Physics 101: Foundations of Theoretical Physics Spring 2019 Homework #3 Due Friday, February 22 Problem 1: Devise two of your own examples of classical systems, different from those introduced in class, and being sure to specify both the kinematics and the dynamics of each model. Your examples can be purely mathematical or inspired by systems in the real world, and should involve configuration spaces with at least four elements and nontrivial dynamics. Identify any selection rules, superselection rules, or attractors in your models. (For the purposes of this problem, your models don't need to have have all these features, nor do you need to identify all those that your model does possess.) p Problem 2: Letting i ≡ −1 be the imaginary unit, compute the average value of the function f (θ) = eiθ = cos θ + i sin θ over the circumference of a unit circle: 1 2π hfiunit circle ≡ dθ f (θ) : 2π ˆ0 Problem 3: (a) Find all the possible values of ii. (b) Find all the possible values of (−1)−i. (c) Is 1 raised to any imaginary power necessarily equal to 1? Problem 4: For real values of θ, we know that −1 ≤ sin θ ≤ 1. Find a complex number z for which sin z = 2. (Express your answer in terms of mathematical constants and logarithms of real numbers, and do not make any numerical approximations.) Problem 5: As we'll explain later, light, X-rays, radio waves, microwaves, infrared rays, ultraviolet rays, and gamma rays are all forms of electromagnetic radiation characterized by different wavelengths, meaning the distance between successive wave crests. A blackbody (not generally to be confused with a black hole) is an object that perfectly absorbs and emits electromagnetic radiation at all wavelengths. When studying radiation from a hot object, regarding the object as being a blackbody often serves as a good first approximation. At the beginning of the twentieth century, and in one of the groundbreaking conceptual triumphs that helped usher in the era of quantum theory, Max Planck correctly deduced the precise spectrum of radiation produced by a blackbody at a given temperature. As we'll show in detail later in this course, a key ingredient was Planck's postulate that electromagnetic radiation consists of discrete \quanta" of energy called photons. As we'll explain later, energy quantifies how much work a system is capable of performing. Based on earlier experiments carried out by the physicist John Tyndall, Josef Stefan proposed in 1879 that an object radiat- ing as a blackbody emits energy E (measured in Joules) at a power P = dE=dt|that is, a rate of energy emitted per unit time|proportional to the object's surface area A multiplied by the fourth power T 4 of the object's temperature T (measured in kelvins). In 1884, Boltzmann used thermodynamic arguments to provide a theoretical basis for Stefan's formula, arriving at what is now called the Stefan-Boltzmann law: P = AσT 4; 1 where the Stefan-Boltzmann constant σ is given by π2k4 J σ ≡ ' 5:670373 × 10−8 ; 60~3c2 m2 · s · K4 and where k is Boltzmann's constant, k ≡ 1:3806488 × 10−23 J=K; c is the speed of light, c = 299 792 458 m=s; and ~ is the reduced Planck constant defined in terms of the Planck constant h by h ≡ ' 1:054572 × 10−34 J · s: ~ 2π Blackbodies emit electromagnetic radiation at all wavelengths|but not equally among wavelengths. In 1896, the German experimental physicist Wilhelm Wien showed empirically that the radiation from a blackbody is peaked at a particular wavelength λmax that dominates the radiation and depends on the blackbody's temperature T according to a rule that's now called Wien's law, b λ ' ; max T where the constant of proportionality b, known as the Wien displacement constant, is given by b ' 2:89777 × 10−3 m · K: Wien was awarded the Nobel Prize in Physics in 1911 for his discovery. (a) Treating the human body as being a blackbody of surface area A = 2 m2 and effective surface temperature T ' 33◦ C ' 300 K, use the exact Stefan-Boltzmann law to calculate the rate P of radiated energy loss by a human being in the empty vacuum of outer space, and use Wien's law to find the peak emission wavelength λmax, expressing your final answers numerically with the correct units. Qualitatively speaking, where in the electromagnetic spectrum does the peak wavelength of this radiation lie? (b) Repeat part (a) for the sun, treating the sun as being a spherical blackbody of radius R ' 695 700 km and temperature T ' 5800 K. (c) Consider a blackbody of surface area A that cools down with time according to the time-dependent tem- perature function T (t) = T0 exp (−rt), where T0 ≡ T (0) is the object's temperature at the initial time t = 0. What is the total amount of energy E radiated by the object from t = 0 to t = 1? Express your answer in terms of A, σ, T0, and r, and be sure to check limiting cases in your formula for E. Problem 6: As we'll explain later, a system's entropy is the logarithm S ≡ k log W of the number W of states consistent with the system's overall appearance. As we'll also discuss later, a spherical black hole of total mass-energy Mc2 has radius 2GM R = c2 and, as first deduced by Bekenstein, an associated entropy kc3 A SBH ≡ ; G~ 4 where G is Newton's gravitational constant and where A = 4πR2 is the surface area of the black hole's event horizon. For most familiar systems, the system's entropy is positively related to the system's total energy U. For example, if we coarse-grain appropriately to obtain a finite configuration space, a box containing a gas of particles has 2 far more allowed configurations at high temperature than it does if its temperature is close to absolute zero. In general, as we'll explain later, the temperature of a system (measured in units of kelvins) is fundamentally defined to be the rate at which the system's energy U grows as a function of the system's entropy S: @U T ≡ : @S One can show that in the case of a box containing a gas of particles, this fundamental definition of T turns out to be proportional to the average kinetic energy of each particle. (a) Regarding U = Mc2 as being the black hole's mass-energy, Hawking surmised from the nonzero entropy SBH 6= 0 that black holes should have a nonzero temperature TH 6= 0. Follow Hawking by showing from the definition of temperature that c3 T = ~ : H 8πGkM Hawking argued that we can interpret this temperature as corresponding to a small amount of radiation be- ing emitted continuously by even the most isolated black hole. Loosely speaking, particle-antiparticle pairs quantum-mechanically emerging from the vacuum of space near the event horizon can split apart before they can recombine, with one member of the pair falling into the black hole and the other escaping and carrying away a net mass-energy. Perfectly isolated black holes therefore gradually evaporate with time, although as their mass-energy M de- creases, their Hawking temperature TH and the evaporation rate both increase. For typical black holes in our universe, however, this evaporation is overwhelmed by the rate at which black holes absorb matter from nearby stars and gas, and as material falls into the black hole, friction causes the material to heat up to temperatures that vastly exceed the black hole's underlying Hawking temperature. A black hole's Hawking temperature is therefore extremely difficult to observe directly. (b) Assuming a perfectly isolated black hole and working in the blackbody approximation in which we treat the black hole as being an object that emits Hawking radiation at all wavelengths at an overall power P = −dU=dt determined by the Stefan-Boltzmann formula, where the minus sign in this expression for P refers to the fact that the evaporating black hole is losing energy U with time, show that the rate at which the black hole loses mass is given by dM c4 1 = − ~ dt 15360πG2 M 2 and that the total time tev it takes for the black hole to evaporate fully is given in terms of the black hole's initial mass M0 by the formula 2 3 5120πG M0 tev = : ~c4 (c) Using m ' 1:99 × 1030 kg for the mass of our sun, what would be the radius R of a black hole with a mass equal to 10 solar masses? Compute the value of the black hole's entropy S and use S ∼ k log W to . BH BH estimate the total number W of the black hole's microstates as a power of 10. Compare the black hole's entropy with the entropy ∼ 50 J=K of an ideal gas in a one-liter box, and compare the number of microstates with the approximate number ∼ 1080 of atoms in the known universe. Compute the value of the black hole's Hawking temperature TH and determine how this temperature compares with room temperature. Assuming that the black hole is perfectly isolated, how long (in years) would it take to evaporate completely? Using Wien's law, determine the peak wavelength of its emitted radiation. Where in the electromagnetic spectrum does this peak wavelength lie? (d) A blue whale has a mass of approximately 105 kg. Repeat part (c) for a black hole with a mass of 10,000 blue whales.