Monday, 4 July 2016

quantum mechanics - Set of states |phinrangle in the density operator rho=sumlimitsnpn|phinranglelanglephin|


The set of quantum states {|ϕn} in the definition of the density operator ρ=npn|ϕnϕn|

need not be orthonormal, and need not form a basis. But unfortunately, in the examples that I have seen so far, the states {|ϕn} were both orthonormal and forms a basis.


Example 1 In the Stern-Gerlach (SG) set-up, the state of the silver atoms coming out of the oven and before passing through the magnetic field, is imperfectly known because Sz remained unmeasured. Therefore, on the ignorance ground, such an ensemble will be represented by ρ=12(||+||).

Note that, in this case, the states | and | are orthonormal and forms the Sz-basis.


Example 2 Consider an unpolarized light moving in the z-direction so that its polarization must be in the xy-plane. Since we do not know the state vector, it is described by the density operator ρ=12(|xx|+|yy|)

where |x and |y describe plane polarized states along the x and y-axes respectively.





Question Can someone suggest an example of a mixed ensemble where the states {|ϕn} need not be orthonormal and need not form a basis? I'm not looking for the trivial example where the desity operator describes a pure state.




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