On the lifetime-width relation for a decaying state and the uncertainty principle (decay law/line shape/characteristic time/equivalent width/time coherence)

M. Bauer, Pier A. Mello

Published Jan 1, 2016• 0 citations

Abstract

A new formulation of the uncertainty rela- tion of position and momentum, and of energy and time, is presented. The connection between the lifetime of excited states and the energy width of these states, which does not follow from the usual uncertainty relation, is shown to be a consequence of the expression here derived.

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On the lifetime-width relation for a decaying state and the uncertainty principle

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On the lifetime-width relation for a decaying state and the uncertainty principle (decay law/line shape/characteristic time/equivalent width/time coherence)

M. Bauer, Pier A. Mello

Published Jan 1, 2016• 0 citations

OpenAlex

Abstract

Related papers

On the lifetime-width relation for a decaying state and the uncertainty principle

M. Bauer, Pier A. Mello

Proceedings of the National Academy of SciencesFeb 1, 197615 citations

New form of the time-energy uncertainty relation

Eric A. Gislason, Nora H. Sabelli, John W. Wood

A new form of the time-energy uncertainty principle is presented. It has the form \ensuremath{\tau} \ensuremath{\Delta}E\ensuremath{\ge}3\ensuremath{\pi} ${5}^{1/2}$ /25, where \ensuremath{\tau} is the average lifetime of a decaying state, and \ensuremath{\Delta}E is the energy spread of the state computed from \ensuremath{\Delta} ${E}^{2}$ =〈 ${E}^{2}$ 〉-〈E ${〉}^{2}$ . The particular state for which the above relationship becomes an equality is identified, and it is proved that all other states give a greater \ensuremath{\tau} \ensuremath{\Delta}E product. This general result should have a wide range of applications.A new form of the time-energy uncertainty principle is presented. It has the form \ensuremath{\tau} \ensuremath{\Delta}E\ensuremath{\ge}3\ensuremath{\pi}${5}^{1/2}$/25, where \ensuremath{\tau} is the average lifetime of a decaying state, and \ensuremath{\Delta}E is the energy spread of the state computed from \ensuremath{\Delta}${E}^{2}$=〈${E}^{2}$〉-〈E${〉}^{2}$. The particular state for which the above relationship becomes an equality is identified, and it is proved that all other states give a greater \ensuremath{\tau} \ensuremath{\Delta}E product. This general result should have a wide range of applications.

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The uncertainty principle for energy and time

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It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.It is generally thought desirable that quantum theory entail an uncertainty relation for time and energy similar to the one for position and momentum. Nevertheless, the existence of such a relation has still remained problematic. Here, it is shown that the problem is due to a confusion between the position coordinates of a point particle (a material system) and the coordinates of a point in space: The time coordinate should be put on a par with the space coordinates, not with the canonical position coordinates of a material system. Whereas quantum mechanics incorporates a Heisenberg uncertainty relation between the canonical position coordinates and their conjugate momenta, there is no reason why a Heisenberg relation should hold between the space coordinates and the canonical momenta, or between the time coordinate and the energy of the system. However, uncertainty relations of a different kind exist between the space coordinates and the total momentum of the system and between the time coordinate and the total energy. These relations are completely similar and may be taken together to form a relativistically covariant set of uncertainty relations. The relation between the time coordinate and the energy implies the well-known relation between the lifetime of a state and its energy spread.

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A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.A concise review of various mathematical formulations of the uncertainty relations in quantum mechanics discovered since 1927 is given. Besides the traditional Heisenberg inequality, the modifications made by Schrödinger and Robertson, as well as generalizations to sets of several noncommuting operators, are considered. The "entropic" inequalities and "local" uncertainty relations, together with inequalities which connect the so-called total width and the mean peak width of a wave function, are discussed. Inequalities for the products of higher order moments of the coordinate and momentum are presented. Inequalities making the uncertainty relations more accurate when the "purity" of a quantum state is fixed are demonstrated. Diverse formulations of the energy-time uncertainty relations are considered.

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Francesco Giacosa

We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time $t=0$ . This expression is a function of the time $t$ at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time $t$ comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type $1/t$ dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time $t=0$. This expression is a function of the time $t$ at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time $t$ comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type $1/t$ dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.

arXiv:1305.4467Physical Review ANov 27, 201317 citations

A new form of the time-energy uncertainty principle is presented. It has the form \ensuremath{\tau} \ensuremath{\Delta}E\ensuremath{\ge}3\ensuremath{\pi}

{5}^{1/2}

/25, where \ensuremath{\tau} is the average lifetime of a decaying state, and \ensuremath{\Delta}E is the energy spread of the state computed from \ensuremath{\Delta}

{E}^{2}

=〈

{E}^{2}

〉-〈E

{〉}^{2}

. The particular state for which the above relationship becomes an equality is identified, and it is proved that all other states give a greater \ensuremath{\tau} \ensuremath{\Delta}E product. This general result should have a wide range of applications.

We derive the expression for the energy uncertainty of the final state of a decay of an unstable quantum state prepared at initial time

t=0

. This expression is a function of the time

t

at which a measurement is performed to determine if the state has decayed and, if yes, in which one of the infinitely many possible final states. For large times the energy spread is, as expected, given by the decay width $\ensuremath{\Gamma}$ of the initial unstable state. However, if the measurement of the final state is performed at a time

t

comparable to (or smaller than) the mean lifetime of the state $1/\ensuremath{\Gamma}$, then the uncertainty in the energy of the final state is much larger than the decay width $\ensuremath{\Gamma}$. Namely, for short times an uncertainty of type

1/t

dominates, while at long times the usual spread $\ensuremath{\Gamma}$ is recovered. Then we turn to a generic two-body decay process and describe the energy uncertainty of each of the two outgoing particles. We apply these formulas to the two-body decays of neutral and charged pions and to the spontaneous emission process of an excited atom. As the last step, we study a case in which the nonexponential decay is realized and show that for short times eventual asymmetric terms are enhanced in the spectrum.