### Energy-time uncertainty relation

Sep. 16th, 2017 10:13 am**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.

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.