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physics
the physics energy
Questions and Answers of
The Physics Energy
Verify eqs. (36.14)–(36.17)
The function is a standing wave solution analogous to eq. (31.10)for a surface gravity in an ocean of depth h. Following the derivation in the text, construct the kinetic and potential energy per
Show that the functiondescribes a (plane) gravity wave propagating in the x̂ direction in a uniform ocean of depth h. In particular, show that ψ(x, t) satisfies eq. (31.7)and the boundary condition
Show that the displacement s given by (31.9)satisfies the constraints eq. (31.4)and (31.5).
The Savonius VAWT shown in Example 30.2 is mainly driven by drag. As shown in Figure 30.7(b), the Savonius design is capable of reaching roughly half of the Betz limit, and for tip-speed ratio close
In the text we explained the shape of Figure 30.12 by taking and . What about the other limits: at fixed v and at fixed Ω? Why does vanish in those limits?
Why does the stream of water in Figure 29.3(a) get thinner as it falls? Why, if left to fall long enough, would it break up into droplets?
Derive eq. (20.4)from eq. (20.3) and the approximations quoted in the text.
The Maxwell–Boltzmann distribution (18.10)to show that the mean kinetic energy of a particle in agas at temperature T is 3/2 kBT.
On the basis of the pVT-diagram, Figure 12.8, construct a sketch of the phase diagram for water in the VT-plane (specific volume on the horizontal axis, temperature on the vertical) for temperatures
Reproduce the graphs in Example 35.1 by performing the numerical calculations on a computer. Find a way of improving the assumptions and redo the calculation.
Investigate the Darrieus and closely related giromill VAWT designs described in Example 30.3. What are their histories and their advantages and disadvantages relative to the conventional HAWT design?
Estimate the wind in a tropical region at latitude (30 + y)◦ North as v = 2yx̂ m/s, where x̂ is a unit vector pointing eastward. Use the formula from Problem 27.9 to estimate the wind stress on
If a pitcher throws a baseball at 44 m/s (100 mph), estimate the deviation in position, both in magnitude and direction, due to the Coriolis force when the ball reaches the batter after traveling
Give an estimate of the collection efficiency for the best double-junction cell formed from the materials in the previous problem.
Prove that a linear compound parabolic concentrator realizes the maximum possible concentration for a given acceptance angle θ. Suggestion: following the notation of Figure 24.11, work in a
Redo the computation of Problem 24.7 assuming incoming radiation at I = 750 W/m2, and fixing Tb = 70 ◦C, Tg = 280 K. Show that the efficiency decreases significantly with the reduction in incoming
contributes radiation exposure when it occurs in the human body and in rocks and soil. Why does the internal exposure come dominantly from electrons and the exposure from decays in rocks and soil
Walk your way around the pV- and ST-diagrams, Figure 11.3, for the Otto cycle, explaining the functional form of the curves, p(V) and T(S). Why is each step vertical in one of the diagrams?
Consider the following ideal gas engine cycle with only three steps: {12} – isothermal compression at T-from V1 to V2 ; {23} – isobaric (constant pressure) heating T- from T+ to , where T+ is
The project studies the paper [501]. Recall that the avalanche dynamics evolves on two time scales. The first, denoted by t, counts the numbers of energy units added to the pile.In between each
We consider the outbreak of a transmittable disease in an infinite population. The disease is characterised by its ability η ∈ [0,1) to infect. Each individual has a specific level of resistance R
Study numerically the fluctuations in the process in Eq. (13.1) with η(t) given by Eq. (13.2). Try, from the changing time dependence of the fluctuations (the variance and the autocorrelations), to
Simulate the process in Eqs. (13.3) and (13.4) in the vicinity of the point at (X̅,0) and try to use as an early warning signal the logarithmic time dependence considered in Eq. (1) of
Being able to forecast approaching transitions or tipping points in complex systems is of great importance and very difficult. Since no well-established consensus concerning the most viable approach
Consider a population of N agents on a network. Let nb(i) denote the set of neighbours of agent i, i.e. if j ∈ nb(i) then j is a neighbour of i. The opinion xi of an agent can assume one of the
The voter model, see e.g. [141], and the Axelrod model of cultural trait dynamics [31] have inspired very many agent-based studies of the dynamics of opinions and cultural evolution. This project
The time between events in SOCmodels is typically exponentially distributed, indicating no correlations between events. This is often in contrast to observed behaviour. Solar flares is an example of
Simulate the Manna model on a square lattice. The model is defined in the following way, see [276, 354]. Each site can contain zero or one particle. New particles are added to randomly chosen sites.
Carefully go through the arguments leading to Eq. (12.23). Next derive Eq. (12.27).Equation (12.23)Equation (12.27) 1+1 0 1+1 z'dz = J P(t) =
Consider the logistic mapDerive analytically an expression for the fixed point x* given by x* = ƒ(x*) and the point xtan for which ƒ(x) has a tangent parallel to the identity, i.e. ƒ(xtan) = 1.
Machine learning, chaos and intermittency are discussed in [237]. Startwith the abstract and discussion sections and then skim read the paper with the aim of making a brief overview of how machine
Derive the different expressions for the entropies S[ p] corresponding to the different functional forms of W(N) in Box 9.1.
(a) Discuss what it can be that makes the Lempel–Ziv complexity of the EEG signal higher the more aware the brain.(b) Do we expect our brain activity to produce more new ‘thought patterns’ if
Consider three stochastic variables Xt, Yt and Zt which assume the values −1 or 1 for t = 1,2,3, . . .. We assume that Z can swap freely between −1 and 1 with constant probability μZ at each
Show that the normalisation constant A in Eq. (10.57) is given byEq. (10.57)Generalise the expression in Eq. (10.57) to the non-symmetric random walker for which p+ ≠ p− and derive expressions
Let n(x,t) denote the density of random walkers at position x ∈ Z at time t ∈ Z. At each time step t a walker makes a move x → x + Δ, where Δ = + 1 or Δ = −1 with probability 1/2. (a)
Consider a two-valued signal ƒ(t) ∈ {−A,A}. Assume that the probability that ƒ(t) switches during the time interval dt is constant and given by νdt. (a) Sow that the autocorrelation function
Consider a discrete-time random walker moving on the set X = {na|n ∈ Z}, where a is the spacing between the discrete positions. Let p+, p0 and p− denote the probability that the walker from time
Let n(x,t) be determined by the diffusion equationwhere g(x,t) is a given function.(a) Use Fourier transform to express n(x,t) in terms of the Fourier transform an(x,t) an(x,t) n(x, t) =y. Y + B- +
Consider a population consisting of individuals that are either of opinion A or opinion B. Initially the entire population is of opinion A. New information arrives, which is scrutinised by everyone
Consider the merger process described in [472]. The process is controlled by the merger probabilitywhere α = 3/2 is considered in the paper.(a) Simulate the process for α = 3/2, say α = 1 and α =
Simulation of a population of N agents. Each agent i is allocated a willingness parameter wi ∈ [0,1] and possesses one of two opinions A or B. Focus on two choices for the distribution of wi. f(p).
This project is of direct relevance to the concepts of a giant component in Sec. 8.4.The process consisting of percolation on a lattice is another paradigmatic model, which conceptually is of great
Consider a network given by the adjacency matrix aij = aji = 1 with probability p and aij = aji = 0 with probability 1 − p.(a) In the limit of large N and fixed pN, compute the degree distribution
Think of throwing a coin. Let X = 1 when the outcome is heads and X = −1 when the outcome is tails. We assume the coin is biased and that PX(1) = p and PX(−1) = 1−p. Compute the entropy of X
An example of dependent variables with zero correlation coefficient. We consider thefollowing two cases. Let X be uniformly distributed(I) on the set {0,1,2, . . . ,N};(II) on the set {−N, − N +
Consider two Ising spins in the canonical ensemble with a Hamiltonian given by H = −S1S2. Compute the correlation coefficient CX1X2 and the mutual information I(X1;X2) between the two variables X1
Derive the expressions for the mutual information given in Eq. (9.20).Equation (9.20) I(X; Y)H(X)+ H(Y)- H(X, Y) = H(X, Y) - H(XY) - H(YX) =H(X) H(XY) - = H(Y) H(YX).
Use the relation between the transfer entropy and mutual information to derive Eq. (9.27).Equation (9.27) = DTEX-X, TEXX-TEX(-)X)-
We generalise the set of Eqs. (9.14) to include an additional dependence between X and Z:Equation 9.14 x(t) = $1(t) y(t)=ax(t z(t) =By(t 1) + 2(1) 1)+3(1).
Show that in the limit q → 1 the Tsallis entropy in Eq. (9.36) becomes equal to the entropy in Eq. (9.31). And show that in the same limit the q-exponential in Eq. (9.38) becomes an ordinary
Consider a general branching process. Let g(s) denote the generator function for the branching probabilities and let gZn (s) denote the generator function for the size Zn of the nth generation.
We had stated in this chapter that the general Rényi entropies do not satisfy subadditivity; only the von Neumann entropy does. In Example 12.2, we constructed an explicit counterexample for the
The von Neumann entropy satisfies a further inequality among three systems called strong subadditivity. \({ }^{20}\) As with subadditivity, strong subadditivity is only a property of the von Neumann
In our analysis of hydrogen in Chap. 9, we discussed the period of recombination, the time in the history of the universe at which protons and electrons became bound and formed neutral hydrogen. We
We had established an intriguing relationship between the path integral of the previous chapter and the partition function here through "complexification" of the time coordinate. In this problem, we
We had mentioned that from the partition function \(Z\), all possible thermodynamical quantities can be determined. In this problem, we will use the harmonic oscillator's partition function of Eq.
An extremely intriguing feature of quantum entanglement is a property that has been called the "monogamy of entanglement": a particle can only be maximally entangled with precisely one other
In Example 12.1, we introduced the Hong-Ou-Mandel interferometer and presented an analysis of thinking about the photons produced by the laser as classical electromagnetic waves. In this exercise, we
Let's consider the harmonic oscillator immersed in a heat bath of temperature \(T\). For any Hermitian operator \(\hat{A}\), we can define its thermal average denoted as
In Sec.12.4.3, we motivated and introduced Bell's inequalities for unambiguous observation of quantum entanglement. There we demonstrated that a classical, random density matrix of two spins
The quaternions are three quantities i, j, and k such that, along with the real number 1, they form the basis of a four-dimensional space over the real numbers. The objects i, j, and k have
The Hong–Ou–Mandel interferometer is an experiment in which two identical photons are emitted from a laser toward separate mirrors that then reflect onto a beam splitter and can subsequently be
Let’s see if we can make some more sense of the statement of subadditivity, and in particular, demonstrate that some particular Rényi entropies violate it.In this example, we will explicitly
Let’s go back to our spin-1/2 friends and construct the density matrix for a collection of spin-1/2 particles. Let’s say that we have a partially polarized beam for which a fraction of 3/4 are
As we explicitly calculated the path integral of the harmonic oscillator in the previous chapter, it is useful to calculate the partition function for the harmonic oscillator to compare the explicit
In the next section, we will explicitly calculate the path integral for the harmonic oscillator. This will be highly non-trivial, and its calculation will involve many moving parts. Here, however, we
After calculating the path integral for the harmonic oscillator, you might need to take a breath or two. To relax, in this example, we will calculate the path integral for the free particle, using
The appearance of the classical Lagrangian and action in the path integral might be surprising given how we had constructed quantum mechanics throughout this book, but it was actually hidden in plain
In the evaluation of the path integral for the harmonic oscillator, we had left the classical action unevaluated. It's high time to fix that.(a) From the Lagrangian for the harmonic
In evaluating the path integral of the harmonic oscillator, we had to perform a Gaussian integral for which the factor in the exponent was a Hermitian matrix \(\mathbb{A}\) sandwiched between an
As emphasized in this chapter, calculating the path integral for even very simple quantum systems is highly non-trivial. In Example 11.2, we calculated the freeparticle path integral one way, and
In this chapter, we only considered path integrals for one-dimensional systems, but in some cases it is straightforward to calculate the path integral for a system in three spatial dimensions. As an
In the calculation of the path integral, we of course have to sum over all possible paths between the initial and final positions. However, it can be useful to consider what contributions to the path
Consider the familiar double-slit experiment: a quantum mechanical particle of mass \(m\) starts at some position to the left of the slits at time \(t=0\). As time evolves, the wavefunction of the
While we have focused on the Lagrangian and action for point particles in this chapter, it is possible to formulate the Schödinger equation itself through the principle of least action with a
Let’s see how this formalism works in an explicit example. Let’s consider a potential that is the infinite square well that has a little bump in the center, as shown in Fig. 10.1. That is, the
Let’s test this out for the infinite square well, where we will set the width a = 1 of the well for simplicity. That is, the potential we consider isWe know what the ground-state energy is in this
While this seems pretty magical, the biggest challenge will be inverting the Hamiltonian in the first place. As we saw with the variational method, an example can be helpful in demonstrating the
The WKB approximation and the Bohr–Sommerfeld quantization condition can be used to calculate the energy levels of the hydrogen atom, as well, and was actually the system in which it was initially
While the Bohr–Sommerfeld condition sometimes gets the energy eigenvalues exactly correct, it can also be used for systems where the exact solution is not known. In this example, we will estimate
It's useful to see how our quantum perturbation theory works in a case that we can solve exactly. Let's consider a two-state system in which the Hamiltonian
Let's see how the variational method works in another application. Let's assume we didn't know the ground-state energy of the quantum harmonic oscillator and use the variational method to determine
While we introduced the variational method and the power method both as a way to approximate the ground state of some system, they both can be used to approximate excited states as well, with
Let's now study the power method for estimating the ground-state energy, applied to the quantum harmonic oscillator. For this problem, we will work with the
The anharmonic oscillator is the quantum system that is a modification to the harmonic oscillator, including a term quartic in position:\[\begin{equation*}\hat{H}=\frac{\hat{p}^{2}}{2 m}+\frac{m
We had introduced quantum mechanical perturbation theory as analogous to the Taylor expansion. As such, one might expect that as higher and higher orders are calculated, a better approximation of the
The WKB approximation and the Bohr-Sommerfeld quantization condition worked perfectly for calculating the energy eigenvalues of the infinite square well and the hydrogen atom. Does it work for the
We can quantitatively study this claim, that for the hydrogen atom, we do not need to invert the Hamiltonian to use the power method to estimate its groundstate energy. In this exercise, we'll just
In our introduction of the hydrogen atom, we assumed that the electron orbited the proton with a speed significantly smaller than the speed of light, \(c\). However, we can include the effect of a
We have evaluated the ground-state wavefunction of hydrogen in Eq. (9.48), and in this example, we will construct the first excited state n = 2, ℓ,m = 0 wavefunction. This can then be used, along
The Zeeman effect is the phenomenon of splitting energy levels with different angular momentum due to the presence of a weak external magnetic field, named after Pieter Zeeman who first observed
Our analysis of the hydrogen atom simply extends to any element which has been ionized to have a single electron orbiting the nucleus. In this problem, we will consider such an atom, whose nucleus
In introducing the Laplace-Runge-Lenz vector and the canonically quantized hydrogen atom, there were a number of Poisson brackets for which we just stated the result, without explicit calculation.
The virial theorem is a statement about the relationship between the kinetic and potential energies for a system bound in a potential that is a power of relative distance between particles. For such
The form of the potential of the hydrogen atom is of course special, because it originates from Coulomb's law of electric force through canonical quantization. However, we can imagine a general form
We explicitly constructed the ground-state and first excited-state wavefunctions for the hydrogen atom in this chapter, but we can go further. The LaplaceRunge-Lenz operator commutes with the
The visible light spectrum ranges from wavelengths of \(380 \mathrm{~nm}\) (violet) to \(750 \mathrm{~nm}\) (red). If a hydrogen atom were excited, what color light would it emit and therefore appear
In Example 9.2, we introduced the Zeeman effect as a splitting of the energy levels of states in hydrogen due to the presence of a weak, external magnetic field. In that example, we just studied the
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