16 - Physics 6210/Spring 2007/Lecture 16 Lecture 16...

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Physics 6210/Spring 2007/Lecture 16 Lecture 16 Relevant sections in text: § 2.2, 2.3 Ehrenfest’s Theorem The Heisenberg equations are appealing because they make formal contact with the Hamilton equations of classical mechanics. In classical mechanics functions on phase space represent the observables, and the time rate of change of an observable A is controlled by the Poisson bracket with the Hamiltonian: dA dt = { A, H } . The formal correspondence with quantum mechanics is made via { A, B } ←→ 1 i ¯ h [ A, B ] , where the observables are represented by functions on phase space on the left and operators on the right. This formal correspondence implies that expectation values will, in a suitable approximation, follow classical trajectories, a result known as Ehrenfest’s theorem . To derive this theorem in the Heisenberg picture is quite easy. Take the expectation value of the quantum form of Newton’s second law, d 2 X i ( t ) dt 2 = - ∂V ∂x i ( X ( t )) , and use the time independence of the state vector to obtain (exercise) d 2 dt 2 h X i ( t ) = h F i ( t ) , where F is the force. This result is Ehrenfest’s theorem. Exercise: How would you derive this equation in the Schr¨ odinger picture? It is often said that Ehrenfest’s theorem shows that expectation values obey the clas- sical dynamical laws. This slogan is not quite true. In particular, the expectation value of position does not necessarily obey Newton’s second law. A true version of Newton’s second law for the expectation value would read m d 2 h X i i ( t ) dt 2 = - ∂V ( h ~ X i ( t )) h X i i . 1
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Physics 6210/Spring 2007/Lecture 16 Of course, this latter equation is not what quantum mechanics gives, in general. To get this last result we need h ∂V ( ~ X ( t )) ∂X i i = ∂V ( h ~ X i ( t )) h X i i , whose validity depends upon the state vector being used as well as on the form of the potential energy operator. A simple example where this equality does
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This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

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16 - Physics 6210/Spring 2007/Lecture 16 Lecture 16...

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