Questions and Answers of Modern Classical Physics

Another interesting 1-dimensional map is provided by the recursion relation(a) Consider the asymptotic behavior of the variable xn for different values of the parameter a, with both xn and a being
One of the first discoveries of chaos in a mathematical model was by Lorenz (1963), who made a simple model of atmospheric convection. In this model, the temperature and velocity field are
To measure a very weak sinusoidal force, let the force act on a simple harmonic oscillator with eigenfrequency at or near the force’s frequency, and measure the oscillator’s response. Examples
Consider a cell with volume V , like those of Fig. 5.1, that has imaginary walls and is immersed in a bath of identical, nonrelativistic, classical perfect-gas particles with temperature Tb and
Water and its vapor (liquid and gaseous H2O) can be described moderately well by the van der Waals model, with the parameters a = 1.52 × 10−48J m3 and b = 5.05 × 10−29m3 determined by fitting
Exercise 5.5 explored the enthalpy representation of thermodynamics for an equilibrium ensemble of systems in contact with a volume bath. Here we extend that analysis to an ensemble out of
Consider a gigantic container of gas made of identical particles that might or might not interact. Regard this gas as a bath, with temperature Tb and pressure Pb. Pick out at random a sample of the
Consider an optically thick hydrogen gas in statistical equilibrium at temperature T. (“Optically thick” means that photons can travel only a small distance compared to the size of the system
Random processes can be stochastic functions of some variable or variables other than time. For example, it is conventional to describe fractional fluctuations in the largescale distribution of mass
Prove Doob’s theorem. More specifically, for any Gaussian-Markov random process, show that P2(y2, t|y1) is given by Eqs. (6.18a,b).(a) Show that the Gaussian process ynew has probability
Suppose that you have a noisy receiver of weak signals (e.g., a communications receiver). You are expecting a signal s(t) with finite duration and known form to come in, beginning at a predetermined
(a) Write down the partition function for a 1-dimensional Ising lattice as a sum over terms describing all possible spin organizations.(b) Show that by separating into even and odd numbered spins, it
(a) If y is a random process with spectral density Sy(f), and w(t) is the output of the finite-Fourier-transform filter (6.58a), what is Sw(f)?(b) Sketch the filter function |K̃(f)2 for this
Highly stable clocks (e.g., cesium clocks, hydrogen maser clocks, or quartz crystal oscillators) have angular frequencies ω of ticking that tend to wander so much over very long timescales that
(a) Explain why, physically, when the Brownian motion of a particle (which starts at x = 0 at time t = 0) is observed only on timescales △τ >> τr corresponding to frequencies f r , its
Estimate how long it would take a personal computer to calculate the partition function for a 32 × 32 Ising lattice by evaluating every possible state.
(a) Show that the kernels K(τ) in Eq. (6.47) produce the indicated outputs w(t). Deduce the ratio Sw(f)/Sy(f ) =|K̃(f)|2 in two ways: (i) By Fourier transforming each K(τ); (ii) By setting y =
(a) Show that for shot noise, y(t) = ∑i F(t − ti), the spectral density Sy(f ) is given by Eq. (6.68b). Show that the relaxation time appearing in the correlation function is approximately the
Let u and ν be two random processes. Show thatwhere R denotes the real part of the argument. Su+v(f) = Su(f) + S₂(f) + Suv (f) + Svu (f) = Su(f) + S, (f)+2RSuv (f), (6.43)
(a) Show that for shot noise with identical pulses that have the infinitely sharply peaked shape of Eq. (6.45), the power spectrum has the flicker form Sy ∝ 1/f for all f .(b) Construct
Consider an L-C-R circuit as shown in Fig. 6.15. This circuit is governed by the differential equation (6.72), where F´ is the fluctuating voltage produced by the resistor’s microscopic degrees of
By a method analogous to that used for the elementary fluctuation-dissipation theorem, derive the generalized fluctuation-dissipation theorem [Eqs. (6.86)].Consider a thought experiment in which the
Derive Eqs. (6.96) for A and B from the Fokker-Planck equation (6.94), and then from Eqs. (6.96) derive Eqs. (6.97).Equations Ә - P2 Ət = Ә ду -[A(y)Pz] + 1 02 2 ду2 -[B(y) P2]. (6.94)
(a) Write down the explicit form of the Langevin equation for the x component of velocity v(t) of a dust particle interacting with thermalized air molecules.(b) Suppose that the dust particle has
Consider an electron that can transition between two levels by emitting or absorbing a photon; and recall that we have argued that the stimulated transitions should be microscopically reversible.
Consider a classical simple harmonic oscillator (e.g., the nanomechanical oscillator, LIGO mass on an optical spring, L-C-R circuit, or optical resonator briefly discussed in Ex. 6.17). Let the
Show that the Fokker-Planck equation can be interpreted as a conservation law for probability. What is the probability flux in this conservation law? What is the interpretation of each of its two
Derive the group and phase velocities (7.10)–(7.13) from the dispersion relations (7.4)–(7.7). w = 22 (k)=Ck = C|k], (7.4)
(a) Show that the prototypical scalar wave equation (7.17) follows from the variational principlewhere L is the lagrangian density(b) For any scalar-field lagrangian density L(ψ, ∂ψ/∂t , ∇ψ,
(a) In connection with Eq. (7.35b), explain whyis the tiny volume occupied by a collection of the wave’s quanta.(b) Choose for the collection of quanta those that occupy a cross sectional area A
Consider a 1-dimensionalwave packetwith dispersion relation ω = Ω(k). For concreteness, let A(k)be a narrow Gaussian peaked around(a) Expand α as α(k) = αo − xoκ with xo a constant, and
Consider the nonrelativistic Schrödinger equation for a particle moving in a time dependent, 3-dimensional potential well:(a) Seek a geometric-optics solution to this equation with the form ψ =
Consider two spherical mirrors, each with radius of curvature R, separated by distance d so as to form an optical cavity (Fig. 7.9). A laser beam bounces back and forth between the two mirrors. The
(a) Work through the derivation of Eq. (7.73) for the scaled time delay in the vicinity of the cusp caustic for our simple example [Eq. (7.68)], with the aid of a suitable change of variables.(b)
Show that Hamilton’s equations for the standard dispersionless dispersion relation (7.4) imply the same ray equation (7.48) as we derived using Fermat’s principle. w=2(k)=Ck = C[k], (7.4)
Derive the quasi-spherical solution (7.42) of the vacuum scalar wave equation −∂2ψ/∂t2 + ∇2ψ = 0 from the geometric-optics laws by the procedure sketched in the text.
Consider a simple refracting telescope (Fig. 7.7) that comprises two converging lenses, the objective and the eyepiece. This telescope takes parallel rays of light from distant stars, which make an
Consider sound waves propagating in an atmosphere with a horizontal wind. Assume that the sound speed C, as measured in the air’s local rest frame, is constant. Let the wind velocity u = uxex
A microscope takes light rays from a point on a microscopic object, very near the optic axis, and transforms them into parallel light rays that will be focused by a human eye’s lens onto the
A particle travels in 1 dimension, along the y axis, making a sequence of steps △yj (labeled by the integer j ), each of which is △yj = +1 with probability 1/2, or △yj = −1with probability
Modify your computer program from Ex. 5.17 to deal with the 2-dimensional Ising model augmented by an externally imposed, uniform magnetic field [Eqs. (5.80)]. Compute the magnetization and the
Explain why, for any (stationary) random process,Use the ergodic hypothesis to argue thatThereby conclude that, for a Markov process, all the probability distributions are determined by the
Write a simple computer program to compute the energy and the specific heat of a 2-dimensional Ising lattice as described in the text. Examine the accuracy of your answers by varying the size of the
Figure 8.17 depicts a two-lens system for spatial filtering (also sometimes called a “4f system,” since it involves five special planes separated by four intervals with lengths equal to the
In LIGO and other GW interferometers, one potential source of noise is scattered light. When the Gaussian beam in one of LIGO’s cavities reflects off a mirror, a small portion of the light gets
Our discussion of microlensing assumed a single star and a circularly symmetric potential about it. This is usually a good approximation for stars in our galaxy. However, when the star is in another
(a) Calculate the 1-dimensional Fourier transforms [Eq. (8.11a) reduced to 1 dimension] of the functions f1(x) ≡ e−x2/2σ2, and f2 ≡ 0 for x 2 ≡ e−x/h for x ≥ 0.(b) Take the inverse
Use the parallel-transport law (7.103) to derive the relation (7.104). df ds -k (f dk ds (7.103)
An alternative derivation of the lens equation for a point-mass lens, Eq. (7.88), evaluates the time delay along a path from the source to the observer and finds the true ray by extremizing it with
A Martian Rover is equipped with a single gyroscope that is free to pivot about the direction perpendicular to the plane containing its wheels. To climb a steep hill on Mars without straining its
The van der Waals equation of state (P + a/v2)(v − b) = kBT for H2O relates the pressure P and specific volume (volume per molecule) v to the temperature T ; see Sec. 5.7. Figure 5.8 makes it clear
Consider a cusp catastrophe created by a screen as in the example and described by a standard cusp potential, Eq. (7.73). Suppose that a detector lies between the folds, so that there are three
Consider an elliptical gravitational lens where the potential Ψ is modeled byDetermine the generic form of the caustic surfaces, the types of catastrophe encountered, and the change in the number of
Suppose that a large black hole forms two images of a background source separated by an angle θ. Let the fluxes of the two images be F+ and F− < F+. Show that the flux from the source would be
As we have emphasized, representing light using wavefronts is complementary to treating it in terms of rays. Sketch the evolution of the wavefronts after they propagate through a phase-changing
Consider a self-focusing optical fiber discussed in Ex. 7.8, in which the refractive index iswhere r = |x|.(a) Write down the Helmholtz equation in cylindrical polar coordinates and seek an
(a) Suppose that you have two thin sheets with transmission functions t = g(x, y) and t = h(x, y), and you wish to compute via Fourier optics the convolution(b) Suppose you wish to convolve a large
In a transmission electron microscope, electrons, behaving as waves, are prepared in near-plane-wave quantum wave-packet states with transverse sizes large compared to the object (“sample”) being
Use the convolution theorem to carry out the calculation of the Fraunhofer diffraction pattern from the grating shown in Fig. 8.6.Fig. 8.6
Sketch the Fraunhofer diffraction pattern you would expect to see from a diffraction grating made from three groups of parallel lines aligned at angles of 120° to one another (Fig. 8.7).Fig. 8.7.
(a) Use an amplitude-and-phase diagram to explain why a zone plate has secondary foci at distances of f/3, f/5, f/7 . . . .(b) An opaque, perfectly circular disk of diameter D is placed perpendicular
Derive a formula for the energy-flux diffraction pattern F(x) of a slit with width a, as a function of distance x from the center of the slit, in terms of Fresnel integrals. Plot your formula for
Conceive and carry out an experiment using light diffraction to measure the thickness of a hair from your head, accurate to within a factor of ∼2.
Consider the scattering of light by an opaque particle with size a >> 1/k. Neglect any scattering via excitation of electrons in the particle. Then the scattering is solely due to diffraction
Derive and plot the Airy diffraction pattern [Eq. (8.18)] and show that 84% of the light is contained within the Airy disk. ψ(θ) α Disk with diameter D kᎠᎾ 2 x jinc e-ikxedΣ (8.18)
How closely separated must a pair of Young’s slits be to see strong fringes from the Sun (angular diameter ∼0.5◦) at visual wavelengths? Suppose that this condition is just satisfied, and the
A circularly symmetric light source has an intensity distribution I (α) = I0 exp[−α2/(2α02)], where α is the angular radius measured from the optic axis. Compute the degree of spatial
An FM radio station has a carrier frequency of 91.3 MHz and transmits heavy metal rock music in frequency-modulated side bands of the carrier. Estimate the coherence length of the radiation.
We developed the theory of real-valued random processes that vary randomly with time t (i.e., that are defined on a 1-dimensional space in which t is a coordinate). Here we generalize a few elements
The longest radio-telescope separation available in 2016 is that between telescopes on Earth’s surface and a 10-m diameter radio telescope in the Russian RadioAstron satellite, which was launched
We have defined the degree of coherence γ12(a, τ) for two points in the radiation field separated laterally by a distance a and longitudinally by a time τ. Under what conditions will this be given
An example of a Michelson interferometer is the Far Infrared Absolute Spectrophotometer (FIRAS) carried by the Cosmic Background Explorer satellite (COBE). COBE studied the spectrum and anisotropies
Modern mirrors, etalons, beam splitters, and other optical devices are generally made of glass or fused silica (quartz) and have dielectric coatings on their surfaces. The coatings consist of
Consider monochromatic electromagnetic waves that propagate from a medium with index of refraction n1 into a medium with index of refraction n2. Let z be a Cartesian coordinate perpendicular to the
Study the step-by-step buildup of the field inside an etalon and the etalon’s transmitted field, when the input field is suddenly turned on. More specifically, carry out the following steps.(a)
Show that the PDH method for locking a laser’s frequency to an optical cavity works for modulations faster than the cavity’s response time,and even work for Ω »1/τresponse.More specifically,
When a thin layer of oil lies on top of water, one sometimes sees beautiful, multicolored, irregular bands of light reflecting off the oil layer. Explain qualitatively what causes this.
A common technique used to reduce the reflection at the surface of a lens is to coat it with a quarter wavelength of material with refractive index equal to the geometric mean of the refractive
(a) Use the Helmholtz-Kirchhoff integral (8.4) or (8.6) to compute all four pieces of the holographically reconstructed wave field. Show that the piece generated byis the same (aside from overall
Simplify the analysis by treating each Gaussian light beam as though it were a plane wave. The answers for the phase shifts will be the same as for a true Gaussian beam, because on the optic axis,
How would you record a hologram if you want to read it out via reflection? Draw diagrams illustrating this, similar to Figs. 10.6 and 10.8.Figure 10.6Figure 10.8.
A holographic lens, like any other hologram, can be described by its transmissivity t(x, y).(a) What t(x, y) will take a reference wave, impinging from the θo direction (as in Fig. 10.8) and produce
By expressing the field as either a Fourier sum or a Fourier integral complete the argument that leads to Eq. (9.58). W(t) W (t +T)= K²Y(t)¥*(t) × V(t + t)¥*(t +T) + K²V (1)V*(1 +t) ×
Is it possible to construct an intensity interferometer (i.e., a number-flux interferometer) to measure the coherence properties of a beam of electrons? What qualitative differences do you expect
A Sagnac interferometer is a rudimentary version of a laser gyroscope for measuring rotation with respect to an inertial frame. The optical configuration is shown in Fig. 9.12. Light from a laser L
Derive the factor 1+ (f/fo)2 by which the spectral density of the shot noise is increased at frequencies fz fo
Information on CDs, DVDs, and BDs (compact, digital video, and blu-ray disks) is recorded and read out using holographic lenses, but it is not stored holographically. Rather, it is stored in a linear
A device much ballyhooed in the United States during the presidency of Ronald Reagan, but thankfully never fully deployed, was a futuristic, superpowerful X-ray laser pumped by a nuclear explosion.
Fill in the details of the derivation of all the equations in the section describing the optical frequency comb.
Derive the evolution equations (10.48) for three-wave mixing. You could proceed as follows.(a) Insert expressions (10.27) and (10.28) into the general wave equation (10.30) and extract the portions
Consider a wave propagating through a dielectric medium that is anisotropic, but not necessarily—for the moment—axisymmetric. Let the wave be sufficiently weak that nonlinear effects are
A child, standing in a swing, bends her knees and then straightens them twice per swing period, making the distance ℓ from the swing’s support to her center of mass oscillate asis the swing’s
Consider the secondary wave generated bythe holographic reconstruction process of Fig. 10.8, Eq. (10.7), and Ex. 10.2.(a) Assume, for simplicity, that the mirror and reference waves propagate nearly
Green laser pointers, popular in 2016, have the structure shown in Fig. 10.13. A battery-driven infrared diode laser puts out 808-nm light that pumps aNd:YVO4 laser crystal (neodymium-doped yttrium
Derive the solution (10.54) to the evolution equations (10.47) for frequency doubling, and verify that it has the claimed properties. d'Az ik dz 2 = (A₁)², d'A₁ dz =ik AzA;
Explain why the nonlinear susceptibilities for an isotropic medium have the forms given in Eq. (10.25).What are the corresponding forms, in an isotropic medium, of χij klmn and χijklmnp? Xij =
Derive Eqs. (10.34b) and (10.34c) for the amplitudes of waves 1 and 2 produced by three-wave mixing. dA(¹) dz dA (²) dz = i^/ / dijk(³) A(?) at w₁ = 03 — @₂, k₁ = k3 — k₂i jkAA(2)* =
Estimate the magnitude of the e-folding length for an optical parametric amplifier that is based on a strong three-wave nonlinearity.
A homogeneous, isotropic, elastic solid is in equilibrium under (uniform) gravity and applied surface stresses. Use Eq. (11.30) to show that the displacement inside it, ξ(x), is biharmonic, i.e., it
A microcantilever, fabricated from a single crystal of silicon, is being used to test the inverse square law of gravity on micron scales (Weld et al., 2008). It is clamped horizontally at one end,

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