دانلود کتاب استدلال مکانیک کوانتومی – نظریه عملگر و نوسانگر هارمونیک

It was von Neumann’s insight that Heisenberg’s abstract approach to quantum mechanics, which eventually lead to its the standard formulation, finds its mathematical support in operator theory (OT), in the theory of densely-defined, linear and self-adjoint operators (DLSO’s) in Hilbert spaces. In particular, the description and interpretation of the result of the measurement process in quantum theory are closely linked to the spectral theorems for DLSO’s, see, e.g., Sect. in the Appendix.
As a consequence of von Neumann’s insight, in addition to its solid experimental basis, quantum theory acquired a subtle and deep mathematical foundation. Roughly speaking, it can be said that quantum theory is standing on2solid legs, the mathematics of DLSO’s and the experiment (Fig. 1.1).
Regrettably, von Neumann’s insights are hardly visible in standard quantum theory text-books, thereby obstructing a complete understanding of the theory. For example, unlike indicated in standard textbooks, an observable in quantum theory needs to be a DLSO, not just a DLHO, i.e., a densely-defined, linear and “Hermitian” operator.
Roughly speaking a DLHO is an operator that can be moved inside the scalar product of a complex Hilbert space, from right to left without change. More precisely, to every DLHO there corresponds an adjoint operator that extends that DLHO. A DLSO is a DLHO whose adjoint coincides with the DLHO. Self-adjointness is a strong property that guarantees that the spectrum of the operator in question is part of the real numbers and the availability of spectral theorems that are crucial for the interpretation in quantum theory. The existence of non-real spectral values observables is not acceptable for observables. In addition, with every formal partial differential operator, i.e., an operator that maps sufficiently differentiable functions into other such functions, there comes the question of which function space to use for its representation. The choice of representation space crucially affects the properties of the resulting operator. In quantum field theory, it is very well known that there are in-equivalent representations of the algebra of canonical commutation rules, leading to physically different theories. So, at least in the case of physical systems with infinitely many degrees of freedom, like those considered in field theory, even this representation step is not trivial.