- #1
eoghan
- 207
- 7
Hi!
During a lesson, the prof said that the Fock space could be non-separable. When can this happen?
During a lesson, the prof said that the Fock space could be non-separable. When can this happen?
Those eigenstates don't have a finite norm of the usual kind required in a Hilbert space. Some authors use a generalization, known as "rigged Hilbert space" or "Gel'fand triple" instead. Ballentine sect. 1.4 gives a gentle introduction.eoghan said:So, if for example I use the eigenstates of the position operator to label the state of a particle, then I have an infinite uncountable basis vectors(the particle can be in any |x> state with x real and non countable) and the space is non separable?
This goes beyond what I described. I didn't say a continuum of modes like you'd get with eigenvectors of x. As strangerep points out, such states would not be normalizable.So, if for example I use the eigenstates of the position operator to label the state of a particle, then I have an infinite uncountable basis vectors(the particle can be in any |x> state with x real and non countable) and the space is non separable?
A non-separable Fock space is a type of mathematical space used in quantum mechanics to describe the state of a system with an infinite number of degrees of freedom. It is a generalization of a separable Fock space, which only applies to systems with a finite number of degrees of freedom.
A separable Fock space can be constructed from a countable basis, meaning that its states can be represented by a countable set of wavefunctions. In contrast, a non-separable Fock space cannot be represented by a countable set of wavefunctions, and instead requires an uncountable basis.
Non-separable Fock spaces are important in quantum mechanics because they allow for the description of systems with an infinite number of degrees of freedom, such as a quantum field. They also play a crucial role in the study of entanglement and the development of quantum information theory.
The uncertainty principle states that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. In a non-separable Fock space, the number of particles can be infinite, making it impossible to know both the position and momentum of all particles simultaneously. This is due to the uncountable basis of the non-separable Fock space.
No, non-separable Fock spaces cannot be visualized or easily understood intuitively. They are abstract mathematical spaces that are used to describe complex quantum systems and do not have a direct physical representation. However, they are a crucial tool in the study of quantum mechanics and have been successfully applied in various areas of physics, including quantum field theory and quantum information theory.