Download Time-Energy Uncertainty Principle: Heisenberg Picture and Time Evolution of Operators and more Study notes Quantum Mechanics in PDF only on Docsity! 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 Ek and eigenvectors |k〉. Any state can be written as |ψ〉 = ∑ k ck|k〉. Assuming this is the initial state at t = t0, the state at time t is given by U(t, t0)|ψ〉 = ∑ k cke − ih̄Ek(t−t0)|Ek〉. 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 |ψ〉 by ∆A ( or ∆E). From the uncertainty relation we have in the initial state ∆A∆E ≥ 1 2 |〈[A,H]〉|. Recall our previous result which relates time evolution of expectation values to commuta- tors; we get 1 2 〈[A,H]〉 = h̄ 2 d dt 〈A〉. Therefore: ∆A∆E ≥ h̄ 2 | d dt 〈A〉|. 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 | ddt〈A〉| . With ∆t so-defined we then have ∆t∆E ≥ h̄ 2 . So, the shortest possible time scale that characterizes a significant change in the system is given by ∆t∆E ∼ h̄. 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 doesn’t change in time since the energy probability dis- tribution doesn’t 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- tions of the time-energy uncertainty principle can become ludicrous. What I am speaking of here are things like the oft-heard “You can violate conservation of energy if you do it for a short enough time”, or “The uncertainty of energy is related to the uncertainty in time”. We shall come back to these bizarre sounding statements and see what they really mean a little bit later. For now, beware of such slogans. As a nice example of the time-energy uncertainty principle, consider the spin precession problem we studied last time. Recall that we had a uniform, static magnetic field B = Bk̂ along the z axis. The Hamiltonian is H = eB mc Sz. We studied the time dependence of the spin observables when the initial state was an Sx eigenvector. It is not hard to compute the standard deviation of energy in the initial state |Sx,+〉. Using H2 = ( eBh̄ 2mc )2 I, we have (exercise) ∆E = h̄ 2 eB mc = 1 2 h̄ω, so that we expect a significant change in the state when 1 2 ω∆t ∼ 1. Thus the frequency ω controls the time scale for changes in the system, as you might have already ascertained from, e.g., the probability distributions Prob(Sx = ± h̄ 2 )(t) = { cos2(ω2 t) sin2(ω2 t) . Heisenberg picture Let us now see how to describe dynamics using the Heisenberg picture, in which we encode the time evolution into the operator representatives of the observables rather than in the state vectors. The idea is that time evolution is mathematically modeled by allowing 2
