Measurement of noncanonical variables

Aharonov, Yakir; Safko, John L.

doi:10.1016/0003-4916(75)90222-5

Cited by 32 publications

(19 citation statements)

References 7 publications

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“…To illustrate the treatment, we can show how to measure the electric field impulsively [6], as did Bohr and Rosenfeld. Thus the derivation of measurement interactions is simpler.…”

Section: Measuring the Electric Fieldmentioning

confidence: 99%

Measurement and Compensation

Aharonov

Rohrlich

2005

Quantum Paradoxes

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The last chapter presents von Neumann's model for measurements of canonical quantities. The model has many applications, and Sect. 7.4 applies it to resolve the velocity paradox of Sect. 7.1. However, we may not be comfortable with the resolution of the paradox. Velocity is not a canonical quantity, but it is gauge invariant: mv = p−eA/c. (See Prob. 4.3.) The model implies that a velocity measurement lasting a time T yields v with uncertainty ∆v ≥ /mT . (See Prob. 7.10.) Yet the same model implies that we can measure momentum instantaneously, with no minimum uncertainty, and without changing it. We can also measure p − eA/c instantaneously, with no minimum certainty, and without changing it; A is a canonical variable. But p − eA/c equals mv (except during the measurement) so why can't we measure velocity instantaneously? Likewise, in the laboratory we measure electric and magnetic fields, E and B, not the potentials A and V that appear in the Hamiltonian, Eq. (4.7). The magnetic field B = ∇ × A is a canonical quantity, but the electric field is not (although it is gauge invariant), because it depends on the derivative of A with respect to time:The question of measuring E troubled such physicists as Landau, Peierls, Bohr and Rosenfeld. Rosenfeld arrived at Bohr's institute in early 1931 just as the question had come to a head. He ran into Gamow and asked what was new. Gamow answered with a drawing he had just made. It showed Landau, tightly bound to a chair and gagged, while Bohr stood before him with upraised finger, saying "Please, please, Landau, can I just get a word in!" Landau and Peierls had come a few days before with a new paper to show Bohr, "but he does not seem to agree," said Gamow airily, "and this is the kind of discussion which has been going on all the time." Peierls had left the previous day, "in a state of complete exhaustion," added Gamow. Landau stayed on for a few weeks, and Rosenfeld discovered that "Gamow's representation of the situation was only exaggerated to the extent usually conceded to artistic fantasy" [1]. Landau and Peierls had considered measuring the electric field by sending a charged test particle through it [2]. The electric field deflects the test charge; the change in the momentum of the charge indicates the field strength. But an accelerated charge radiates and loses an unknown part of its momentum to the electromagnetic field. (See Prob. 8.1.) We can suppress the radiation by reducing the charge on the test particle; then the momentum of the particle changes more slowly, but the measurement lasts longer. An impulsive, accurate measurement of the electric field is impossible. Bohr felt uneasy, but could not refute the claim of Landau and Peierls.Ultimately, Bohr and Rosenfeld did refute the claim; it took them nearly three years [3], but they showed how to measure electric and magnetic fields instantaneously [4]. Bohr and Rosenfeld did not treat noncanonical observables in general. We will see that in the von Quantum Paradoxes: Quantum Theory for the Perplexed. Y. Aharonov...

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“…To illustrate the treatment, we can show how to measure the electric field impulsively [6], as did Bohr and Rosenfeld. Thus the derivation of measurement interactions is simpler.…”

Section: Measuring the Electric Fieldmentioning

confidence: 99%

Measurement and Compensation

Aharonov

Rohrlich

2005

Quantum Paradoxes

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“…whereF (p) = dqF (q, p) is the momentum marginal of the phase space POVM (18). It should be pointed out that the defined below uncertainty relation is given between the measured time of arrival and the measured energy of the chosen system, and as a result, the criticism of Aharonov and Bohm [4] and Peres [30] concerning those approaches where one tried to construct the uncertainty principle between the time of arrival detected by the filter and the energy of the investigated particle (two commuting quantities) does not apply here.…”

Section: Operational Time-energy Uncertaintymentioning

confidence: 99%

Erratum: Operational time of arrival in quantum phase space [Phys. Rev. A 60, 2689 (1999)]

Kochański¹,

Wódkiewicz²

2000

Phys. Rev. A

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An operational time of arrival is introduced using a realistic position and momentum measurement scheme. The phase space measurement involves the dynamics of a quantum particle probed by a measuring device. For such a measurement an operational positive operator valued measure in phase space is introduced and investigated. In such an operational formalism a quantum mechanical time operator is constructed and analyzed. A phase space time and energy uncertainty relation is derived. PACS number(s): 03.65.Bz, 42.50.Dv

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“…Мы примем вне этого интервала времени. КСС может быть представлена свободной частицей с массой М. Полный гамильтониан в простейшем случае представляют в виде [16] В случае достаточно большой массы М оператор можно считать постоянным в течение конечного времени взаимодействия Тогда в интервале из (6) получим (Дисперсия наблюдаемой в состоянии быть меньше, чем в начальном состоянии, если в началь ном состоянии Обычно считается, что неопределенность возмущения наблюдаемой зана с погрешностью измерения наблюдаемой ношением, тождественным соответствующему соотно шению неопределенностей. Однако это не так [10].…”

Section: эволюция состояния системы в процессе измере нияunclassified

“…Однако это обратное действие прибора не исключается из самой координаты. Из соотношения (7), а) следует, что среднеквадратическое (СКВ) возмущение координаты в момент Не принимая во внимание начальную неопределен ность импульса p(0), получим из (13), (16), что полная дисперсия координаты в момент будет равна Следовательно, результат повторного измерения коор динаты свободного тела через время t после первого измерения не может быть предсказан точнее, чем с СКВ погрешностью [9]. Эту величину называют СКП неопределенности координаты тела через время t после ее измерения.…”

Section: неопределенность координаты после измеренияunclassified

Standard quantum limits of measurement error and methods of overcoming them

Vorontsov¹

1994

Usp. Fiz. Nauk

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Measurement of noncanonical variables

Cited by 32 publications

References 7 publications

Measurement and Compensation

Measurement and Compensation

Erratum: Operational time of arrival in quantum phase space [Phys. Rev. A 60, 2689 (1999)]

Standard quantum limits of measurement error and methods of overcoming them

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