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38649 results in Cambridge Textbooks

Page 1331 of 1933

  • 7 - Measurement Postulate and Paradoxes of Quantum Mechanics

  • Ravinder R. Puri
  • Book:
    Non-Relativistic Quantum Mechanics
    Published online:
    16 July 2025
    Print publication:
    17 April 2017, pp 143-148

  • Summary

    The measurement postulate (postulate 4) relates the quantum theoretic predictions with experimental observations. Those predictions are found to be in conformity with experimental observations. That postulate, however, has been the subject of debate right since its conception. The issues debated are: (i) whether the measurement postulate is consistent with the postulate specifying the time evolution (postulate 5) and (ii) whether it denies the objective reality per the criterion of the Einstein, Podolsky and Rosen (EPR). The said issues are highlighted in the form of paradoxes by the thought experiment (Gedanken experiment in German) of Schrödinger, and that of EPR. See [40] for collection of articles and papers on the problem of measurement. See in particular [42] and [43].

    The Measurement Problem

    We recall first the measurement postulate. Let a system S (hereafter called the object) be subject to the measurement of an observable . Let the eigenvalues of be non-degenerate with as the corresponding orthonormal eigenvectors. Let the object be in the state jsi before the measurement, where

    According to the measurement postulate, the measurement of would make the state of the object collapse to one of the eigenstates of and the outcome of the measurement would be the eigenvalue corresponding to the eigenstate to which the state of the object has collapsed. Hence, if the outcome of the measurement is the eigenvalue kk then we know that the state of the object after the measurement is, written symbolically as.

  • Appendix I - Hot Research Topics

  • Sandeep Bhowmik
  • Book:
    Cloud Computing
    Published online:
    31 August 2020
    Print publication:
    17 April 2017, pp 384-387

  • Summary

    A lot of research is happening in and around the field of cloud computing in order to achieve more efficient solutions to different issues and to gain optimal results from them. This section introduces a few, among many, hot research topics related to the wide domain of cloud computing.

    OPTIMAL MANAGEMENT OF CLOUD RESOURCE

    Resource management in cloud computing is the prime responsibility of the IaaS cloud service providers who build and manage the data centers. IaaS cloud provider has two major resource management objectives. First, to make the environment reliable as well as robust for optimal and efficient utilization of the data center infrastructure. The second is related to the SLA, which defines the relation with the consumer party.

    The first objective deals with issues like load balancing, fault tolerance, energy conservation etc. Here, the load should be distributed in a way that in a data center, the utilization rate remains in balance among resources of a similar type. To make a system fault tolerant, the allocation of the resources should happen in a way that the effect of the failure of some resource component remains insignificant. Energy conservation is another important issue while managing resources at a data center. Optimal energy use reduces computing cost and it should also be minimized to reduce the carbon footprint.

    A cloud provider may opt for a number of approaches to handle these issues. It can optimize these parameters by taking a combined approach, or it can work on every metric individually. The cloud provider may use some consistent set of combinations and/or may also decide to employ varied performance goals for individual operational situations. For instance, energy minimization may be a higher priority during low-load conditions, while the service performance may be the higher priority during high load. A balance should be maintained among all these choices without affecting the user expectations. Finding out an optimal balancing point between performance and energy conservation is a critical concern of the research in such cases.

    For resource management objectives related to the Service Level Agreements (SLA), the cloud providers generally adopt an approach called service differentiation. In this approach, different levels of service performances are delivered to different consumers depending on the service level agreements made with them.

  • 17 - Approximation Methods

  • Ravinder R. Puri
  • Book:
    Non-Relativistic Quantum Mechanics
    Published online:
    16 July 2025
    Print publication:
    17 April 2017, pp 337-376

  • Summary

    Not many problems in quantum mechanics are analytically exactly solvable. The methods of solving the equations approximately therefore play an important role. Though construction of approximate solution would depend on a particular problem, there are methods, discussed in this chapter, which are of general interest as they are applicable to widely occurring situations. These methods can be divided into two categories: one for time-independent and the other for time-dependent Hamiltonians. These categories can be divided further into two classes: one in which the Hamiltonian can be divided in to a strong and a weak part allowing the weak part to be treated as a perturbation, and the other in which certain gross features of the interaction suggest approximation methods for its treatment. The quasi-classical approximation for stationary states, and the adiabatic and sudden approximations for time-dependent Hamiltonians are the examples of non-perturbative techniques.

    WKB Approximation

    The quantum theory is expected to approach the classical theory in the limit of de Broglie wavelength becoming small compared with the distances over which the potential varies. In that limit, the Schrödinger equation reduces to the Hamilton–Jacobi equation of classical mechanics. The method outlined in this section consists in finding that solution of the Schrödinger equation which contains lowest order quantum effects when the conditions for the classical approximation hold. We will see that this approximation closely reproduces the quantization rule of the old quantum theory. Consider the time-independent Schrödinger equation of a particle of mass m in the time-independent potential V(r),

  • 1 - Introduction

  • Sandeep Bhowmik
  • Book:
    Cloud Computing
    Published online:
    31 August 2020
    Print publication:
    17 April 2017, pp 1-18

  • Summary

    The catchword Cloud Computing may sound new to the arena of computing as well as information technology; although the concept behind it is not completely new. In a broad sense, it is a step (forward) to access the web-based services through Internet; or in general terms, cloud computing represents an advance in the field of computing technology where management of computing services is outsourced to greater extents and in multiple dimensions. To be specific, computing facilities are being made available as services offered by reputed vendors.

    Computing technology has been evolved over the years. There have been steady developments in the field of computing hardware, software architecture, web technology and network communications over the last decade. The speed of internetworks has increased day by day and it has also become cheaper. All these developments contributed in settinging the stage for the initiation of the revolutionary concept of ‘cloud computing’.

    Cloud computing has changed the way computation happens. It provides the means for smarter ways to do business and accordingly it makes life simpler. This chapter tries to highlight the shortcomings in the traditional computing approaches and it attempts to iterate how the stage was ready for the arrival of this new model of computing.

    Cloud computing has evolved over the years and has changed the way people live and do business over the World Wide Web.

    WHAT IS THE BUZZ ABOUT

    Cloud computing provides the means for users to easily avail computing facilities whenever and wherever required. They need not worry about setting up infrastructure, purchasing new equipment or investing in the procurement of licensed software. Rather they can access any volume, large or small, of computing facilities in exchange for some nominal payment. It is a new model of computing which has become possible through integration of advanced computing models, sophisticated web technologies and modern network communication technologies (especially high-speed Internet).

    Cloud computing became a hot topic for large computing vendors (companies like Amazon, Google, Microsoft etc.) only from mid-2008, but the concept itself is quite old. People have different perceptions regarding this new computing model. To some, it is a new and radical innovation.

  • 5 - Postulates of Quantum Mechanics

  • Ravinder R. Puri
  • Book:
    Non-Relativistic Quantum Mechanics
    Published online:
    16 July 2025
    Print publication:
    17 April 2017, pp 99-124

  • Summary

    This chapter builds the formal structure of quantum mechanics on the basis of five postulates. These postulates unify and formalize the concepts of matrix and wave mechanics introduced in Chapter 1. The first three postulates provide the framework for mathematical modelling of isolated systems and observables. The fourth postulate establishes relationship between theoretical description and experimental observations. Describing time evolution of a system is the content of the fifth postulate. The formalism is then extended to describe a composite system in terms of its constituent subsystems.

    The formalism for describing a subsystem of a composite system, leading to the concept of density operator, is developed in Chapter 6.

    Postulate 1: On Representing an Isolated System

    An isolated system is described by a vector in the Hilbert space. The vectors differing only by a multiplying constant represent the same physical state.

    A vector representing the state of a system is also referred to as a state vector. The implications of this postulate are

    • The concept of the Hilbert space has been introduced in Chapter 2. Following the notation therein, the state of an isolated system according to the postulate in question is represented by a ket . A particular state is characterized by a label inside the ket. Thus, stands for a particular state of a given system.

  • 5 - Cloud Computing Services

  • Sandeep Bhowmik
  • Book:
    Cloud Computing
    Published online:
    31 August 2020
    Print publication:
    17 April 2017, pp 76-94

  • Summary

    As discussed in the previous chapter, the NIST model of cloud computing treats the computing methodology from two perspectives as deployment and as service. While cloud can be deployed in four possible ways, the services it offers can be categorized primarily into three different types.

    Apart from three primary cloud services, computing vendors deliver a number of specialized cloud services which are considered as special services under these three primary cloud services. In order to understand how cloud computing can be of value to consumers, it is important to understand the services it offers. This chapter focusses on all of these service delivery models.

    There are many vendors who provide different cloud services in the market today. Along with proprietary public cloud services, many open-source initiatives are also available which mainly focus on to create private as well as hybrid cloud environments. Few among those open-source initiatives have also been discussed in this chapter.

    SERVICE DELIVERY MODELS

    Three categories of computing services that people consume from the days of traditional computing have been already mentioned in the first chapter. They are:

  • ▪ Infrastructure Service

  • ▪ Platform Service

  • ▪ Software Application Service

    • Cloud computing talks about delivering these facilities to consumers as computing services through network/internetwork. The benefit for the consumers is that they can avail these facilities over Internet anytime, as much as required, sitting at their own locations in a cost-effective manner. They only need to have a simple and suitable access device (like PC, laptop, tablet, mobile etc.) to access these services. Using these simple devices anyone can access any kind of computing infrastructure, platform or software application on payment-as-per-actual-usage basis.

    Cloud computing offers computing infrastructure, platform and application delivered ‘as-a-service’. Those services are considered as primary cloud computing services and are referred to as:

  • ▪ Infrastructure-as-a-Service (IaaS)

  • ▪ Platform-as-a-Service (PaaS)

  • ▪ Software-as-a-Service (SaaS)

    • These services are generally pronounced as ‘i-a-a-s’, ‘pass’, and ‘saas’ respectively. They are the driving forces behind the growth of cloud computing. Clubbed together these three service models are commonly referred as SPI (Service-Platform-Infrastructure) model. Cloud service providers arrange these services for the cloud consumers. The NIST reference architecture represents these services under ‘service orchestration’ component of the providers (Figure 5.1).

  • 3 - Finite Dimensional Spaces

  • Ravinder R. Puri
  • Book:
    Non-Relativistic Quantum Mechanics
    Published online:
    16 July 2025
    Print publication:
    17 April 2017, pp 61-86

  • Summary

    In Chapter 2, we showed that the vectors in the Hilbert space can be represented by column vectors and the operators by n × n matrices. The algebra of operators in a finite dimensional space is thus equivalent with the algebra of finite dimensional matrices. In this chapter we summarize some results of relevance to us in matrix algebra.

    Matrix Algebra

    Consider the space of finite dimension n spanned by the orthonormal basis f﹛ |k﹜. As discussed in Chapter 2, a vector jui in the space is represented by the column vectorue constituted by as its elements, whereas its dual huj is represented

    †. Similarly an operator AO is represented by the n × n matrix AQ constituted by as its elements. We restate below in matrix notation some notions of abstract operator algebra introduced in Chapter 2.

    1. In terms of their representation by column vectors, the scalar or inner product hujvi between jui and is given by

    where ﹛ṵ﹜ and ﹛ṵ﹜ are the elements ofṵ and ṵ , respectively. It may be verified that the definition of the scalar product given above is in accordance with the axioms of the scalar product.

    Let ﹛ui﹜ be an orthonormal basis spanning the given space. By virtue of (3.1) and (3.2), the completeness and the orthonormality relations,

Page 1331 of 1933