Dirac Notation

9 Key excerpts on "Dirac Notation"

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  How to Be a Quantum Mechanic

  • Charles G. Wohl(Author)
  • 2022(Publication Date)
  • CRC Press

    (Publisher)

  5. MATHEMATICAL FORMALISM 5.1. Vector Spaces. Dirac Notation 5.2. States as Vectors 5.3. Operators 5.4. Successive Operations. Commutators 5.5. Operators as Matrices 5.6. Expectation Values 5.7. More Theorems 5.8. Revised Rules Problems We develop much of the mathematical formalism of quantum mechanics. As Eugene Wigner said, “The miracle of the appropriateness of the language of mathematics for the for- mulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” We begin by introducing in an informal way a unifying notational scheme invented by Paul Adrien Maurice Dirac. Our first use of Dirac Notation is to extend the domain of vectors to include the states that describe a quantum system. This lets us carry over the visual language of vector geometry (length, direction, orthogonality, orthogonal unit axes, components, and so on) to those states. A vector to represent the general state of a harmonic oscillator will require an infinite number of axes and components (Chapter 6); a vector to represent the general spin state of a spin-1/2 particle will require just two (Chapter 10). Whereas the states of quantum systems are represented by functions or vectors, the physical properties we can measure (“observables”) are represented by operators. Thus far we have used the position and momentum operators ˆ x and ˆ p, and the energy operator ˆ H constructed from them. The operators that represent observables are both linear and Hermi- tian. These properties (to be defined) lead to two easy-to-prove and extremely useful results: (1) The eigenvalues of linear, Hermitian operators are real; (2) the eigenstates that belong to eigenvalues that differ are orthogonal. A crucial property of the operators for observables is that the order of the operations can matter: The operators do not all commute with one another. Most fundamentally, the operators ˆ x and ˆ p do not commute. Full understanding of the formalism will only come from using it.

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  Introduction to Quantum Effects in Gravity

  • Viatcheslav Mukhanov, Sergei Winitzki(Authors)
  • 2007(Publication Date)
  • Cambridge University Press

    (Publisher)

  It turns out that the space of quantum states of a system with one degree of freedom is exactly the space of square-integrable functions q, where q is a 2 This is easy to prove by considering the trace of a commutator. If ˆ A and ˆ B are arbitrary finite-dimensional matrices, then Tr  ˆ A ˆ B = Tr ˆ A ˆ B − Tr ˆ B ˆ A = 0 which contradicts equation (2.20). In an infinite-dimensional space, this reasoning no longer holds because the trace is not well-defined for an arbitrary operator. 2.4 Hilbert spaces and Dirac Notation 21 generalized coordinate of a system (for example, position of a particle moving in one dimensional space). In this case the function q is called the wave function. In the case of two degrees of freedom the wave function depends on both coor- dinates q 1 and q 2 characterizing the state of the corresponding classical system,  =  q 1 q 2 . In quantum field theory, the “coordinates” are field configurations x ≡  x and the wave function depends on infinitely many “coordinates”  x ; in other words, it is a functional,  x. The Dirac Notation Linear algebra is used in many areas of physics, and the Dirac Notation is a convenient shorthand for calculations in both finite- and infinite-dimensional vector spaces. To denote vectors in abstract linear space, Dirac proposed to use symbols such as a, x,   , which he called “ket”-vectors. Then the symbol 2 v− 3i w, for example, denotes a linear combination of the vectors v and w. Dual space In a vector space V one can define linear forms, which act on a vector to produce a (complex) number; f  V → C. A linear form is called covector or “bra”-vector and denoted by f . A complex number produced by linear form f  as a result of acting on a vector v is denoted by f v (the mnemonic rule is: “bra”-vector acting on “ket”-vector makes a “bracket”, which is a complex number).

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  Introduction to Quantum Mechanics 2

  Wave-Corpuscle, Quantization and Schrodinger's Equation

  • Ibrahima Sakho(Author)
  • 2020(Publication Date)
  • Wiley-ISTE

    (Publisher)

  In 1928, Dirac established the relativistic wave equation in order to unify quantum mechanics and the theory of special relativity. The development of this relativistic theory of quantum mechanics made it possible to predict in 1931 the existence of a positron (positive electron), which was discovered in 1932 by the American physicist Carl David Anderson (1905–1991). In 1930, Dirac introduced linear operator algebra as a generalization of Heisenberg’s and Schrödinger’s theories. He also introduced the notions of ket and bra known as Dirac’s notations, greatly simplifying the mathematical formalism of quantum mechanics. In 1933, he was awarded the Nobel Prize for physics, which he shared with Schrödinger for their important contributions to quantum mechanics. The symbol 〈|〉 is known as bracket. This explains the origin of the names ket for the symbol |〉 and bra for the symbol 〈|. The ket |〉 and bra 〈| notations are known as Dirac’s notations to honor Paul Dirac who introduced them to quantum mechanics. 2.2.3. Properties of the scalar product The scalar product has been defined for two square-summable wave functions [1.2] and its properties have also been established [1.3]. They are established here once again in the space of states. Using Dirac’s notations, the scalar product of ket |Ψ〉 and ket |Φ〉 is defined by the relation: [2.15] This scalar product verifies the following properties: [2.16] If 〈Φ|Ψ〉 = 0, then ket |Φ〉 and ket |Ψ〉 are orthogonal. If |Φ〉 = |Ψ〉 then the squared norm of the ket is equal to 〈Ψ|Ψ〉. If the ket |Ψ〉 is normed to unity, then 〈Ψ|Ψ〉 = 1. 2.2.4. Discrete orthonormal bases, ket component Discrete orthonormal bases have been previously defined [2.3] for the space of square-summable wave functions

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  A Mathematical Introduction to Dirac's Formalism

  • S.J.L. van Eijndhoven, J. de Graaf(Authors)
  • 1986(Publication Date)
  • North Holland

    (Publisher)

  Also, in our interpretation a central role is played by our new mathematical notion of Dirac basis. The mentioned concepts of Dirac basis and of bracket together constitute the p i l l a r s on which we build a mathematical apparatus that t o a great extent founds the bold claims on which Dirac has based h i s principles of quantum mechanics. This Page Intentionally Left Blank 285 CHAPTER I D I R A C ' S FORMALISM ACCORDING TO D I R A C AND ITS RELATIONS WITH L I N E A R ALGEBRA This chapter contains a description of Dirac's formalism according t o Dirac's own original introduction. From our point of view Dirac claims t h a t several concepts of the theory of f i n i t e dimensional vector spaces remain valid i n a certain unspecified type of i n f i n i t e dimensional vector spaces, replacing sums by integrals, discrete bases by continuum bases, Kronecker deltas by D i r a c deltas, etc. W e approach Dirac's formalism having these re- lations with linear algebra i n mind. So it is natural t o summarize certain well-known concepts of linear algebra presenting them i n bracket notation. 1.1. Some elementary concepts of linear algebra L e t v denote a complex f i n i t e dimensional vector space. It means t h a t v is a complex vector space for which there exists a number d E I N and a linear isomorphism which sends v onto Cd. The number d is called the dimension of v. In the sequel we take d fixed. The elements of v are denoted by la> (or la, ... a,>) where the l a b e l ( s ) a (or a l , ..., a ) can be chosen so t h a t cer- t a i n special properties of the considered vector are indicated. If a t 6 and la> and Ib> belong t o v, then also ala> and la> + Ib> belong t o v. We observe that, i n general, it does not make sense t o write laa> or la+ b> in- stead of ala> or la> + Ib> and even when it does, the vectors laa> and la+ b> may not equal ala> o r la> + Ib>.

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  Quantum Mechanics

  Principles and Formalism

  • Roy McWeeny(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER 5

  GENERAL THEORY OF REPRESENTATIONS

  5.1. Dirac Notation. Discrete case

  In Chapter 4 the postulates were formulated generally enough to provide a basis for the whole of non-relativistic quantum mechanics, but so far we are familiar with only one mathematical realization of the vectors and operators that occur throughout the theory. In Schrödinger language the vectors are elements of a Hilbert space comprising all (well-behaved) functions of particle coordinates, and the operators are partial differential operators working on these functions. Other possibilities clearly exist, however. The commutation relations (4.30) are equally well satisfied by the association

  (5.1)

  and this suggests a description in terms of functions of the momentum components, the operators associated with momentum becoming the multipliers, those associated with spatial coordinates becoming differential operators working on the momentum variables. We shall find presently that the Schrödinger and momentum languages are indeed equivalent and equally acceptable. First, however, we shall examine in a general way the possibility of passing from one type of description to another; this transformation theory is most easily formulated by starting from the matrix representation of the operator equations and using the Dirac Notation (used so far only to indicate scalar products), which must now be explained in its full generality.

  Let us introduce a discrete basis {Φ i } such that

  (5.2)

  The key equations referring to (i) the action of an operator on a vector, and (ii) the product of two operators, may then be transcribed into a matrix form fully discussed in Section 3.5. The basic equations become

  (5.3a)

  (5.3b)

  where the three statements on each line are entirely equivalent. In the matrix form (second statement) c and c′ are sets of expansion coefficients representing Ψ and Ψ′ (collected into column matrices), while A, B and C are square matrices representing the operators A, B and C . The matrix equations may be written in subscript form (third statement) where A ij , for example, is the element in the i th row and j th column of matrix A

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  Relativistic Quantum Chemistry

  The Fundamental Theory of Molecular Science

  • Markus Reiher, Alexander Wolf(Authors)
  • 2014(Publication Date)
  • Wiley-VCH

    (Publisher)

  Obviously, the situation with Dirac’s theory of the electron requires revision and extension, which allows us to keep the virtues of Dirac’s equation and to find a formulation which is more consistent with respect to the conceptual difficulties just mentioned. This will lead to the formulation of quantum electrodynamics, as we shall see in chapter 7. We then understand that the original Dirac equation must not be considered as a quantum mechanical wave equation, but rather as the equation of motion for a classical fermionic field, which might subsequently be quantized by the methods of second quantization (meaning quantization of fields). Then, in quantum field theory no negative-energy states and related pathologies show up due to the beneficial protocol of normal ordering. It is interesting to note that even today, when hole theory and quantum field theory are compared, situations have been found where they do not only differ from a conceptual point of view, but lead to different results in the presence of external perturbations [106, 107]. However, before we delve deeper into these issues, we first consider the Dirac electron in an external field as an essential prerequisite of all later chapters.

  Further Reading

  F. Schwabl, [79]. Advanced Quantum Mechanics .

  In his very useful book on advanced topics in quantum mechanics Schwabl presents a very extensive discussion of relativistic quantum mechanics as needed for the molecular sciences (rather than with a strong focus on particle physics). It can be recommended to readers who are starting to learn about relativistic quantum mechanics and also to those who want to quickly look up certain topics.

  P. Strange, [108]. Relativistic Quantum Mechanics — with Applications in Condensed Matter and Atomic Physics .

  This book is an excellent introduction and an advanced text at the same time. It will not fail the beginner and provides a lot of additional material necessary to better understand relativistic quantum mechanics applied to spherically symmetric systems, i.e., to atoms, but also to solid-state physics.

  B. Thaller, [109]. The Dirac Equation .

  Thaller’s book is a comprehensive discussion of the Dirac equation from a rather mathematical point of view. Ten years after publication of this work, Thaller contributed a review-like article [110] to Schwerdtfeger’s collection of review articles on relativistic quantum mechanics [34].

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  Pauli And The Spin-statistics Theorem

  • Ian Duck, E C George Sudarshan(Authors)
  • 1998(Publication Date)
  • World Scientific

    (Publisher)

  First he defines a Fock state |rir 2 ---r n ---rjv) which corresponds to particle 1 in state 7*1 and so on. In order to describe identical Bose-Einstein particles, the state vector must be symmetrized on the N particles and normalized. These states are related to the occupation number state (the TV-states) Ni,N 2 ---N s ) corresponding to Ni particles in state 1, etc. Dirac derives the proportion-ality factor between these states by an elegant combinatorial argument in Eqn.16. The states satisfy a Schrodinger equation with a Hamiltonian oper-ator which Dirac eventually constructs by imposing canonical commutation relations on suitably chosen variables. His first choice, the wave function expansion coefficient a r of Eqn.4, obeys a Schrodinger equation with a time dependent interaction Hamiltonian H , in what is now called the interaction representation. The expansion coefficients a r satisfy the equation iha r = ^2V rs a s (1) s and -iha* = Y^ a *s V sr s where V* 3 = V sr . With the identification Q r — a r and P r = iha*, these have the Hamiltonian form with H 1 = Y,-jrP r V ra Q„ (2) 150 and Qr^dHi/dPr, P r = ~dH 1 /dQ r . (3) Interpreted as c-number amplitudes, the probability of the system being in state V is |a r | 2 which, properly normalized, is the occupation number N r . Dirac next introduces new real canonical variables -N r and a phase — (j) r -by a classical contact transformation Q r = N r 2e-i,h , P r = ihN r 2e l °V h , (5) rs and the equations of motion for the new variables N r and —

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  Elementary Particle Physics in a Nutshell

  • Christopher G. Tully(Author)
  • 2011(Publication Date)
  • Princeton University Press

    (Publisher)

  2 Dirac Equation and Quantum Electrodynamics

  Consider a spin particle, for example, an electron at rest. The rotations of the electron spin are described by the SU(2) group and the states of the system are labeled by the values of the total spin, or specifically S2 , and the projection of the spin onto an arbitrary axis of quantization, for example, Sz , the component along the z-direction. Now take the electron and Lorentz boost into a frame moving to the right (along the x-axis). How do we construct a wave function of a relativistic electron?

  The answer to this question will lead us to one of the most profound equations in elementary particle physics, the Dirac equation. The Dirac equation introduces the first of several symmetry extensions to the structure of matter, namely, the antiparticles.

  2.1 Natural Units and Conversions

  By choosing natural units, where

  the quantities of mass, inverse length, and inverse time can be described by a single dimensional unit. The choice in this text is to measure all quantities in units of GeV. The conversion to units of meters and seconds is handled, for the most part, by inserting values of c and where needed:

  The charge of the electron is denoted e with e < 0. In general, the conventions of the Peskin and Schroeder [1] text are followed for all calculations.

  2.2 Relativistic Invariance

  A wave function in nonrelativistic quantum mechanics describes the probability of finding a particle at time t in a volume element d3 x centered at position x by the quantity The normalization of the wave function is defined to be unity when integrated over all space, or a periodic box of volume V for plane-wave solutions,

  indicating that in a fixed snapshot of time the particle must be somewhere. The generators of infinitesimal translations

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  Free Theory

  • Anders Bengtsson(Author)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  3.3.1 Baseline quantum mechanics Two basic ingredients in any quantum theory are the operators and the states , nei-ther of which are themselves generally accessible to direct observation. Together they constitute what we mean by a quantum system . Closer to measurable quantities are the matrix elements of operators evaluated between pairs of states. Another, almost defin- 3.3 Quantum mechanics and quantum field theory | 129 ing, property of quantum systems is the linear superposition of states into new states. The mathematical structure that has turned out to encode these features of quantum systems in general in a successful way, are the Hilbert spaces (see Section 3.7.6). Hilbert space in few lines A Hilbert space is a complex linear vector space (superposition possible) with a metric (distance mea-sure between vectors) that derives from an inner product (matrix elements). Linear operators can be applied to the vectors and matrix elements can be computed using the inner product. There is a notion of continuity in the sense of nearness of vectors as measured by the metric. Relying on the Hilbert space concept, states of a quantum system are represented by equivalence classes of vectors Ψ, called rays . The inner product between two rays Ψ and Φ is denoted by ⟨ Φ , Ψ ⟩ and it evaluates to a complex number. The rays are nor-malized in the sense that ⟨ Ψ , Ψ ⟩ = 1 and Ψ and Ψ ? belong to the same ray if and only if Ψ ? = c Ψ with c a complex number with | c | = 1. A ray, that is, a state, can be repre-sented by any of its vectors Ψ belonging to the ray. The following properties for the inner product between states, are fundamental ⟨ Φ , Ψ ⟩ ∗ = ⟨ Ψ , Φ ⟩ (3.63) ⟨ Ψ , Ψ ⟩ ≥ 0 with ⟨ Ψ , Ψ ⟩ = 0 ⇔ Ψ = 0 (3.64) ⟨ Φ , c 1 Ψ 1 + c 2 Ψ 2 ⟩ = c 1 ⟨ Φ , Ψ 1 ⟩ + c 2 ⟨ Φ , Ψ 2 ⟩ (3.65) ⟨ c 1 Φ 1 + c 2 Φ 2 , Ψ ⟩ = c ∗ 1 ⟨ Φ 1 , Ψ ⟩ + c ∗ 2 ⟨ Φ 2 , Ψ ⟩ (3.66) where the two last equations express how the norm behaves under superposition of states.

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