Energy-time uncertainty relation

Sep. 16th, 2017 10:13 am
leblon: (Default)
[personal profile] leblon
It is well-known that the energy-time uncertainty relation has a different status than the usual momentum-coordinate uncertainty relation. Heisenberg introduced both uncertainty relations, but while the p-x relation can be formulated and proved rigorously, the energy-time relation is more subtle, and in fact some often-mentioned formulations are wrong. The problem is that in QM time is not a dynamical variable, but a parameter. So the accuracy of measuring the time coordinate can be arbitrarily good, regardless of what we know about the energy of the system.

Consider some popular formulations of the energy-time relation.

(1) When measuring an energy of a system, the accuracy of the measurement cannot exceed h/t, where t is the duration of the measurement.

(2) When preparing a system in a particular state, the uncertainty of the energy of this state will be at least h/t, where t is the preparation time, and h is the Planck constant.

These two formulations are essentially equivalent, since measuring the energy of the system is the same as preparing a state where the energy has a definite value. I think Landau-Lifshits textbook states (1) as a viable formulation of the energy-time uncertainty relation. But as shown by Aharonov and Bohm, (1) (and therefore (2)) are incorrect. It is possible to set-up a non-demolition measurement of energy which takes an arbitrarily short time and has an arbitrarily good accuracy.

(3) If some property of a system changes substantially on a time scale t, then the energy of the state has uncertainty at least h/t.

A version of this was stated by Bohr and Wigner. This is the formulation which "explains" why an unstable particle (resonance) does not have a definite energy. It is a bit hard to make this principle precise, and in fact there are many slightly different formulations. But it can be proved rigorously.

(4) If an internal (dynamical) clock of a system has accuracy t, then the energy of the system is uncertain, with uncertainty being at least h/t.

This is more or less equivalent to (3).

There is a well known story (told, for example, in R. Peierls's wonderful book "Surprises in theoretical physics") about Einstein inventing a counter-example to (1), and Bohr refuting him using Einstein's own General Relativity Theory. In retrospect, Bohr's refutal, while correct, seems beside the point, since (1) is not true in general.



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