# Phase Space and two dimensional Hilbert Space

I always had this doubt,but i guess i never asked someone. What's the main difference between the Classical phase space, and the two dimensional Hilbert Space ?

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dextercioby
Homework Helper
What reading have you got on both things?

my brief understanding for each one is :

The Classical phase space allowed me to describe the dynamics of a system by its eigenvalues and eigenvectors, given from a Hamiltonian .

The two dimensional Hilbert Space, its quitely the same thing,but here i work with eigenstates that my system can assume,restricted by rules of this vectorial space ,where my observables are represented by operators acting on eigenstates.

i don't know if i make myself clear,but my question remains : There's no difference between them ?

dextercioby
Homework Helper
Do you have a reference for this ?
[...]The Classical phase space allowed me to describe the dynamics of a system by its eigenvalues and eigenvectors, given from a Hamiltonian .[...]

Actually, i wrote this from my previous knowledge acquired from undergraduate course in physics.

dextercioby
Homework Helper
I am sorry, you need to do some more reading. We cannot answer questions coming from misunderstandings whose clearing you cannot grasp.

PeterDonis
Mentor
2019 Award
i wrote this from my previous knowledge acquired from undergraduate course in physics.
What textbook were you using? Can you give a reference from it that explains where your statement in post #3 comes from?

(And in case it isn't apparent, the reason we are asking is that your description of classical phase space and dynamics doesn't look right; it looks like a description of quantum Hilbert space and dynamics. So it seems like you are confusing the two.)

Gean Martins
A. Neumaier
2019 Award
A 2-dimensional space (but a symplectic space, not a Hilbert space) describes classical linear dynamics of a single particle in 1 dimension. In contrast, a 2-dimensional Hilbert space describes quantum linear dynamics of a single spin degree of freedom. The interpretation of a vector in the two spaces is also quite different: Each 2-dimensional vector in the classical symplectic space has real coordinates and describes position and momentum of a moving particle, while a 2-dimensional vector in the quantum Hilbert space has complex coordinates and is just a representative of a ray describing a point on the Bloch sphere, corresponding to a pure spin 1/2 state of a particle at rest.

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Gean Martins
I've used basically two books for my readings on Classical Mechanics ,they are : Classical dynamics of particles and systems - Stephen T. Thornton and Jerry B. Marion. ; And Mecânica Analítica - Nivaldo Lemos;

Maybe, i might have expressed badly on my definition of a Phase Space, so i don't want to relate the ideas that i've pass with the books that i used, they are great for advanced introduction on Classical Mechanics. I want to thank you all for the answers, especially A . Neumaier , that's an answer that i was expecting to get . Thank you good sir.

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