Physics
Quantum Representation
Quantum representation refers to the mathematical framework used to describe the behavior of quantum systems. It involves representing physical quantities, such as the state of a particle or the outcome of a measurement, using mathematical objects like wave functions or state vectors. This representation is essential for understanding and predicting the behavior of quantum systems, and it forms the basis of quantum mechanics.
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11 Key excerpts on "Quantum Representation"
eBook - PDF- R Mirman(Author)
- 1995(Publication Date)
- WSPC(Publisher)
Chapter V Representations V.l THE PHYSICALLY MOST RELEVANT PART OF GROUP THEORY A physicist or chemist (and often a mathematician) interested in groups usually is interested in applications. A group itself is abstract, a set of symbols plus rules relating them. What is generally needed are con-crete objects, statefunctions, their transformations, quantum numbers, quantities like angular momentum, and so on, and for a mathematician other entities, or often the same, perhaps with different names. How are these related to groups describing systems? Such physical quantities are not groups, but their representations, specifically the objects that representations consist of, basis states, called in quantum mechanics wavefunctions (or statefunctions, a better term), and the matrices that carry out the group transformations by act-ing on them. It is not the abstract symbols that we (usually) want, but these matrices and basis states (the objects given by the transforma-tion matrices of a representation acting on any one, say a unit vector). They are often explicit functions of the relevant parameters, such as angles; spherical harmonics are examples. For applications (not only to physics), we have to develop means of classifying and finding the representations of the relevant groups: matrices and basis states and functions of them, and their labels. Here we study what a representation is, its relationship to its group, and to the physical (and mathematical) quantities and properties of a system described by the group. The purpose of the present chapter is to introduce the concepts and explain their meaning and relevance, putting them into context. Vocabulary is emphasized; it is important to 146 V.2. HOW REPRESENTATIONS ARE DISGUISED 147 first thoroughly understand the terminology. In the following chapters we develop needed mathematical tools.
eBook - PDFThe Meaning of the Wave Function
In Search of the Ontology of Quantum Mechanics
- Shan Gao(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
2 On the other hand, a mathematical representation may represent different phys- ical ontologies, and one also needs to explain why one would choose one ontology rather than another. Although wave function realism is the straightforward way of thinking about the wave function realistically, there are also other possible realistic interpretations of the wave function. For example, as argued by Monton (2002, 2013) and Lewis (2013), the wave function of an N-body quantum system may rep- resent the property of N particles in three-dimensional space. 3 Moreover, according to their analysis, this interpretation is devoid of several potential problems of wave function realism (see later discussion). Thus, it seems that even if one chooses the Hilbert space formulation one should not simply assume wave function realism, either. To sum up, as Maudlin (2013) concluded, studying only the mathematics in which a physical theory is formulated is not the royal road to grasp its ontology. As I have argued in the previous chapters, we also need to study the connection between theory and experience in order to grasp the ontology of a physical theory. Next, I will analyze the second concrete motivation to adopt wave function real- ism, the existence of quantum entanglement. Indeed, quantum mechanical states are distinct from classical mechanical states in their nonseparability or entangle- ment. The wave function of an entangled N-body system, which is defined in a 3N-dimensional space, cannot be decomposed of individual three-dimensional wave functions of its subsystems. Schr¨ odinger took this as the defining feature of quantum mechanics (Schr¨ odinger, 1935b). However, contrary to the claims of Ney (2012), this feature does not necessarily imply that the 3N-dimensional space is 2 For a more detailed analysis of this problem, see Maudlin (2013).
eBook - PDF- Vishnu S. Mathur, Surendra Singh(Authors)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
However, S being a rectangular matrix, does not admit an inverse. 2.5 Coordinate Representation As all the position observables ˆ q 1 , ˆ q 2 , · · · ˆ q f , for a system with f degrees of freedom, commute with each other, we can set up a representation in which the position observables are diagonal. This representation is called the coordinate representation . In this representation, the basis states h q 1 , q 2 · · · q f | are the simultaneous eigenstates of ˆ q 1 , ˆ q 2 · · · ˆ q f belonging to the eigenvalues q 1 , q 2 · · · q f , each eigenvalue varying continuously over a certain range. Thus the basis bras, in this case, do not form a denumerable set of states (as, for example ξ (1) , ξ (2) , · · · etc.) but a continuous set of states. Consequently, the representatives h q 1 , q 2 · · · q f | A i of a state | A i , in this representation do not form a discrete set of numbers 44 Concepts in Quantum Mechanics which could be written in the form of a column vector. The representative numbers, in this case, vary continuously with the continuous variation of the eigenvalues of the position observables. So we can regard h q 1 , q 2 , · · · , q f | A i , the coordinate representative of state | A i , as a function of the eigenvalues of the position observables. This function h q 1 , q 2 · · · q f | A i ≡ Ψ A ( q 1 , q 2 · · · q f ) is called the wave function and is the most common means of representing the physical state | A i of a system. 2.5.1 Physical Interpretation of the Wave Function We have seen that, in the ˆ ξ , ˆ η , ˆ ζ , · · · representation, the quantity | C r | 2 = |h ξ ( r ) | A i| 2 is interpreted as the probability of getting the result ξ ( r ) , when a measurement of ˆ ξ is made on the system in state | A i .
eBook - PDFZero To Infinity: The Foundations Of Physics
The Foundations of Physics
- Peter Rowlands(Author)
- 2007(Publication Date)
- World Scientific(Publisher)
88 Chapter 4 Groups and Representations Some parts of 4.3 to 4.7 are based on collaborative work done with BRIAN KOBERLEIN . The Dirac nilpotent and group representations allow many mathematical reformulations of the basic structures, using reversals of properties and a potentially infinite hierarchy of dualities. The formal representations can be extended to include all the group structures of interest in physics. At the same time, there are at least three significant visual representations, which show the interchangeability of the dual processes of conjugation, complexification and dimensionalization outlined in chapters 1 and 2, and allow the possibility of incorporating the already extensive results of mathematical topology into the description of fundamental physics processes. 4.1 The Dirac Equation and Quantum Field Theory The Dirac equation, in its nilpotent form, allows an explicit treatment of mass as a ‘fifth’ dimension, on a par with the four of space and time. The five-dimensional nature can even be represented in the compact ‘5-vector’ form γ μ D μ ψ = 0, where μ = 0, 1, 2, 3, 5, and m t D + ∇ + ∂ ∂ = μ . In quaternion terms, however, mass is effectively only a third dimension, the three of space being reduced to one in the product γ . ∇ . In effect, the equation presents us with a three-dimensional system, whose units are represented by the three imaginary quaternion operators i , j , k . As these terms refer to the energy, momentum and mass operators, the equation, in a sense, incorporates three degrees of freedom associated with the parameter mass. Mass conservation is shown with respect to time and space variation, and with respect to mass itself. Groups and Representations 89 So, the equation seems to be suggesting a quasi-‘three-dimensionality’ in mass (and even, indirectly, for a quantum system, in time, cf 8.3 1 ), which is somehow linked to the existence of three independent parameters of measurement.
eBook - PDF- F. Constantinescu, E. Magyari, J.A. Spiers(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
113 Problems in Quantum Mechanics where the implied time-dependence of ip(q') is not shown explicitly. In the Schrödinger picture in {q} representation the rate of change with time of p(q') is given by the usual Schrödinger equation. It should be noted that although the symbol q, suggestive of the generalized coordi-nates of classical Lagrangian and Hamiltonian mechanics, is traditionally used in abstract argument to denote position variables, extreme care is necessary in practice if the q's are used to denote anything other than the Cartesian coordinates of individual particles of a system. 2.2. THE MOMENTUM REPRESENTATION {/?} The base vectors of this representation, p) = ΡΊΡ 2 ΡΖ)> are the simultaneous eigenvec-tors of the complete set of observables pi, p 2 , p$ (the momenta conjugate to the coordinates #1, #2, qs). The orthonormality and closure relations are similar to those given in (V.9), (V. 10). The state vector ψ) is represented by the one-column continuous matrix (ρ'ψ) = Φ(//), called the wavefunction in the {p} representation (the implied time-dependence of Φ(/?') is not shown explicitly). The following relations hold between these two representations n = 1, 2, 3 (ρ'ΐηΦ) = ϊ*-^ΓΦ(ρ') (V.ll) . (V.12) The time-dependence of this wavefunction is of the form -iE(t-t 0 ) ψ(Ε, α; t) = ψ(Ε, a; t 0 )e h . (V.13) 114
eBook - PDF- Alan C. Tribble(Author)
- 2018(Publication Date)
- Princeton University Press(Publisher)
8 Quantum Mechanics 8.1 Fundamental Postulates of Quantum Mechanics Like Newton's laws or Maxwell's equations, the laws of quantum mechanics cannot be derived. The justification for quantum mechanics is its agreement with observation. The mathematical justification for quantum mechanics evolve from a number of fundamental assumptions that are summarized, without elaboration, as follows. 8.1.1 State Functions We shall make the assumption that the state of a particle at time t is completely describable by some function v|/ that we shall call the state function (wave function) of the particle or system. Only those state functions that are physically admissible correspond to realizable physical states. We also make the plausible and physically necessary assumptions that the probability of finding a particle is large where v;/ is large and small where \\J is small. I f P(x,t)dx is the relative probability of finding the particle at time t within a volume dx centered about x, then P(x,t) = |v|/(x,0| 2 = \|/*(JC,/)V(*,0, (8-0 and the absolute probability is P (x,t)dx = - p ^ . (8.2) J P(x,t)dx For normalized state functions, Jv|/*V|/dx: = 1, (8.3) and 192 • Quantum Mechanics p{x,t) = \\i*{x,t)\\f{x,t) (8.4) is the probability amplitude. An important property of the state function is that i f §i describes one possible state and 2 describes a second possible state. A third state can be formed from a linear combination of ty x and (j> 2 . In general, v^, (8.5) where are arbitrary constants and are independent state functions. This principle of superposition is necessary to support experimental observations of interference. 8.1.2 Operators Each dynamic variable that relates to the motion of the particle can be represented by a linear operator. We assume that the eigenvalues of a physical operator form a complete set. One of the eigenvalues co ^ is the only possible value of a precise measurement of the dynamical variable represented by Q.
eBook - PDF- Bernard Dugué(Author)
- 2018(Publication Date)
- Wiley-ISTE(Publisher)
1.2. Quantum states or how nature communicates with physicists Let us start with a general idea that is applicable to every physical experience; two notions are always present: state and measurement. In a sense, a state is like a “dynamic and kinematic inventory”. With variables and equations that determine dynamics and kinematics, that is, the arrangement of forces and the movement of particles. A measurement is a number obtained during observation. It may be a speed, a position, an angle, a mass, a trajectory, a translational or torsion force, kinetic momentum, a particle detected with a spin, etc. The analysis of the “epistemological work” around the notions of state and measurement makes it possible to highlight the irreducible difference between classical physics and quantum dynamics. Firstly, the definition of a quantum system or a quantum state with wave functions is completely different from the descriptions used in mechanics. Second, we have to be aware that while in classical mechanics the notions used for describing the state of a system are also implemented for describing observations (speed, angles and positions); in quantum mechanics, this is no longer the case. I will now illustrate with the example of a function defined as the Lagrangian in classical mechanics. To calculate it, we have to subtract the kinetic energy from the potential energy and its formula can be summarized as L = L (p, q). In this formula, it is sufficient to know the potential and six parameters, three for p, the position, and three for q, the impulse, the latter being calculated from m mass and dp/dt speed. In a way, the Lagrangian sums up the state of the system, as it evolves along a trajectory by obeying the principle of least action. It is clear that the parameters used for describing the state are also the parameters used in experimental measurement (mass, speed and
- R. Guy Woolley(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
It is essential to recognise that in the infinite-dimensional case any two representa- tions are in general unitarily inequivalent and hence a choice has to be made for the 179 5.4 Quantisation for a Field Hilbert space H. The discussion so far has been based on a formal analogy with the Schrödinger representation of quantum mechanics. Clearly, the representation ought to be chosen so that the Hamiltonian can be written as a well-defined operator in H; this is easier said than done, however, and the infinities that have plagued quantum field theory since its inception can be understood as a reflection of the difficulty in finding a physically significant space. Our main concern is with low-energy phenomena for which a ‘non-relativistic’ treatment is expected to be appropriate. Consequently, eve- rything we do really implies that a high-energy (or momentum) cut-off will be applied, and this will yield a useful regularised theory, provided the answers do not depend on the cut-off. However that may be, in practice an especially useful representation is afforded by the Fock space construction which we sketch briefly here; it will be dis- cussed in detail for the quantised electromagnetic field in Chapter 7. We confine our attention to a field satisfying Bose–Einstein statistics. A boson field operator ψ(u) can be defined in terms of the basic variables, X and P, by setting ψ(u) = 1 √ 2¯ h ( X(u)+ iP(u) ) , (5.175) which is such that its commutation relation with its complex conjugate ψ * is [ψ(u), ψ(u 0 ) * ] = δ (u - u 0 ). (5.176) The corresponding smeared operators are ψ( f ) = Z ψ(u) f (u) d 3 u, (5.177) and similarly for ψ(u 0 ) * . The Fock space can then be defined by the statement that there exists a vacuum vector, Φ F , satisfying ψ( f ) Φ F = 0, (5.178) for all f (u) in the Schwartz space. The operator ψ( f ) is called the field annihilation operator; its conjugate, ψ( f ) * is the field creation operator.
eBook - PDFQuantum Physics
An Introduction
- J Manners(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
That is to say, it is concerned mainly with formalism. It starts by reviewing and elaborating those parts of the formalism that were introduced in Chapters 2 and 3. It then extends the formalism by introducing the related concepts of eigenstate and eigenvalue, and by explaining the importance of linear superposition and coherence in quantum mechanics. The section ends with a first look at what is generally regarded as the ‘conventional’ interpretation of quantum mechanics and at some of the issues that it raises. 2.1 Quantum systems The starting point of any application of quantum mechanics is the identification and specification of the system to which the formalism will be applied. Such a system is often referred to as a quantum system, though its specification generally involves purely classical concepts. You have already met many examples of quantum systems. The simplest, considered in Chapter 2, consisted of a particle of mass m that was free to move in one dimension. In that chapter you were also introduced to a variety of other one-dimensional systems in which the potential energy of the particle was described by a function £p0t(x) that was either a well, a barrier or a step. Some of these potential energy functions are illustrated in Figure 4.1. Figure 4.1 Some of the one-dimensional potential energy functions that were considered in the discussion of quantum mechanics in Chapter 2. A more complicated quantum system formed the main subject of Chapter 3 — the hydrogen atom. In that case, the system consisted of an electron (a particle of mass me, charge -e, etc.) moving in three dimensions under the influence of a positively charged nucleus. The details of the nucleus were ignored; all of its effects on the electron were accounted for by assuming that the electron’s potential energy was described by the function £ ^ 0 ) = -e 2/ ( 4 n £br), with the radial coordinate r being measured from the centre of the nucleus.
- Michael A. Parker(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
However, the observations made in the two portions of the fi gure must agree. This brings out another point for comparing and contrasting operators and vectors in the quantum theory. Regardless of the representation, an operator must contain all possible outcomes to an obser-vation or operation. We can understand this point of view using the basis vector expansion of an operator found in Chapter 3 and Section 5.1. For example using the energy basis set, the Hamiltonian ^ H ¼ X All E E j f E ih f E j¼ X All E E j E ih E j consists of all possible results of the observation because of the sum over all the energy eigenvalues E . However, the ‘‘ wave functions ’’ are written as a speci fi c sum over the basis set; only a certain combination of basis vectors appears in the sum for the wave functions. For example, the wave function j C i¼ X Some E b E j f E i¼ X Some E b E j E i contains information on only speci fi c eigenvalues E . Even if it contains all eigenvalues, the sum refers to only one certain mixture (i.e., one vector in the Hilbert space) because of the speci fi c values of b chosen. In summary, operators contain all possible results of a measurement while vectors represent speci fi c instances of the system in question. The ‘‘ interaction representation ’’ assigns some time dependence to the operators and some to the wave functions. We will fi nd this representation especially suited for an ‘‘ open ’’ system. First, consider a ‘‘ closed system ’’ for which the number of particles and the total energy contained within the system remain constant. Basically one assumes that the Schrödinger ’ s equation has been solved Observers move Universe stationary Heisenberg picture |2 |1 |ψ Schrodinger picture u Observers stationary |ψ(t) |2 |1 Universe moves FIGURE 5.39 Cartoon representation of the Schrödinger and Heisenberg pictures. 332 Solid State and Quantum Theory for Optoelectronics
- N N Bogolubov, N N Bogolubov, Jr.(Authors)
- 2014(Publication Date)
- WSPC(Publisher)
Furthermore, because of the of the Method of Approximate )f the second quantized unwieldy calculations involved, a majority of textbooks on quantum physics and statistical physics simply list the results of the second quantization method, which as a rule are of a /ery formal nature, and do not provide any understanding of the physical meaning of this important method. In the approach that we have adopted, given to the writing of dynamical quantities due attention has been in the second quantized representation for 1-body, 2-body and s-body quantities using a series of new lemmas. Systems consisting of severa bosons have also been considered. Equations evolution of operator functions have been di< 1 types of fermions and of motion for the time scussed and 'self-consistent' field equations developed in the operator form. A series of ideas, analogous to the second quantization method, kinetic theory of gases, has been discussed • which can be used in the in detail. CHAPTER 1 MATRIX REPRESENTATION OF SYMMETRICAL DYNAMICAL OPERATORS In this chapter and subsequent ones, certain developments, conventions and discussions parallel closely those of earlier chapters in the first half of the volume. Rather than economise on words, the repetitions have been made, no doubt abbreviated, to provide a summary and review of basic ideas. This creates a wholly self-contained unit in the second half that allows those already acquainted with the basics to start immediately on the elements of Second Quantization. LI Introduction Consider states defined -Qi^i dependent upon gate of spatial the jth particl where x = (q, > q (a) (a a dynamical system of N identical partic by wavefunctions ,...* N ) variables Xy...x„. Here, x-represents coordinates and the supplementary quantum r e: V ) ; q=(q ( l ) ,q ( 2 ) ,q ( 3 ) ) .
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