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Time-evolution Operators in a Position-measurement Apparatus

誠司 小杉, Seiji KOSUGI

Published Feb 25, 2010• 0 citations
OpenAlex

Abstract

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.

Related papers

Time-Evolution Operators for Position-Measuring Apparatuses (Part2)

誠司 小杉, Seiji KOSUGI

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

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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.

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Fokker–Planck Dynamics, Schrödinger Evolution, and Quantum Measurement Compatibility Conditions, Exact Model Constructions, and Quantitative Tests

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Positive Fokker–Planck dynamics can reproduce stationary quantum densities and open-system phase-space evolution, but this does not by itself establish unitary Schrödinger dynamics or an individual measurement outcome. We analyze the compatibility conditions among these levels in a drift–diffusion–circulation model motivated by the Architecture of Distinction. After fixing physical dimensions, harmonic ground-state matching yields a diffusion–frequency relation, while a multimode common-clock criterion separates spectral correspondence from a shared physical time calibration. An explicit phase–density Hamiltonian supplies the additional information required for local Schrödinger evolution, with global winding conditions stated separately. A Husimi oscillator provides an exact positive three-term quantum example. For measurement, we reconstruct a finite-time completely positive instrument from an explicit contact and registration rules and distinguish apparatus backaction from conditional state update. Numerical checks verify the analytic identities and channel properties. We formulate quantitative consistency tests for harmonic matching, timing–visibility relations, finite record retention, and a separately postulated momentum-diffusion channel. The results identify which quantum structures can be represented within a positive diffusion framework and which require additional phase, operator, apparatus, or interpretive input. No individual outcome or parameter-free departure from quantum mechanics is derived from scalar diffusion alone.

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In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component ŝz by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S + A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetization. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called collapse of the wavefunction or the reduction of the state appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.

arXiv:1406.5178•International Journal of Modern Physics B•Jun 23, 2014•5 citations

Operators of Time Evolution S(t, t0) and Sα(t, t0)

WORLD SCIENTIFIC eBooks•Apr 16, 2020•0 citations
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.
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.
Positive Fokker–Planck dynamics can reproduce stationary quantum densities and open-system phase-space evolution, but this does not by itself establish unitary Schrödinger dynamics or an individual measurement outcome. We analyze the compatibility conditions among these levels in a drift–diffusion–circulation model motivated by the Architecture of Distinction. After fixing physical dimensions, harmonic ground-state matching yields a diffusion–frequency relation, while a multimode common-clock criterion separates spectral correspondence from a shared physical time calibration. An explicit phase–density Hamiltonian supplies the additional information required for local Schrödinger evolution, with global winding conditions stated separately. A Husimi oscillator provides an exact positive three-term quantum example. For measurement, we reconstruct a finite-time completely positive instrument from an explicit contact and registration rules and distinguish apparatus backaction from conditional state update. Numerical checks verify the analytic identities and channel properties. We formulate quantitative consistency tests for harmonic matching, timing–visibility relations, finite record retention, and a separately postulated momentum-diffusion channel. The results identify which quantum structures can be represented within a positive diffusion framework and which require additional phase, operator, apparatus, or interpretive input. No individual outcome or parameter-free departure from quantum mechanics is derived from scalar diffusion alone.
In textbooks, ideal quantum measurements are described in terms of the tested system only by the collapse postulate and Born's rule. This level of description offers a rather flexible position for the interpretation of quantum mechanics. Here we analyse an ideal measurement as a process of interaction between the tested system S and an apparatus A, so as to derive the properties postulated in textbooks. We thus consider within standard quantum mechanics the measurement of a quantum spin component ŝz by an apparatus A, being a magnet coupled to a bath. We first consider the evolution of the density operator of S + A describing a large set of runs of the measurement process. The approach describes the disappearance of the off-diagonal terms ("truncation") of the density matrix as a physical effect due to A, while the registration of the outcome has classical features due to the large size of the pointer variable, the magnetization. A quantum ambiguity implies that the density matrix at the final time can be decomposed on many bases, not only the one of the measurement. This quantum oddity prevents to connect individual outcomes to measurements, a difficulty known as the "measurement problem". It is shown that it is circumvented by the apparatus as well, since the evolution in a small time interval erases all decompositions, except the one on the measurement basis. Once one can derive the outcome of individual events from quantum theory, the so-called collapse of the wavefunction or the reduction of the state appears as the result of a selection of runs among the original large set. Hence nothing more than standard quantum mechanics is needed to explain features of measurements. The employed statistical formulation is advocated for the teaching of quantum theory.