Formulation of quantum mechanics

[gentsagree]

From Wiki:

"...the possible states of a quantum mechanical system are represented by unit vectors (called state vectors). Formally, these reside in a complex separable Hilbert space—variously called the state space or the associated Hilbert space of the system—that is well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space. The exact nature of this Hilbert space is dependent on the system—for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes"

In what sense the states are points in the projective space? Is this because they are only defined up to the phase rotation (i.e. a U(1) gauge redundancy)? Does this mean that when I think of a theory admitting gauge symmetry I can think of the states of this theory as living not in a Hilbert space, but a projective Hilbert space, technically?

Also, what is it meant by the last sentence, that "state space for the spin of a single proton is just the product of two complex planes" ?
Could you comment on this?

Thanks!

 

[bhobba]

OK - I will tell you what the more advanced books say - which is correct. Its from Ballentine - QM - A Modern development:
https://www.amazon.com/dp/9814578584/?tag=pfamazon01-20

I suggest if you are interested in such things get a hold of the book and read the first three chapters.

First we will start with the axioms of QM:

Axiom 1
Associated with each measurement we can find a Hermitian operator O, called the observations observable, such that the possible outcomes of the observation are its eigenvalues yi.

Axiiom 2 - called the Born Rule
Associated with any system is a positive operator of unit trace, P, called the state of the system, such that expected value of of the outcomes of the observation is Trace (PO).

Note - the state of a system is not an element of a Hilbert space - it is an operator. How they come into it I will explain.

A state is called pure if its of the form |u><u|. A state is called mixed if its the convex sum of pure states ∑ pi |ui><ui|. It can be shown all states are either pure or mixed. Its the pure states that can be mapped to the Hilbert space - the u in |u><u| can be mapped to normalised vectors but there is an ambiguity in doing so because |cu><cu| = |u><u| if c is simply a phase factor. It can be extended further for all vectors in the Hilbert space by |u/||u||><u/||u||| - hence the states become rays in the Hilbert space - but the length is on no consequence. However always bear in mind the state is really an operator - this is simply a mapping.

Indeed this mapping introduces a gauge symmetry in the states and that is important in EM:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

Cant help you with the last sentence - don't know what they mean by that myself - but I think its probably got to do with the Bloch Shpere:
http://en.wikipedia.org/wiki/Bloch_sphere

Thanks
Bill

 

[dextercioby]

[...], while the state space for the spin of a single proton is just the product of two complex planes"[...]

[...] Also, what is it meant by the last sentence, that "state space for the spin of a single proton is just the product of two complex planes" ?
Could you comment on this?

Thanks!

There's a trick with the spin of the photon, it's not a well-defined concept/observable. What I think the author meant to say is that the the states of well-defined polarization are members in a 2-dimensional Hilbert space which is $\mathbb{C}^2 = \mathbb{C}\times\mathbb{C}$

 

[vanhees71]

There's one more subtlety. The first quoted sentence from the Wikipedia is not the whole truth. A (pure) quantum state is not represented by a unit vector in Hilbert space but by a ray in Hilbert space. This is of utmost importance, because this gives you more freedom within the theory to choose states. E.g., for this reason the half-integer representations of angular momentum, which are representations of the covering group SU(2) of the usual rotation group SO(3), which occurs in the very beginning of classical mechanics through the space-time structure in both non-relativistic (Galilei-Newton) and special-relativistic (Einstein-Minkowski) spacetime. For sure, the possibility of half-integer-spin representations is very important since all the visible matter in the universe consists, according to the very (or even annoyingly) successful Standard Model of elementary-particle physics, of spin-1/2 fermions, and this in turn is utmost important for the stability of bulk matter, forming stars, planets, and last but not least us :-)!

 

[Fredrik]

In what sense the states are points in the projective space? Is this because they are only defined up to the phase rotation (i.e. a U(1) gauge redundancy)? Does this mean that when I think of a theory admitting gauge symmetry I can think of the states of this theory as living not in a Hilbert space, but a projective Hilbert space, technically?

It has nothing to do with gauge invariance. In every quantum theory, if v is a state vector and c is a complex number, then v and cv represent the same state. The set of pure states is not in bijective correspondence with the set of state vectors. It's in bijective correspondence with the set of 1-dimensional subspaces. This set is what they call the projective space.

Also, what is it meant by the last sentence, that "state space for the spin of a single proton is just the product of two complex planes" ?
Could you comment on this?

They're just saying that it's a 2-dimensional complex Hilbert space. This makes it isomorphic to $\mathbb C^2$, the cartesian product of two copies of $\mathbb C$.