i\'m not really grasping when you can tell if a problem in this case should be u
ID: 2053761 • Letter: I
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i'm not really grasping when you can tell if a problem in this case should be used as relativistic calculations or in general calculations. is there a certain energy cap that has to be met?Explanation / Answer
Energy–time uncertainty principle Other than the position-momentum uncertainty relation, the most important uncertainty relation is that between energy and time. The energy-time uncertainty relation is not, however, an obvious consequence of the general Robertson–Schrödinger relation. Since energy bears the same relation to time as momentum does to space in special relativity, it was clear to many early founders, Niels Bohr among them, that the following relation should hold:[7][8] but it was not always obvious what ?t precisely meant. The problem is that the time at which the particle has a given state is not an operator belonging to the particle, it is a parameter describing the evolution of the system. As Lev Landau once joked "To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!" [26] Nevertheless, Einstein and Bohr understood the heuristic meaning of the principle. A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must accurately be defined, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy. For example, in spectroscopy, excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and each time they decay the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth. The broad linewidth of fast decaying states makes it difficult to accurately measure the energy of the state, and researchers have even used microwave cavities to slow down the decay-rate, to get sharper peaks.[27] The same linewidth effect also makes it difficult to measure the rest mass of fast decaying particles in particle physics. The faster the particle decays, the less certain is its mass. One false formulation of the energy-time uncertainty principle says that measuring the energy of a quantum system to an accuracy ?E requires a time interval ?t > h / ?E. This formulation is similar to the one alluded to in Landau's joke, and was explicitly invalidated by Y. Aharonov and D. Bohm in 1961 [28]. The time ?t in the uncertainty relation is the time during which the system exists unperturbed, not the time during which the experimental equipment is turned on, whereas the position in the other version of the principle refers to where the particle has some probability to be and not where the observer might look. Another common misconception is that the energy-time uncertainty principle says that the conservation of energy can be temporarily violated – energy can be "borrowed" from the Universe as long as it is "returned" within a short amount of time.[29] Although this agrees with the spirit of relativistic quantum mechanics, it is based on the false axiom that the energy of the Universe is an exactly known parameter at all times. More accurately, when events transpire at shorter time intervals, there is a greater uncertainty in the energy of these events. Therefore it is not that the conservation of energy is violated when quantum field theory uses temporary electron-positron pairs in its calculations, but that the energy of quantum systems is not known with enough precision to limit their behavior to a single, simple history. Thus the influence of all histories must be incorporated into quantum calculations, including those with much greater or much less energy than the mean of the measured/calculated energy distribution. In 1932 Dirac offered a precise definition and derivation of the time-energy uncertainty relation in a relativistic quantum theory of "events".[30] But a better-known, more widely used formulation of the time-energy uncertainty principle was given in 1945 by L. I. Mandelshtam and I. E. Tamm, as follows.[31] For a quantum system in a non-stationary state ? and an observable B represented by a self-adjoint operator , the following formula holds: where sE is the standard deviation of the energy operator in the state ?, sB stands for the standard deviation of B. Although, the second factor in the left-hand side has dimension of time, it is different from the time parameter that enters Schrödinger equation. It is a lifetime of the state ? with respect to the observable B. In other words, this is the time after which the expectation value changes appreciably.
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