State Vector Concepts in Quantum Mechanics: A Beginner's Guide

[Seanskahn]

hello everyone.

my first post here. I apologize, if i make something stupid.

I am a third year undergraduate. I took a course in quantum mechanics last semester, and i am taking one more course this semester, and i must admit, i am not very fluent in the realm of quantum mechanics.

my question is rather basic. I was rereading the book "Quantum mechanics, concepts and applications" by Nouredine Zettili. WHile many of my colleagues say this is a "basic" book, i like it, because it helps me to build concepts.

When studying the state vectors, i reminded myself about the phase space in classical physics. a link here: http://en.wikipedia.org/wiki/Phase_space

I was wondering, the reason one wants to deal with the phase space is that, it is a complete space which encloses the position and momnetum vectors. But to the best of my knowledge, it does not include any mathematical operation (addition and multiplication can be applied on it, but does the definition of the phase space include these?).

My question is, is the reason why one wants to calculate state vector the same as above, that because it encloses the position and momentums of the system? also, as the name implies, the state vector is not the space of all possible momentum/positions. is there the notation of such a space which encloses all momentum and positions? does that space encloses the mathematical operations? [bzw. operators]

sorry for the entry level question, but often the professor is too busy to answer it.

 

[Fredrik]

I was wondering, the reason one wants to deal with the phase space is that, it is a complete space which encloses the position and momnetum vectors. But to the best of my knowledge, it does not include any mathematical operation (addition and multiplication can be applied on it, but does the definition of the phase space include these?).

You're right that addition and multiplication isn't defined on a classical phase space.

My question is, is the reason why one wants to calculate state vector the same as above, that because it encloses the position and momentums of the system?

This is the point of view I like best: We don't usually think of classical mechanics in terms of probabilities, but we can if we want to. Classical mechanics assigns trivial probabilities (either 0 or 1) to possible results of experiments. If we try to modify the mathematical structure of classical mechanics to allow non-trivial probabilities, we will almost inevitably end up with quantum mechanics. The simplest idea about how to introduce non-trivial probabilities turns out to be quantum mechanics.

In classical physics, the states of the system are the points of phase space. Observables are real-valued functions on phase space. For example, the energy observable is the function that takes a state to the energy that the system has when it's in that state. This means that experimentally verifiable statements correspond to subsets of phase space. For example, if f is the energy observable, the statement "If I measure the energy of this system, I will get a result in the set E", corresponds to the preimage f^-1(E). This is the set of states in which the observable has a value in E. Every experimentally verifiable statement can be represented by the preimage of the set mentioned in the statement. A classical theory assigns the probability 1 to an experimentally verifiable statement when the state is the subset, and 0 when it isn't.

I don't think there can be a simpler way to introduce non-trivial probabilities than to let the experimentally verifiable statements be represented by subspaces of an inner product space instead of subsets of some other structure. If a pure state is represented by a one-dimensional subspace R, and an experimentally verifiable statement is represented by some subspace V, then we assign probability 0 to the statement represented by V if R is orthogonal to V, and probability 1 if R is a subspace of V. In any other case, we assign a non-trivial probability using the inner product. Gleason's theorem tells us that there's only one way to do that.

This approach to QM makes the theory look like a toy model that someone came up with just to show that it's possible to define a theory that assigns non-trivial probabilities. I think that's what I like the most about it.

is there the notation of such a space which encloses all momentum and positions?

As I'm sure you know, a quantum particle has neither a well-defined momentum nor a well-defined position, and if you prepare a state with a very low "uncertainty" in one of these observables, the uncertainty in the other will be very high. So the quantum counterpart to the phase space would have to be something completely different. The points in phase space represent states, so you should probably be looking at the set of states in QM. There are many different ways to represent states mathematically, for example

1. A state operator. See Ballentine.
2. A function that takes observables to probability measures on the Borel sets of the real numbers, or a probability measure on the lattice of subspaces. See Varadarajan. (Pages 48-52, which can't be previewed at Google Books).

The pure states can be represented by one-dimensional subspaces, or by positive normalized linear functionals on the algebra of observables. See Araki for the latter option.

 

[Seanskahn]

hi Fredrik

Thank you very much for the reply. i find it awesome that our "intuitive consiousness" also agrees with the classical notion of trivial probabilities. (i apologize, the sentence sounds philosophical)

AND MANY MANY THANKS FOR THE BIBILIOGRAPHY; after I posted I saw I forgot to ask for them. Therefore I was thinking I would ask the answerer to refer me to some bibliography. You already posted them. Many thanks for them.

with friendly greetings
s

 

[Fredrik]

You should definitely read the first few pages of Araki, where he explains the concepts of "state" and "observable" (better than anywhere else I've seen). It's important to understand the real-world concepts, and not just how to represent them mathematically. Once you understand those concepts, it's time to start thinking about how to represent real-world concepts mathematically. There are three main approaches to QM:

1. The standard (Hilbert space) approach, in which pure states are represented by the one-dimensional subspaces (rays) of a complex separable Hilbert space.
2. The algebraic approach, in which observables are represented by a C*-algebra (and pure states by linear functionals on that algebra).
3. The quantum logic approach, in which experimentally verifiable statements are represented by some sort of lattice (which is usually assumed to be isomorphic to the lattice of closed subspaces of a complex separable Hilbert space).

(Those are not exact definitions; I left out some technical details, mainly because I don't remember them all).

The Hilbert space approach is the easiest, and I think the quantum logic approach is the hardest. Varadarajan's book is the most difficult book I own. The algebraic approach is difficult too, since it requires more knowledge of functional analysis than the Hilbert space approach. This thread has some references to books on functional analysis. This one has a few references to books and an article on quantum logic (in post #4, by Meopemuk), which could be good alternatives to Varadarajan. "The logic of quantum mechanics" by Beltrametti and Cassinelli is another standard reference, but it's out of print.

Regarding the books I mentioned, Ballentine is a standard textbook suitable for a second course in QM. Araki is an advanced book about the algebraic approach to QM and quantum field theory. Strocchi also emphasizes the algebraic approach.

I'll add one more. I really like Isham's book. It's a much easier read than any of the ones mentioned above, and is a good place to begin if you find them too hard. I should probably mention that I have only read little bits of the books I referenced. Isham is the only one I read cover to cover.

 

[dextercioby]

For the Hilbert space approach (using vectors and linear operators in a Hilbert space) I would warmly reccomend E. Prugovecki's <Quantum Mechanics in Hilbert Space>. It has a thorough treatment of HS methods and an axiomatization based on the mathematical methods. No RHS, though.

If you can't get this book, you may search for the (I believe still free) lecture notes of Professor Teschl at TU Wien (in English). The axiomatization here is missing though.

Or you may consider other famous (but old and normally found only in libraries) treatments: Jauch, Mackey, von Neumann, Emch (algebraic approach), Piron (algebraic approach), Bunge, etc.

 

[Seanskahn]

thanks for the books.

As for the language, either German or English would work with me.

Fredrik, i will pick up the book of akira today