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Time-Evolution Operators of an Object-Probe System for Position-Measuring Apparatuses (Part3)

Seiji KOSUGI

Published Feb 25, 2014• 0 citations
OpenAlex

Abstract

位置の測定装置における対象粒子とプローブの相互作用が,相対座標の関数であるポテンシャルで表わされる場合に,相互作用後の粒子とプローブの位置演算子を求め,線形性の仮説が成立していることを示した.von NeumannやOzawaは,相互作用後の演算子を求める際に,運動エネルギーを無視するという近似をおこなっている.運動量を保存する相互作用の場合に,運動エネルギーを無視しないで正確に相互作用後の位置演算子と運動量演算子を求めた.ここでは相互作用の強さを表すパラメータで,とはそれぞれ対象粒子とプローブの相互作用前の位置演算子,とはそれぞれの相互作用前の運動量演算子である.そして,この近似をしなければ,相互作用後の位置演算子はとに依存してしまうので,この相互作用を用いて位置の測定ができないことを明らかにした.

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Time-evolution Operators in a Position-measurement Apparatus

誠司 小杉, Seiji KOSUGI

1.測定装置においては測定結果は一意的でなければならない.2.対象粒子とプローブの相互作用は運動量と角運動量を保存しなければならない. 相互作用後の粒子とプローブの位置演算子xt,Xtが相互作用前の演算子xo,Xoの関数で与えられるとき,1と2の条件がそれらの関数形をどのように限定するか考察した.本論文の結果は,すべての粒子の位置xoに対して,その測定値の平均がxoの平均に等しいという条件を課すことによって,線形性の仮説を証明できることを示唆している.

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相互作用後の粒子とプローブの位置演算子xt,Xtが,相互作用前のそれらの位置演算子x0,X0の線形結合で与えられるという線形性の仮説を証明した.その際に位置の測定装置が満たさなければならないと考えた条件は次の4つである.1.測定結果は一意的に決定されなければならない.2.粒子とプローブの相互作用は運動量を保存しなければならない.3.粒子とプローブの時間発展演算子Uはユニタリである.4.位置x0の測定誤差ε(x0) は対象粒子の初期波動関数φ0(x0)に依存してはならない.条件4はある特定のプローブの確率密度関数に対してだけでなく,それを少し変化させた確率密度関数に対しても成立していなければならない.条件4の代わりに,測定値の平均は測定対象である粒子の位置x0の平均に等しくなければならないという条件を用いても,線形性の仮説を導出することができる.

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位置の測定装置における対象粒子とプローブの相互作用が,相対座標の関数であるポテンシャルで表わされる場合に,相互作用後の粒子とプローブの位置演算子を求め,線形性の仮説が成立していることを示した.von NeumannやOzawaは,相互作用後の演算子を求める際に,運動エネルギーを無視するという近似をおこなっている.運動量を保存する相互作用の場合に,運動エネルギーを無視しないで正確に相互作用後の位置演算子と運動量演算子を求めた.ここでは相互作用の強さを表すパラメータで,とはそれぞれ対象粒子とプローブの相互作用前の位置演算子,とはそれぞれの相互作用前の運動量演算子である.そして,この近似をしなければ,相互作用後の位置演算子はとに依存してしまうので,この相互作用を用いて位置の測定ができないことを明らかにした.
1.測定装置においては測定結果は一意的でなければならない.2.対象粒子とプローブの相互作用は運動量と角運動量を保存しなければならない. 相互作用後の粒子とプローブの位置演算子xt,Xtが相互作用前の演算子xo,Xoの関数で与えられるとき,1と2の条件がそれらの関数形をどのように限定するか考察した.本論文の結果は,すべての粒子の位置xoに対して,その測定値の平均がxoの平均に等しいという条件を課すことによって,線形性の仮説を証明できることを示唆している.
相互作用後の粒子とプローブの位置演算子xt,Xtが,相互作用前のそれらの位置演算子x0,X0の線形結合で与えられるという線形性の仮説を証明した.その際に位置の測定装置が満たさなければならないと考えた条件は次の4つである.1.測定結果は一意的に決定されなければならない.2.粒子とプローブの相互作用は運動量を保存しなければならない.3.粒子とプローブの時間発展演算子Uはユニタリである.4.位置x0の測定誤差ε(x0) は対象粒子の初期波動関数φ0(x0)に依存してはならない.条件4はある特定のプローブの確率密度関数に対してだけでなく,それを少し変化させた確率密度関数に対しても成立していなければならない.条件4の代わりに,測定値の平均は測定対象である粒子の位置x0の平均に等しくなければならないという条件を用いても,線形性の仮説を導出することができる.
In a quantum measurement setting, it is known that environment-induced decoherence theory describes the emergence of effectively classical features of the quantum system–measuring apparatus composite system when the apparatus is allowed to interact with the environment. In [E. A. Galapon, Europhys. Lett. 113, 60007 (2016)], a measurement model is found to have the feature of inducing exact decoherence at a finite time via one internal degree of freedom of the apparatus provided that the apparatus is decomposed into a pointer and an inaccessible probe, with the pointer and the probe being in momentum-limited initial states. However, an issue can be raised against the model: while the factorization method of the time-evolution operator used there is formally correct, it is not completely rigorous due to some unstated conditions on the validity of the factorization in the Hilbert space of the model. Furthermore, no examples were presented there in implementing the measurement scheme in specific quantum systems. The goal of this paper is to re-examine the model and confirm its features independently by solving the von Neumann equation for the joint state of the composite system as a function of time. This approach reproduces the joint state obtained in the original work, leading to the same conditions for exact decoherence and orthogonal pointer states when the required initial conditions on the probe and pointer are imposed. We illustrate the exact decoherence process in the measurement of observables of a spin-1/2 particle and a quantum harmonic oscillator by using the model.
An argument is presented that the consistent description of the evolution of a system is possible only with the statistical operator and not with the wave function. A reconstruction of the logic of quantum mechanics is proposed with the statistical operator as the basic quantity to describe the state in one-to-one correspondence. The problem of measurement is reconsidered in the light of this new logic. It is shown for the first time how the irreversible change in the measuring apparatus leads to the truncation of von Neumann's chain between the object and the subject.
We tackle the dynamical description of the quantum measurement process by explicitly addressing the interaction between the system under investigation and the measurement apparatus, the latter ultimately considered as a macroscopic quantum object. We consider arbitrary positive-operator-valued measures (POVMs) such that the orthogonality constraint on the measurement operators is relaxed. We show that, as with the well-known von Neumann scheme for projective measurements, it is possible to build up a dynamical model holding a unitary propagator characterized by a single time-independent Hamiltonian. This is achieved by modifying the standard model so as to compensate for the possible lack of orthogonality among the measurement operators of arbitrary POVMs.