A simple operation is to lengthen or to shorten a vector by a factor a without changing its orientation. The simplest operator of this kind is the identity operatorII^. The operators introduced so far have a clear geometric meaning. One should use these operators as building blocks to establish other operators which may not have distinctive geometric properties. The building blocks we have in mind are projectors. Selfadjoint matrices and selfadjoint operators have many distinctive properties which are crucial to the probabilistic formulation of quantum mechanics. Since selfadjoint operators are represented by selfadjoint matrices one can solve their eigenvalue problem in terms of their representative selfadjoint matrices to obtain similar results, e.g., the contents of Theorems for selfadjoint matrices also carry over to selfadjoint operators. Selfadjoint operators with non-negative eigenvalues can be singled out to form a distinctive group with many useful properties.
A simple operation is to lengthen or to shorten a vector by a factor a without changing its orientation. The simplest operator of this kind is the identity operatorII^. The operators introduced so far have a clear geometric meaning. One should use these operators as building blocks to establish other operators which may not have distinctive geometric properties. The building blocks we have in mind are projectors. Selfadjoint matrices and selfadjoint operators have many distinctive properties which are crucial to the probabilistic formulation of quantum mechanics. Since selfadjoint operators are represented by selfadjoint matrices one can solve their eigenvalue problem in terms of their representative selfadjoint matrices to obtain similar results, e.g., the contents of Theorems for selfadjoint matrices also carry over to selfadjoint operators. Selfadjoint operators with non-negative eigenvalues can be singled out to form a distinctive group with many useful properties.
Many selfadjoint operators important to physical applications, e.g., number operators and some first order and second order differential operators, possess a discrete spectrum. For a selfadjoint operator with a discrete spectrum the spectral function reduces to a sum in the form. An arbitrary symmetric operator does not possess a spectral function and the spectral theorem does not apply to such an operator. Hence the relation between selfadjoint operators and probability distributions presented above does not apply to symmetric operators. This is the reason that the probabilistic nature of quantum mechanics is formulated in terms of selfadjoint operators, not symmetric operators. In the momentum representation the momentum appears as a multiplication operator in the same way that the position acts as a multiplication operator in the coordinate representation. It follows that we can treat the momentum in the momentum representation in the same way as position in the coordinate representation.
The density matrix ρ is defined and its basic properties are derived in detail: ρ is a Hermitian non-negative operator with unit trace, its eigenvalues are non-negative and lie in the closed interval [0, 1]; if ρ is a projector, it projects onto a one-dimensional sub-space, the trace of ρ 2 is ≤ 1, the equality holding only if ρ is a projector. A necessary and sufficient condition for ρ to be a projector is that all vectors in its definition are identical up to a phase. Pure cases are sets that can be described by a state vector, unlike mixtures . The density matrix of a mixture is not a projection operator, and this may be viewed as its distinguishing feature with respect to pure cases. The mean value of an observable can be reexpressed with the help of the density matrix, and a Schrödinger equation for density matrices holds. Last, but not least, pure cases cannot unitarily evolve into mixtures . The applications of the formalism presented in this chapter deal with orientation of spin-½ particles, polarization of pairs of photons, thermal-equilibrium states. In the second part, quantum entanglement is presented, with emphasis on the existence of state vectors for entangled pairs of particles that belong to the tensor product of Hilbert spaces H 1 ⊗ H 2 , but are not of the simple product form ψ 1 ⊗ ψ 2 . The only correct interpretation of such states has a statistical nature, associated to series of repeated measurements performed by two observers on the constituents of the entangled pair. Hidden-variable theories, with the associated Bell inequalities, are introduced, and a way of using entangled quantum states is described which makes it possible to transfer the polarization state of a photon to another photon. This property provides the foundation of the modern investigations of quantum teleportation.
This chapter develops a basic working knowledge of operators — objects central to quantum mechanics. The fact that quantum operators are linear determines how they may be manipulated, so linearity forms the chapter's first topic. The adjoint of an operator is then introduced. This provides the basis for Hermitian or self-adjoint operators, which are discussed for both discrete eigenvalue systems and wavefunctions. Projection operators, which ‘project out’ a certain part of a quantum state, are then introduced. These form the basis for the identity operator — a sum over projection operators — which changes the representation in which a state is expressed without changing the state itself. Finally, unitary operators are briefly discussed, including the defining unitarity condition, the fact that such operators preserve inner products, and their physical meaning as effecting transformations in space and time. Unitary operators are discussed in greater detail in Appendix D.
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Quantum theory is the soul of contemporary physics. It was discovered in an adventurous way, under the urge to solve the puzzles posed by atomic spectra and blackbody radiation. But after its invention, it immediately became clear that it was not just a theory of specific physical systems: it was rather a new language of universal applicability. Already in 1928, the theory had received solid mathematical foundations by Hilbert, von Neumann, and Nordheim, and this work was brought to completion in the monumental work of von Neumann, in the form that we still study nowadays. The theory is extraordinarily successful, and its predictions have been confirmed to an astonishing level of precision in a large spectrum of experiments. However, almost 90 years after von Neumann's book, quantum theory remains mysterious. Its mathematical formulation – based on Hilbert spaces and self-adjoint operators – is far from having an intuitive interpretation. The association of physical systems to Hilbert spaces whose unit vectors represent pure states, the representation of transformations by unitary operators and of observables by self-adjoint operators – all such postulates look artificial and ad hoc . A slightly more operational approach is provided by the C * -algebraic formulation of quantum theory – still, this formulation relies on the assumption that observables form an algebra , where the physical meanings of the multiplication and the sum are far from clear. In short, the postulates of quantum theory impose mathematical structures without providing any simple reason for this choice: the mathematics of Hilbert spaces is adopted as a magic blackbox that “works well” at producing experimental predictions. However, in a satisfactory axiomatization of a physical theory the mathematical structures should emerge as a consequence of postulates that have a direct physical interpretation. By this we mean postulates referring, e.g., to primitive notions like physical system, measurement, or process, rather than notions like, e.g., Hilbert space, C * -algebra, unit vector, or selfadjoint operator. The crucial question thus remains unanswered: why quantum theory? Which are the principles at the basis of the theory? A case that is often invoked in contrast is that of Special Relativity theory, which directly follows from the simple understandable principle of relativity.