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Unformatted text preview: Physics 6210/Spring 2007/Lecture 15 Lecture 15 Relevant sections in text: 2.2 Compatible/Incompatible observables in the Heisenberg picture Let us also note that the notion of compatible/incompatible observables is left undis- turbed by the transition to the Heisenberg picture. This is because the commutator trans- forms as (exercise) [ A ( t ) , B ( t )] = [ U AU, U BU ] = U [ A, B ] U. (Note any two operators related by a unitary/similarity transformation will have this commutator property.) Thus, if A and B are (in)compatible in the Schr odinger picture they will be (in)compatible (at each time) in the Heisenberg picture. Note also that the commutator of two observables in the Schr odinger picture which is i times another observable makes the transition to the Heisenberg picture just as any other Schr odinger observable, namely, via the unitary transformation A U AU. Unitary transformations in general Let us note that while our discussion was phrased in the context of time evolution, the same logic can be applied to any unitary transformation. For example, for a particle moving in one dimension one can view the effect of translations as either redefining the state vector, leaving the operator-observables unchanged: | i T a | i , A A or equivalently as redefining the observables, with the state vectors unchanged: A T a AT a , | i | i . Note, in particular, that the position and momentum operators change in the expected way under a translation (exercise you played with this stuff in the homework): T a XT a = x + a, T a P T a = p. Heisenberg equations We saw that the conventional Schr odinger equation is really just a consequence of the relation between the time evolution operator and its infinitesimal generator in the context of the Schr odinger picture: i h d dt U ( t, t ) = H ( t ) U ( t, t ) i h d dt | ( t ) i = H ( t ) | ( t ) i . 1 Physics 6210/Spring 2007/Lecture 15 Given a Hamiltonian, this equation is the starting point for investigating quantum dynam- ics in the Schr odinger picture. We can now ask: what is the analog of this in the Heisenberg picture?picture?...
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