Questions and Answers of The Physics Energy

Coherent states are potentially an interesting basis to consider in which to express states on the Hilbert space. Are they a good basis, satisfying qualities that we desire of a basis on the Hilbert
In this chapter, we had only expressed eigenstates of the harmonic oscillator Hamiltonian through repeated action of the raising operator, \(\hat{a}^{\dagger}\). This gives us a concrete algorithm
In this chapter, we noted that the Hilbert space of the harmonic oscillator corresponds to all those states that can be accessed from the ground state through action by an anatic function of
We studied coherent states, which we had identified as eigenstates of the lowering operator, \(\hat{a}\). A coherent state \(|\psiangle\) is defined by the eigenvalue
Supersymmetry is a proposed extension of the symmetries of space-time beyond that of just familiar Lorentz transformations and translations. A supersymmetrytransformation interchanges fermions and
In Example 6.1, we introduced the Hermitian number and phase operators \(\hat{N}\) and \(\hat{\Theta}\) constructed from the raising and lowering operators, \(\hat{a}^{\dagger}\) and \(\hat{a}\). In
We had shown that any state \(|\psiangle\) on the Hilbert space of the harmonic oscillator is described by some analytic function \(f\left(\hat{a}^{\dagger}\right)\) of the raising operator acting on
We happen to find that the ground state of the harmonic oscillator (and the coherent states in general) is a minimum uncertainty state, but can we do the converse? That is, can we directly determine
A simple model for radioactive decay of an unstable nuclear isotope is as follows.3 Consider the potential illustrated in Fig. 7.9, in which there is a hard, infinite barrier at the spatial origin, a
We constructed the S-matrix as encoding the reflection and transmission amplitudes for individual momentum eigenstates, but of course these are not physical states on the Hilbert space. In this
Consider a particle in a quantum state \(|\psiangle\) of the infinite square well that is a linear combination of the two lowest-energy eigenstates:\[\begin{equation*}|\psiangle
We discussed how the energy ground state of the infinite square well, \(\left|\psi_{1}\rightangle\), was the state that minimized the uncertainty relation. That is, in this state, the product of the
In this problem, we will work to understand the time evolution of wavefunctions that are localized in position in the infinite square well.(a) It will prove simplest later to re-write the infinite
A closely related system to the infinite square well is a particle on a ring. The Hamiltonian for a particle on a ring has 0 potential energy and the position on the ring can be represented by an
We like to think of the \(E \rightarrow \infty\) limit as the limit in which quantum mechanics "turns into" classical mechanics, but this clearly has limitations. The limitation that we will consider
In Sec. 5.1.3, we explicitly constructed the momentum operator as a matrix in the energy eigenbasis. For the infinite square well, the square of the momentum operator is proportional to the
We had proved the canonical commutation relation \([\hat{x}, \hat{p}]=i \hbar\) in generality in the previous chapter. Of course, then, any matrix realization of the position and momentum operators
The energy eigenstates of the infinite square well are of course not unique as a basis for the Hilbert space, which has no support at the boundaries of the well, where \(x=0\) and \(x=a\). For
We will look at walks on a network. Consider a network of N nodes. Assume that each pair of nodes are connected by a link with probability p. In the following we neglect loops and work in the limit
Consider a probability distribution for event sizes s given bywhere A is a normalisation factor.(a) Explain in what sense power laws are scale invariant.(b) Derive the normalisation constant.(c)
Consider a branching process for which the generator function for the branching probabilities is given by(a) What is the probability that an individual produces k offspring?(b) What is the average
Use the equations for the canonical ensemble to discuss a very simple system consisting of two Ising spins S1 and S2 with energy given by H = −JS1S2, where J > 0. Use the notation x = βJ and
Show that in the canonical ensemble, the entropy given by Eq. (6.11) for the two-spin system in Exercise 1 is given byData From Equation (6.11) S2 S[p]=-Pilnpi i=1
Now consider a general system and show that the entropy given by Eq. (6.11) in the microcanonical ensemble is given by S = ln Ω.Equation (6.11) == 22 S[p]=-pilnpi i=1
We consider again the system in Exercises 1 and 2 and make use of the same notation.We will study the limits T → 0 and T →∞.(a) In the limit T → 0 compute the four probabilities given by Eq.
Consider a particle moving on the lattice structure depicted in Fig. 6.10. We assume it moves to each of its nearest-neighbour sites with equal probability and that it moves in a way such that within
Derive the expression for the eigenvalues given in Eq. (6.75). =ecosh Be-43 + sinh B (cosh + sinh-
Discuss the similarity and differences between a vortex–anti-vortex pair and a pair of ↑↓ and ↓↑. How does the energy of creating a single ↑↓ perturbation scale with system size? How
Consider the one-dimensional Ising model given by the following Hamiltonian:(a) Calculate the energy E1 of the first excited state of the system in which the spin configuration can be described in
Consider a set of particles on a two-dimensional square lattice of linear size L. The particles interact through repulsive central unit forces with their nearest neighbours.Double occupancy is not
We consider two rotors given by the angles θ1 and θ2. (a) Assume their time evolution is given by do =1+(02-01) dt d02 =w2+(01-02). dt (7.22) Write down the equation for the time evolution of the
Go through the details of the calculation leading to K = 8g() Ig" (@)e3 (e-) for < Ec for > c'
Use Eq. (7.7) and Fig. 7.1 to establish the details of how the separation of the rotors into the two classes in Eq. (7.10) works.Figure 7.1Equation (7.7)Equation (7.10) 1 0.5 Case (1) o -0.5 -1 0.5
Consider an infinite linear chain of rotors each described by a phase variable θn, with n ∈ Z. The rotors are coupled to their neighbours and their equation of motion is given byMake use of the
Check that node α in Fig. 8.1 has lα = 14/11 and therefore Cα ≈ 0.79 and that node γ has lγ = 26/11 and accordingly Cγ ≈ 0.42.Figure 8.1 10
Calculate Q in Eq. (8.32) and Eq. (8.34) for each of the two partitionings into clustersgiven in Fig. 8.4.Equation (8.32)Equation (8.34)Figure.8.4 Q[c] = K () 4=1 ie CajeCa kjkj 2L
Apply the Louvain algorithm to the network in Fig. 8.4.Figure 8.4 (a) C C2. 10 C3 12 15 (b) C 10 C2 12
We discussed in Sec. 3.5 how the dynamics of complex systems commonly involve intermittent bursts of activity and the importance of developing methods enabling forecasting of the events. Without
This project is intended as an example of how the structure of a network is related to its stability and at the same time as an illustration of how difficult it can be to settle the actual structure
Consider the financial system of a country like Japan for example and discuss which ingredients we may have to include in a model that can address the reason for the observed recurrent financial
Discuss how one decides whether a model is too simple or unnecessarily complicated.
Imagine that the EU flagship Human Brain Project has succeeded in producing a highperformance computer that can simulate the entire network of 8.6×1010 neurones andtheir 1015 connections.(a) Will it
The stakeholder concept has become popular in management theory. The publisher’s description of the book Stakeholder Theory [146] says:The stakeholder perspective is an alternative way of
The discipline of neuroscience makes use of network theory to identify the structures relating to functionality of the brain. Consider the paper by Vértes et al. [469]. You do not need to study all
The project is concerned with a model of ant foraging. The observational background for the model consists of ants leaving their nets to go and collect food and is described in [365]. The theoretical
Recall the discussion around Fig. 3.2. Though the model was introduced to describe magnetic phase transitions, it has been used much more widely. Applications include, for example, gas theory,
Find an online movie on fireflies such as David Attenborough’s ‘Talking to Strangers’ in the BBC series Trials of Life. Notice how Malaysian fireflies are able to gradually synchronise along
In Sec. 3.5 we discussed various aspects of relations between dynamics at different levels of organisation in a complex system.(a) Think of the human body and try to list the different time scales
Make a list of as many transitions as possible in as many different complex systems as you can think of. For each transition try to suggest ways to check if the transition involves a diverging length
Try to think of networks in different systems such as food webs, interconnected airports, social networks, etc. Use a web search to investigate which type of distribution the number of links attached
Make a summary of the hierarchies of processes, nested within each other, you can identify when you think of a human being from the level of atoms, through cells to organs, to the level of brain and
Discuss similarities and differences between the views concerning complexity and emergence expressed by Weaver [487] and Anderson [18].
Discuss emergence from the viewpoint that reality acually consists of processes and not of ‘things’ [271, 401, 494]. Then, from this position, consider the notion of strong and weak emergence
Consider in general terms similarities and differences between the evolution and emergenceof structure in (1) biological ecosystems and (2) sociological structures and building blocks of society.(a)
We mentioned around Eq. (1.1) and in Sec. 1.1 that without interaction between component, new emergent properties cannot be generated.Data From Equation (1.1)Nevertheless, sometimes people do mention
List examples from biology, sociology and neuroscience/psychology which involve the emergence of some kind of:(a Clustering or grouping amongst components.(b) Intermittency in time.
Make a list of examples of synchronisation across biology, physics, economics, neuroscience, etc.
Consider the emergence of social class and the emergence of the way we perceive ourself. Discuss if it is an example of two interacting emergent phenomena, as seems to be implied in [277].
Discuss and compare the views of complexity as expressed in Herbert Simon’s ‘The architecture of complexity’ [412] and Phil Anderson’s ‘More is different’ [18].
Discuss the different scales relevant to the organisation of:(a) Human society.(b) A forest ecosystem.(c) The brain.(d) The financial system of a country like Italy and like China.
Discuss what might be considered the robust emerging degrees of freedom of:(a) Human society.(b) A forest ecosystem.(c) The brain.(d) The financial system of a country like Italy and like China.
Consider whether the concept of symmetry breaking can be defined in a way to make it of value to our understanding of:(a) Social transitions such as industrialisation.(b) Political revolutions.(c)
The zoology of emergence. In this chapter we listed the following examples of wellestablished classes of emergence:(1) Characteristic scale.(2) Collective degrees of freedom.(3) Transitions.(4)
Discussion of the paper ‘How emergence arises’ by the ecologist de Haan [102]. De Haan makes a number of suggestions for how we understand what emergence is and how we can distinguish between
Eldredge and Gould have argued that the fossil record exhibits intermittent dynamics, which they named punctuated equilibrium [169]. That adaptive evolutionary dynamics is able to generate
This project is somewhat open ended. It studies several papers to develop some insights into the importance of the co-existence of brain structures possessing characteristic scales with
Here are a number of questions which are meant to stimulate thinking and should be illuminated by web search and open access journals.(a) Why is synchronisation a problem in finance (herding), the
This project is concerned with the review article ‘Connectivity and complex systems:Learning from a multi-disciplinary perspective’ [459]. The article is rather long, but a very detailed reading
In our derivation of the generalized uncertainty principle, we needed to assume that the two operators that we considered were Hermitian. What if we relax this assumption? Consider two operators
In this chapter, we derived the canonical commutation relation between the position and momentum operators, but we had previously constructed the derivative operator \(\mathbb{D}\) on a grid of
Consider the following operators that correspond to exponentiating the momentum and position operators, \(\hat{x}\) and \(\hat{p}\) :\[\begin{equation*}\hat{X}=e^{i \sqrt{\frac{2 \pi c}{\hbar}}
Consider two Hermitian operators \(\hat{A}\) and \(\hat{B}\) and consider the unitary operators formed from exponentiating them:\[\begin{equation*}\hat{U}_{A}=e^{i \hat{A}}, \quad \hat{U}_{B}=e^{i
An instanton is a quantum mechanical excitation that is localized in space like a particle. They are closely related to solitary waves or solitons that were first observed in the mid-nineteenth
The Heisenberg uncertainty principle establishes a robust lower bound on the product of variances of Hermitian operators, which depends on their commutator. We established this bound for any
The Ehrenfest's theorem that we quoted in the text was that quantum expectation values satisfy the classical equations of motion. In Sec. 4.2.1, we had also stated that this isn't quite true. If the
We've seen simple examples of expectation value and variance in analyzing the rolls of a fair die and the uniform distribution in Example 4.1. In this exercise, we will study these ideas on another
In this chapter, we introduced normal matrices as those matrices that commute with their Hermitian conjugate. This seems like a weak requirement, but it's very easy to construct matrices for which
In this exercise, we study the unitary matrix constructed from exponentiation of one of the Pauli matrices, introduced in Example 3.3. As Hermitian matrices, exponentiation should produce a unitary
The unitary matrix that implements rotations on real two-dimensional vectors can be written as\[\mathbb{M}=\left(\begin{array}{cc}\cos \theta & \sin \theta \tag{3.153}\\-\sin \theta & \cos
We introduced unitary operators as those that map the Hilbert space to itself (i.e., those that maintain the normalization constraint that we require of all vectors in the Hilbert space). However, we
Consider two states on the Hilbert space \(|uangle\) and \(|vangle\). Construct the operator \(\mathbb{U}_{u v}\) that maps one to another:\[\begin{equation*}|vangle=\mathbb{U}_{u v}|uangle
In this chapter, we introduced completeness as the requirement that an orthonormal basis \(\left\{\left|v_{i}\rightangle\right\}_{i}\)
In this exercise, we continue the study of the two-state system introduced in Example 3.4. Recall that \(|1angle\) and \(|2angle\) are energy eigenstates with energies \(E_{1}\) and \(E_{2}\),
Neutrinos are very low mass, extremely weakly interacting particles that permeate the universe. About a quadrillion will pass through you while you read this problem. There are multiple types, or
In this chapter, we introduced the matrix \(\mathbb{D}\) defined as the derivative acting on a grid with spacing \(\Delta x\). This matrix has the form\[\mathbb{D}=\left(\begin{array}{ccccc}\ddots &
Why didn't we define the derivative matrix on a grid through the familiar "asymmetric" derivative:\[\begin{equation*}\frac{d f(x)}{d x}=\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta
In this exercise, we extend the analysis of the Legendre polynomials presented in Example 2.1.(a) With only the first three Legendre polynomials, \(P_{0}(x), P_{1}(x)\), and \(P_{2}(x)\), this is
Another linear operation that you are familiar with is anti-differentiation, or indefinite integration. In this problem, we will study features of antidifferentiation and relate it to
Differentiation isn't the only operation that can be performed on functions. In this problem, we'll consider the operator \(\hat{S}\) that takes the form\[\begin{equation*}\hat{S}=-i x \frac{d}{d x}
Consider a general \(2 \times 2\) matrix \(\mathbb{M}\) which we can express as\[\mathbb{M}=\left(\begin{array}{ll}a & b \tag{2.99}\\c & d\end{array}\right)\]for some, in general complex, numbers
In this chapter, we considered how two different orthonormal and complete bases lead to distinct representations of a matrix \(\mathbb{M}\). In this problem, we will show explicitly that this change
An orthonormal and complete basis for functions \(g(\theta)\) that are periodic in \(\theta \in[0,2 \pi)\), where \(g(\theta)=g(\theta+2 \pi)\), are the imaginary
Show that the stator magnetic field of a three-phase generator has a constant magnitude and rotates with constant angular frequency. You can assume that the magnetic fields produced by the stator
Repeat the analysis that led to eq. (3.40) for a load with impedance described by a power factor cos ?.Show that the fraction of power lost compared to real power consumed in the load is
If they are visible, find and follow the electric wires that connect your residence to the grid. Can you identify where three-phase power is split into separate phases to serve residential customers?
Figure 30.2 shows the power coefficient for a wind turbine blade as a function of tip-speed ratio. In light of this, explain why a wind turbine constrained to operate at fixed rotor speed is less
Make a list of the properties that you would expect to characterize a liquid used as a transformer coolant. Investigate: what was the dominant coolant used prior to the 1970 s? Why was its use
Discuss the stability of an induction machine operating in the generator domain (see Figure 38.14) For example, suppose more power from the prime mover causes the slip speed to increase. How does
Some utilities charge customers for the apparent power delivered (i.e. including reactive power) rather than real power consumed. Discuss the possible justifications for this.
When you turn on a powerful motor in your home, such as an air conditioner compressor, you will notice the incandescent lights dim briefly. Explain why.

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