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arXiv:quant-ph/0206168 · Journal of Physics A Mathematical and General

On representing observables in quantum mechanics

D A Dubin, M A Hennings, P Lahti, Juha-Pekka Pellonpää

Published Oct 15, 2002• 5 citations
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Abstract

There are self-adjoint operators which determine both spectral and semispectral measures. These measures have very different commutativity and covariance properties. This fact opens additional possibilities with regard to the physical meaning of such a self-adjoint operator and its associated operator measures.
DOI
10.1088/0305-4470/35/43/310

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We are focused on the idea that observables in quantum physics are a bit more then just hermitian operators and that this is, in general, a “tricky business”. The origin of this idea comes from the fact that there is a subtle difference between symmetric, hermitian, and self-adjoint operators which are of immense importance in formulating Quantum Mechanics. The theory of self-adjoint extensions is presented through several physical examples and some emphasis is given on the physical implications and applications.
In the theory of quantum mechanics the Hamiltonian H is typically selfadjoint, i.e., H = H *. The self-adjointness ensures that the spectrum of the Hamiltonian, representing the energy spectrum of H , is real but it is not a necessary condition. In the literature on so-called PT -symmetric quantum mechanics (see, e.g., [BB98], [BBM99], [BBJ03], [Ben04b] and [Ben07]), it is believed that self-adjointness is rather a mathematical requirement than a physically established fact. Therefore, it was considered a surprise that operators exist which are not self-adjoint in the given quantum mechanical Hilbert space, but have real spectrum and that – if e.g. complex eigenvalues were present – they occurred only in complex conjugate pairs.
Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e). The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators. The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.
In nonrelativistic quantum mechanics, the notion of the confined version of a system is proposed and discussed. Confined observables can turn out to be represented by just positive operator valued measures also when their nonconfined equivalents are represented by self-adjoint operators, and this nearly always happens for confined linear momentum observables. However, the self-adjoint operators of the nonconfined system can still be used to compute expected results and variances of confined observables. As a result, the precision with which some confined observables can be measured has limits which ultimately derive, through the Heisenberg commutation rules, from the covariance properties of nonconfined kinematical observables. Two conditions are proposed to characterize the approximation of a nonconfined system by a sequence of confined ones. A nonconfined system is deemed to be approximated by a sequence of confined systems if all probabilities determined by any observable of the nonconfined system with respect to nonconfined states can be approximated by confined counterparts and if the time evolution of any state of the nonconfined system can be approximated in any finite time interval by the time evolution of confined states. A rather general example of this situation is discussed.
We give a characterization of commutative semispectral measures by means of Feller and Strong Feller Markov kernels. In particular: {itemize} we show that a semispectral measure FFF is commutative if and only if there exist a self-adjoint operator AAA and a Markov kernel μ(⋅)(⋅):Γ×B(R)→[0,1]μ_{(\cdot)}(\cdot):Γ\times\mathcal{B}(\mathbb{R})\to[0,1]μ(⋅)​(⋅):Γ×B(R)→[0,1], Γ⊂σ(A)Γ\subsetσ(A)Γ⊂σ(A), E(Γ)=1E(Γ)=\mathbf{1}E(Γ)=1, such that F(Δ)=∫ΓμΔ(λ) dEλ,F(Δ)=\int_Γμ_Δ(λ)\,dE_λ,F(Δ)=∫Γ​μΔ​(λ)dEλ​, \noindent and μ(Δ)μ_{(Δ)}μ(Δ)​ is continuous for each Δ∈RΔ\in RΔ∈R where, R⊂B(R)R\subset\mathcal{B}(\mathbb{R})R⊂B(R) is a ring which generates the Borel σσσ-algebra of the reals B(R)\mathcal{B}(\mathbb{R})B(R). Moreover, μ(⋅)(⋅)μ_{(\cdot)}(\cdot)μ(⋅)​(⋅) is a Feller Markov kernel and separates the points of ΓΓΓ. we prove that FFF admits a strong Feller Markov kernel μ(⋅)(⋅)μ_{(\cdot)}(\cdot)μ(⋅)​(⋅), if and only if FFF is uniformly continuous. Finally, we prove that if FFF is absolutely continuous with respect to a regular finite measure ννν then, it admits a strong Feller Markov kernel. {itemize} The mathematical and physical relevance of the results is discussed giving a particular emphasis to the connections between μμμ and the imprecision of the measurement apparatus.