Meaning of operators for observables.

[alexepascual]

Chasingwind:
Welcome to the thread!.
I agree with Slyboy. I think I have read that Hilbert space is a space of "rays" were the main thing is the "angle" between rays. The sphere was for me just a way to visualize the position of the rays. I was not too concerned with the length of vectors.
I don't remember much about the Bloch sphere. But from what you are saying, I can see some similarity with the chormaticity diagram, in which pure colors are represented on the periphery and mixed colors in the interior.
I am going to do some reading tonight to refresh my memory about the bloch sphere.

 

[slyboy]

The bloch sphere is just the sort of visualization you are talking about for two-level quantum systems. Pure states in a 2-d complex Hilbert space can be parameterized by two angles, so you can visualize them on a sphere in 3-d.

The interior of the sphere represents "mixed" states, which are a generalization of the sort of quantum states we have been discussing. Briefly, there are two ways in which mixed states can arise:

1. If you have a source that emits quantum states, but you are not exactly sure what states are being emitted. You can then describe your ignorance by a classical probability distribution over the pure states. The collection of states along with their classical probabilities is called an ensemble, but more than one ensemble can give the same predictions for quantum measurements. Therefore, we decsribe the state by something called the density operator, which takes the same form for enembles that give the same predictions.

2. If you have a composite system, for example a particle and its environment, then the total system may be in a pure state. However, if you consider the particle on its own then it cannot be described by a pure state, since it might be entangled with the environment. To describe the particle on its own, we do an operation called the "partial trace" to eliminate the environmental degrees of freedom. It turns out that you get exactly the same sort of density operator as in case 1.

Some philosophers make a big deal of the difference between 1 and 2. They call 1 proper mixing and 2 improper mixing. However, as far as the maths of QM is concerned, the two are completely equivalenet.

 

[alexepascual]

Slyboy:
I am already somewhat familiar with mixed states vs. pure states. During my undergraduate studies I worked on a project which consisted in studying the transition from quantum to classical behavior. At that time I was exposed to the density operator, mixed states, decoherence, dissapearence of the off-diagonal elements, etc. This project was kind of a survey of the different ideas. Considering that my knowledge of quantum mechanics was not solid, I can not say that my understanding was good. But I got a rough idea of the main concepts.
With respect to the Bloch sphere, I think I could use it to think about pure states, as long as I stay on the surface of the sphere and don't go inside. When you say that the Block sphere is the kind of visualization that I was talking about, I wonder if you mean the kind of understanding that I was seeking about the meaning of operators from the beginning of this thread. If that is what you mean, I agree, or at least it looks like it might help me gain that understanding. I mentioned at one point I was planning to study two-state systems but I didn't want to get too deep into angular momentum. Well maybe this sphere will allow me to do that. Actually now I remember seing the sphere before, but I never understood it 100% and I didn't remember it was called a Block sphere.
In this particular case, the measurable operators would be represented by the Pauli matrices, so if I got all the elements, the state vectors and the operators, then I can see how they are related and what they represent.
Now, I understand I may loose generality, as I have read that the Pauli matrices have some peculiarities, such as being both Hermitian and Unitary.
So I'll see if I can focus on this on my spare time for the comming week.

Chasingwind:
Thanks for mentioning the Bloch sphere, As I tell Slyboy, I think it will help me understand the connection between state vectors and operators for the case of pure states.

 

[slyboy]

Pure states of a 2-d system can be represented by a density matrix

\rho = \frac{1}{2} \left ( I + \vec{r}\cdot\vec{\sigma}\right )

where I is the identity operator, \vec{\sigma} is a vector with the 3 Pauli matrices as its entries and \vec{r} is a real, 3-dimensional, unit vector. The vector \vec{r} is the Bloch sphere representation of the state.

There are more observables than just the Pauli matices, but any Hermitian operator can be written as:

H = \alpha I + \vec{\beta}\cdot\vec{\sigma}
where \alpha is any real number and \vec{\sigma} is any real 3-d vector.

As I mentioned before, a measurement of an observable can alternaticely be represented by projectors, and in this case the possible projective measurements are given by pairs of projectors

\Pi^{\pm}(\vec{r}) = \frac{1}{2} \left ( I \pm \vec{r}\cdot\vec{\sigma}\right )

where \vec{r} is a unit vector labeling the measurement and the \pm corresponds to the two possible outcomes. Note that the measurement projectors correspond to opposite points on the Bloch sphere.

 

[turin]

... any Hermitian operator can be written as:

...
where ... \vec{\sigma} is any real 3-d vector.

Did you mean "\vec{\beta} is any real 3-d vector?"

... the possible projective measurements are given by pairs of projectors

\Pi^{\pm}(\vec{r}) = \frac{1}{2} \left ( I \pm \vec{r}\cdot\vec{\sigma}\right )

This clarifies quite well how the projectors can depend on the state. Good stuff.

 

[alexepascual]

SlyBoy and Turin:
Thanks for your posts. I can't say anything right now because I have to think about it. I prited it out and I'll study the subject tonight.

 

[alexepascual]

Hi guys,
After this long silence, I wanted to let you know that I have been thinking about the issues discussed in this thread and I have learned a good deal, but I have not gotten to the point I feel I have an answer to my original question. I have some ideas, but I am not sure.
I want to thank you all, your posts have helped me a lot, and have made my search for understanding less lonely. (I can't talk to my wife about QM)
Anyway, this thread is getting kind of long. I am planning on starting at least two new threads on more specific questions I ran into while trying to understand the meaning of operators:

Thread: Understanding the commutator.

Thread: The momentum operator as the generator of translations

So I'll consider this thread practically closed, unless you are a new visitor to the thread who feels there is more to be said. Otherwise, I invite you to my two new threads.