Here, | ( )ψ t > is the Dirac ket state. The corresponding bra state is denoted as < ψ( ) |t . The time-dependent Schrödinger equation (2.1) is one of the most fundamental and important postulates in quantum mechanics.6-8 In this chapter, we introduce the concept of linear vector (wavefunction) and matrix (density operator) spaces to describe the quantum dynamics of matters interacting with external radiations.
Here, | ( )ψ t > is the Dirac ket state. The corresponding bra state is denoted as < ψ( ) |t . The time-dependent Schrödinger equation (2.1) is one of the most fundamental and important postulates in quantum mechanics.6-8 In this chapter, we introduce the concept of linear vector (wavefunction) and matrix (density operator) spaces to describe the quantum dynamics of matters interacting with external radiations.
We summarize below some of the postulates and definitions basic to our formalism, and present some important results based on these postulates. The formalism is purely mathematical in nature with very little physics input, but it provides the structure within which the physical concepts that will be discussed in the later chapters will be framed. State vectors It is important to realize that the Quantum Theory is a linear theory in which the physical state of a system is described by a vector in a complex, linear vector space. This vector may represent a free particle or a particle bound in an atom or a particle interacting with other particles or with external fields. It is much like a vector in ordinary three-dimensional space, following many of the same rules, except that it describes a very complicated physical system. We will be elaborating further on this in the following. The mathematical structure of a quantum mechanical system will be presented in terms of the notations developed by Dirac. A physical state in this notation is described by a “ket” vector, |〉, designated variously as |α〉 or |ψ〉 or a ket with other appropriate symbols depending on the specific problem at hand. The kets can be complex. Their complex conjugates, |〉 * , are designated by 〈| which are called “bra” vectors. Thus, corresponding to every ket vector there is a bra vector. These vectors are abstract quantities whose physical interpretation is derived through their so-called “representatives” in the coordinate or momentum space or in a space appropriate to the problem under consideration.
Abstract Chapter 1 opens with a brief review of some basic features of quantum mechanics, including the Schrödinger equation, linear and angular momentum and the theory of the hydrogenic atom: It also includes complete orthonormal sets of eigenfunctions, the translation operator, current, spin, equation of continuity, gauge transformation, determinant & permanent multiparticle energy eigenfunctions for noninteracting particles and the Pauli exclusion principle. Attention is then focused on Dirac bra-ket notation and complete sets of commuting observables. In this connection, representations and transformation among representations are discussed in detail for the Schrödinger system state vector and the eigenstates, as well as bra-ket matrix elements of operators. Finally, Schwinger’s interpretation of ket-bra matrix operator structures (Schwinger “Measurement Symbols”) in terms of annihilation and creation of systems in eigenstates is introduced.
The content of this chapter may serve as yet another supplemental topic to meet the needs and interests beyond those of a usual course curriculum. Here we shall present an over-simplified, but hopefully totally transparent, description of some of the fundamental ideas and concepts of quantum mechanics, using a pure linear algebra formalism. Vectors in ℂ n and Dirac bracket Consider the vector space ℂ n , consisting of column vectors, and use { e 1 , …, e n } to denote the standard basis of ℂ n . For u , υ ∈ ℂ n with recall that the Hermitian scalar product is given by so that { e 1 , …, e n } is a unitary basis, satisfying ( e i , e j ) = δ ij , i, j = 1, …, n . In quantum mechanics, it is customary to rewrite the scalar product (9.1.2) in a bracket form, 〈 u | v 〉. Then it was Dirac who suggested to view 〈 u |υ〉 as the scalar pairing of a ‘bra’ vector 〈 u | and a ‘ket’ vector |υ〉, representing the row vector u † and the column vector υ. Thus we may use | e 1 〉, …, | e n 〉 to denote the standard basis vectors of ℂ n and represent the vector u in ℂ n as Therefore the bra-counterpart of | u 〉 is simply given as As a consequence, the orthonormal condition regarding the basis { e 1 , …, e n } becomes and the Hermitian scalar product of the vectors | u 〉 and |υ〉 assumes the form For the vector | u 〉 given in (9.1.3), we find that a i = 〈 e i | u 〉, i = 1, …, n . Now rewriting | u 〉 as and inserting (9.1.7) into (9.1.8), we obtain which suggests that the strange-looking ‘quantity’, should naturally be identified as the identity mapping or matrix, which readily follows from the associativity property of matrix multiplication. Similarly, we have Thus (9.1.10) can be applied to both bra and ket vectors symmetrically and what it expresses is simply the fact that | e 1 〉, …,| e n form an orthonormal basis of ℂ n .
Hilbert space and inner product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ( x ) which depends upon the position variable x , or by an alternative function ψ( p ) of the momentum variable p . It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars , which are complex numbers, are the elements of the quantum Hilbert space H . (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function ɑψ( x ) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø( x )+ψ( x ) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ( x ) and ( p ), so that αø( x )+βψ( x ) and αϕ( p ) + βψ( p ) correspond to the same quantum state α|ψ〉+β|ø〉.
Come forth into the light of things, Let nature be your teacher. William Wordsworth , The Tables Turned The main focus of this book is many-particle systems such as electrons in a crystal. Such systems are studied within the framework of quantum mechanics, with which the reader is assumed to be familiar. Nevertheless, a brief review of this subject will provide an opportunity to establish notation and collect results that will be used later on. The postulates Quantummechanics is based on five postulates, listed below with some explanatory comments. The quantum state The quantum state of a particle, at time t , is described by a continuous, singlevalued, square-integrable wave function Ψ( r , t ), where r is the position of the particle. In Dirac notation, the state is represented by a state vector, or ket, ∣Ψ( t )⟩, which is an element of a vector space V. We define a dual vector space V* whose elements, called bras, are in one-to-one correspondence with the elements of V: ket ∣α⟩ ∈ V ⟷ bra ⟨α∣ ∈ V*, as illustrated in Figure 1.1. The bra corresponding to ket c ∣α⟩ is c * ⟨∣, where c *. is the complex conjugate of c . The inner product of kets ∣α⟩ and ∣β⟩ is denoted by ⟨β∣α⟩, and it is a complex number (c-number). Note that the inner product is obtained by combining a bra and a ket. By definition, ⟨β∣α⟩ = ⟨α∣β⟩*.
Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles