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DensityMatrix

DensityMatrix - The Density Matrix Michael Fowler Pure...

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The Density Matrix Michael Fowler 11/19/07 Pure States and Mixed States Our treatment here more or less follows that of Sakurai, beginning with two imagined Stern- Gerlach experiments. In that experiment, a stream of (non-ionized) silver atoms from an oven is directed through an inhomogeneous vertical magnetic field, and the stream splits into two. The silver atoms have nonzero magnetic moments, and a magnetic moment in an inhomogeneous magnetic field experiences a nonzero force, causing the atom to veer from its straight line path, the magnitude of the deflection being proportional to the component of the atom’s magnetic moment in the vertical (field) direction. The observation of the beam splitting into two, and no more, means that the vertical component of the magnetic moment, and therefore the associated angular momentum, can only have two different values. From the basic analysis of rotation operators and the properties of angular momentum that follow, this observation forces us to the conclusion that the total angular momentum of a silver atom is 1 2 = . Ordinary orbital angular momenta cannot have half-integer values; this experiment was one of the first indications that the electron has a spin degree of freedom, an angular momentum that cannot be interpreted as orbital angular momentum of constituent parts. The silver atom has 47 electrons, 46 of them have total spin and orbital momenta that separately cancel, the 47 th has no orbital angular momentum, and its spin is the entire angular momentum of the atom. Here we shall use the Stern-Gerlach stream as an example of a large collection of quantum systems (the atoms) to clarify just how to describe such a collection, often called an ensemble . To avoid unnecessary complications, we only consider the spin degrees of freedom. We begin by examining two different streams: Suppose experimentalist A prepares a stream of silver atoms such that each atom is in the spin state A ψ : ( ) 1 2 A ψ = ↑ + ↓ . Meanwhile, experimentalist B prepares a stream of silver atoms which is a mixture : half the atoms are in state and half are in the state : call this mix B . Question : can we distinguish the A stream from the B stream? Evidently, not by measuring the spin in the z -direction! Both will give up 50% of the time, down 50%. But: we can distinguish them by measuring the spin in the x -direction: the A ψ quantum state is in fact just that of a spin in the x -direction, so it will give “up” in the x -direction every time—

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