14 - Physics 6210/Spring 2007/Lecture 14 Lecture 14...

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Unformatted text preview: Physics 6210/Spring 2007/Lecture 14 Lecture 14 Relevant sections in text: 2.1, 2.2 Time-Energy 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 so-defined 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 time-energy 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 time-energy 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 interpreta-given here....
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14 - Physics 6210/Spring 2007/Lecture 14 Lecture 14...

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