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Unformatted text preview: Physics 6210/Spring 2007/Lecture 14 Lecture 14 Relevant sections in text: 2.1, 2.2 TimeEnergy Uncertainty Principle (cont.) Suppose the energy is discrete, for simplicity, with values E k and eigenvectors  k i . Any state can be written as  i = X k c k  k i . Assuming this is the initial state at t = t , the state at time t is given by U ( t, t )  i = X k c k e i h E k ( t t )  E k i . Let us use an observable A to characterize the change in the system in time (which is, after all, what we actually do). Let us denote the standard deviation of A (or H ) in the initial state  i by A ( or E ). From the uncertainty relation we have in the initial state A E 1 2 h [ A, H ] i . Recall our previous result which relates time evolution of expectation values to commuta tors; we get 1 2 h [ A, H ] i = h 2 d dt h A i . Therefore: A E h 2  d dt h A i . If we want to use A to characterize the time scale for a significant change in the system we can do this by comparing the rate of change of the average value of A to the initial uncertainty in A : t = A  d dt h A i . With t sodefined we then have t E h 2 . So, the shortest possible time scale that characterizes a significant change in the system is given by t E h. Of course, if the (initial) state is stationary that is, an energy eigenvector, then E = 0, which forces t , which makes sense since the physical attributes of the state never change. 1 Physics 6210/Spring 2007/Lecture 14 The timeenergy uncertainty principle is then a statement about how the statistical uncertainty in the energy (which doesnt change in time since the energy probability dis tribution doesnt change in time) controls the time scale for a change in the system. In various special circumstances this fundamental meaning of the timeenergy uncertainty principle can be given other interpretations, but they are not as general as the one we have given here. Indeed, outside of these special circumstances, the alternative interpretagiven here....
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 Spring '07
 M
 mechanics, Energy

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