Hilbert space and Dirac notation

This section introduces the basic mathematics of linear vector spaces as an alternative conceptual scheme for quantum-mechanical wave functions. The concept of vector spaces was developed before quantum mechanics, but Dirac applied it to wave functions and introduced a particularly useful and widely accepted notation. Much of the literature on quantum mechanics uses Dirac s ideas and notation. 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

In general, the ket 0,) is not in the same direction as 0,) nor in the same direction as any other ket ipj), but rather has projections along several or all basis kets. If an operator A acts on all kets 0,) of the basis set, and the resulting set of kets 0,) = Aipi) are orthonormal, then the net result of the 

Although the expressions A, ) and A, ) are completely equivalent, there is a subtle distinction between them. The first, A, ), indicates the operator A being applied to the ket, ). The quantity A ipi) is the ket which results from that application. 

The functions tpi(x) are, in general, complex functions. As a consequence, ket space is a complex vector space, making it mathematically necessary to introduce a corresponding set of vectors which are the adjoints of the ket vectors. The adjoint (sometimes also called the complex conjugate transpose) of a complex vector is the generalization of the complex conjugate of a complex number. In Dirac notation these adjoint vectors are called bra vectors or bras and are denoted by or (/. Thus, the bra (0,j is the adjoint of the ket, ) and, conversely, the ket j, ) is the adjoint (0,j of the bra (0,j 

These bra vectors determine a bra space, just as the kets determine ket space. 

When a ket, ) is multiplied by a constant c, the result c = cii, ) is a ket in the same direction as, ) only the magnitude of the ket vector is changed. However, when an operator A acts on a ket /), the result is another ket 10/) 

Although the expressions A ipi) and are completely equivalent, there is a subtle distinction between them. The first, indicates the operator A 

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Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

Using the Dirac notations a) = ipa( ) and assuming that ipa( ) are or-thonormal functions (a (3) = 5ap we can write the single-particle matrix (tight-binding ) Hamiltonian in the Hilbert space formed by 4>a ( )... 

The inner product (scalar product) (.,. ) in the Hilbert space Y follows from the canonical rules for inner products of direct sums and tensor products of Hilbert spaces like mentioned above. We use common Dirac notation for matrix elements (T O Z) of an operator O cuid vectors F) and Z) in the Hilbert space Y. 

Note that these equations can also be written without resuming to the jU-product notation indicated by the round brackets. Instead one can use common Dirac notation of the inner product in the Hilbert space Y which is indicated by the use of angular brackets like usual. In this case the metric operator fi (defined in Sec. IIB) appears explicitly and, e.g., Eq. (26) becomes... 

In the previous section, it has been shown that the convex structure of D and the energy for ground states evolve into two branches, each one as a two-state level model of N and N l Hilbert spaces as expressed in Eqs. (6) and (7) [4, 12, 13]. The corresponding pure ground state DMs in Dirac notation reads,... 

State vectors and Hilbert spaces. The state of a physical system is represented by its state vector, which is a vector in a complex Hilbert space 1-1. The notation which is used in the following is due to Paul Dirac. According to this notation the state vectors are represented as follows... 

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