12-06-2014, 03:22 PM
Density operators
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Motivation
Let us imagine we have a single spin, associated with a Hilbert space H ' C2. We
now throw a coin. In case of heads, we prepare the spin in j0i, in case of tails, we
prepare it in j1i. That is to say, with the classical probability 1=2 we have j0i, and
with classical probability 1=2 we get j1i. How do we capture this situation? Can we
describe the system by a state vector
j+i = (j0i + j1i)=
p
2? (4.1)
Not quite. This is easy to see: In case of a x measurement, we would always get the
same outcome. But this is different from the situation we encounter here. In fact, when
we make a measurement of x, we would get both outcomes with equal probability. Or
j?i = (j0i ? j1i)=
p
2? (4.2)
Again, this will not work, for the same reason. In fact, no state vector is associated
with such a situation, and for that, we need to generalize our concept of a quantum
state slightly: to density operators. This is, however, the most general quantum state in
standard quantum mechanics, and we will not have to generalize it any further.
Traces
A hint we have already available: Since all operations we can apply to state vectors
act linearly (time evolution and measurement), we already know the following: Let
as assume that we initially have the situation that with probability pj the state vector
j j(t0)i is prepared.
Von-Neumann equation
Now that we have understood what a density operator is, the rest will be a piece of
cake. How do density operators evolve in time? Well, this equation is just inherited
from the Schroedinger equation by linearity. Since it was von-Neumann who first
described this situation well, it is called von-Neumann equation. But really, it is just
the ordinary Schroedinger equation written for density operators. There is no new
physics happening here.