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chap3 - Chapter 3 Foundations II Measurement and Evolution...

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Chapter 3 Foundations II: Measurement and Evolution 3.1 Orthogonal Measurement and Beyond 3.1.1 Orthogonal Measurements We would like to examine the properties of the generalized measurements that can be realized on system A by performing orthogonal measurements on a larger system that contains A . But first we will briefly consider how (orthogonal) measurements of an arbitrary observable can be achieved in principle, following the classic treatment of Von Neumann. To measure an observable M , we will modify the Hamiltonian of the world by turning on a coupling between that observable and a “pointer” variable that will serve as the apparatus. The coupling establishes entanglement between the eigenstates of the observable and the distinguishable states of the pointer, so that we can prepare an eigenstate of the observable by “observing” the pointer. Of course, this is not a fully satisfying model of measurement because we have not explained how it is possible to measure the pointer. Von Neumann’s attitude was that one can see that it is possible in principle to correlate the state of a microscopic quantum system with the value of a macroscopic classical variable, and we may take it for granted that we can perceive the value of the classical variable. A more complete explanation is desirable and possible; we will return to this issue later. We may think of the pointer as a particle that propagates freely apart 1
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2 CHAPTER 3. MEASUREMENT AND EVOLUTION from its tunable coupling to the quantum system being measured. Since we intend to measure the position of the pointer, it should be prepared initially in a wavepacket state that is narrow in position space — but not too narrow, because a vary narrow wave packet will spread too rapidly. If the initial width of the wave packet is Δ x , then the uncertainty in it velocity will be of order Δ v = Δ p/m planckover2pi1 /m Δ x , so that after a time t , the wavepacket will spread to a width Δ x ( t ) Δ x + planckover2pi1 t m Δ x , (3.1) which is minimized for [Δ x ( t )] 2 x ] 2 planckover2pi1 t/m . Therefore, if the experi- ment takes a time t , the resolution we can achieve for the final position of the pointer is limited by Δ x > x ) SQL radicalBigg planckover2pi1 t m , (3.2) the “standard quantum limit.” We will choose our pointer to be sufficiently heavy that this limitation is not serious. The Hamiltonian describing the coupling of the quantum system to the pointer has the form H = H 0 + 1 2 m P 2 + λ MP , (3.3) where P 2 / 2 m is the Hamiltonian of the free pointer particle (which we will henceforth ignore on the grounds that the pointer is so heavy that spreading of its wavepacket may be neglected), H 0 is the unperturbed Hamiltonian of the system to be measured, and λ is a coupling constant that we are able to turn on and off as desired. The observable to be measured, M , is coupled to the momentum P of the pointer.
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