# Time evolution of ensembles

The density operador which describes an ensemble, $$\rho = \sum_{i} w_i |a_i> <a_i|$$ (represented in the basis in which it is diagonal), evolves in time such that $$|a_i(t)> = U(t) |a_i>$$when the enemble remains undisturbed, i.e. the w_i do not change.

But for irreversible processes the w_i may change towards a more random ensemble, driving the density matrix to a diagonal form with equal values for the w_i.

This is what I got after reading chapter 3.4 of Sakurai´s Modern Quantum Mechanics. But what Sakurai does not explain is how the time evolution of the w_i may look like for irreversible processes. May be someone can give a hint or a reference.

The background of my question: I´ve read that a transition from a pure ensemble (density matrix has only one element different from zero) to a mixed ensemble (with several w_i different from zero) is not allowed in QM (I read this in relation with black holes). I would like to understand why.

Thanks.

Another God
Staff Emeritus
Gold Member
Sorry that I can't help you with your question at all, but is it at all possible that you could help me out a little, and explain your question a little?

SImply explain the variable in the equations, explain what the equation is for, what each part of it represents etc... Maybe if you help me understand the equations, I may be able to eventually help you?

(OK, so maybe thats a little unlikely...but I'll try.)

I am not sure whether I am able to be more clear with this question, but I will try (although I am afraid I will still repeat).

As you may know, an ensemble can be described with a density operator $$\rho$$ such that $$\rho = \sum_{i} w_i |a_i> <a_i|$$ (represented in the basis in which it is diagonal), where each of the w_i is a real number representing the relative population of elements in a given coherent state $$|a_i>$$ and such that $$\sum_{i}w_i = 1$$.

In ensembles which remain undisturbed (relative populations remain constant) the $$|a_i>$$ evolve in time affected by the time-evolution operator $$|a_i(t)> = U(t) |a_i>$$ (Schroedinger picture).

This is what I read in Sakurai´s book. Now my question.

I assume (but I am not really sure) that in ensembles which do not remain undisturbed (physical processes which are not reversible), the time evolution may be described as a change of the w_i. The entropy is defined as $$S = - k \sum_{i} w_i ln w_i$$. How does entropy increase otherwise, if the w_i do not evolve?.

Now, if the w_i may evolve, why is a transition between a pure ensemble (only one w_i) and a mixed ensemble (several w_i) not possible?