Classical and quantum chaos by Cvitanovic P., Artuso R., Dahlqvist P.

By Cvitanovic P., Artuso R., Dahlqvist P.

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48. We now make the assumption that the time evolution acts in a linear way, so that each term evolves independently according to Eq. 51. Then |ψ(t) = α0 e−iω0 t |E0 + α1 e−iω1 t |E1 . 53) Not only does the overall phase of this state vary over time, but the relative phases of the two terms will also change, provided ω0 = ω1 . To see some of the implications of this, consider the state |u of the two-level atom given by 1 1 |u = √ |E0 + √ |E1 . 54) 2 2 We will suppose that |u is the state associated with the outcome u of some measurement, and also that the initial state of the atom is |ψ(0) = |u .

In Newtonian mechanics the component Sz can take on a continuous range of values, but as an experimental fact only these two results are possible. There is, of course, nothing special about the z-axis. The same basic fact holds true for measurements of Sx , Sy , or any other component of the spin. How can a component of a particle’s intrinsic angular momentum be measured? This was the problem faced by Otto Stern and Walther Gerlach in the early days of quantum physics. They were testing Bohr’s quantum theory of atomic structure, in which angular momentum has only discrete values.

Light sensors measure the intensities of the two output beams. amplitude. Each beam in our single-photon interferometer experiment will have an amplitude α, and the probability P that a detector would find the photon there (if we were to introduce such a detector) is just P = |α|2 . 1) Suppose at some stage of our interferometer we know for sure that the photon must be in one of two beams, which have amplitudes α and β respectively. Then it follows that |α|2 + |β|2 = 1. Complex amplitudes One important kind of device that we can introduce into a beam is called a phase shifter.

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