General Relativity: Principle of equivalence: Being in a gravitational field is equivalent to being in an accelerated frame of reference: No force acts on the objects on the left. No experiment can distinguish that situation from the one in the center. So, gravity is a pseudo force, similar to a centrifugal “force”. The “force” of gravity is due to the way a mass distorts spacetime around it: Consequences of G. R.: - Bending of light in strong gravitational fields. - Slowing of time in strong gravitational fields. - Gravitational redshift. - Other. -52Example: Clocks on GPS satellites: From , an object raised 20 200 km above Earth gains an energy per unit mass of 7 ΔU/m = 4.77 x 10 Fraction its energy increases: It’s the same for photons: By T = 1/f, period changes by same factor as f. Any other kind of vibration speeds up similarly, so a clock gains 5.3 x 10-10 day each day due to G. R. (46 μs) Nanosecond accuracy is needed, so this must be taken into account. Cosmology: Universe began with a big bang 13.8 billion years ago (+ ½%), and has been expanding since. Hubble's law: v = HR v = speed object moves away from us R = distance H = Hubble constant .022 m/s per light-year Ex 14-3: The Doppler shift of a certain galaxy's spectrum indicates a speed of 3.0 x 106 m/s. How far is it from Earth? The Universe cools as it expands. t = 10-40s: Universe was probably an ultrahot "quark soup," with the four forces indistinguishable. The forces became distinct, one by one, as it cooled. By about 10-12s, temperature & density were down to where the laws of physics are well understood. t = a few minutes: Temperature low enough for hadrons to bind into nuclei. Elements predicted by model match observation. t = a few hundred thousand years: Electrons and nuclei bind into neutral atoms, making universe transparent. Radiation moving freely since then has cooled into the cosmic microwave background. (Observation of this was confirmation of big bang model.) -53Review for Final Exam: 1. Consider the nuclear reaction 2H + 6Li 24He. The masses of these nuclei are: 2H, 3.344019 x 10-27 kg; 6Li, 9.98664 x 10-27 kg; and 4He, 6.64558 x 10-27 kg. Assume the hydrogen and lithium nuclei collide at very low speeds, meaning that their initial energies are entirely in the form of mass. a. What is the total kinetic energy, in joules, of the helium nuclei after the reaction? b. If each 4He gets half of this energy, what is their speed? 2. Assume that a clarinet is a cylindrical tube full of air at room temperature (speed of sound = 343 m/s), 60 cm long, open at one end and closed at the other. a. Find the frequency and wavelength of the lowest note which can be played on it with all the side holes shut. b. Find the frequency and wavelength of the next lowest note. (It’s not just twice the frequency from a.) 3. None of the following reactions and decays can actually occur. In each case, state a physical principle which is being violated. a. p n p p p b. μ- e- + e c. p p e d. π- + p p + π+ 4. In a metal where the Fermi energy is 8 eV, about how many free electrons per unit volume have energies above 9 eV at 300K? (The integral is too ugly to do by hand. Use a calculator or a website. You still need this simplification: Because the exponential is much larger than 1 throughout the range of integration, e(E-Ef)/kT + 1 ≈ e(E-Ef)/kT. Search for “definite integral calculator.” At the time I wrote this, I got good results from solvemymath.com, wolframalpha.com and numberempire.com; mathportal.org gave an error message. You may need to use sqrt( ) for √ and exp( ) for an exponential. If it won’t take infinity as a limit of integration, just use 1000.) 5. A light bulb gives off electromagnetic radiation at a rate of 4.50 joules per second, uniformly in all directions. Find the radiation's average intensity 5.00 m away, and the average electric field there. 6. A beam of electrons is incident on a crystal, parallel to the horizontal rows of atoms shown. If first order diffraction from the planes indicated by dashed lines is observed when the electron's speed is 2.65 x 107 m/s, what is d? 7. A ray of sunlight is refracted and reflected by a spherical raindrop as shown. (Some geometric facts are built into the diagram. For example, the two angles labeled "a" are equal because they are -54base angles of an isosceles triangle.) a. What is angle a? b. What is angle b? c. What is angle c? d. What is θ, the angle between the ray's original and final directions? (Although a complete treatment requires considering incident angles other than 60, you have just calculated the angular radius of a rainbow.) 8. Short answer, 5 points each: a. Give an example of a situation in which quantum mechanical tunneling takes place. b. A π+ meson contains an antired contains. quark. Give the color and flavor of the other quark(s) it c. The wave function for a 2p state in hydrogen is (using spherical coordinates) r e r / 2 a0 cos . For this state, what is ψ2 dV, integrated over all places the 5/ 2 4 2 a0 electron can reach? d. The rod connecting the two objects has neglegible mass. For which system, A or B, is the reduced mass smaller, or are they equal? e. The bubble is 1/4 of a wavelength thick (using the wavelength in the liquid). Will the reflected rays interfere constructively, destructively, or do something in between?