- #1
AspiringResearcher
- 18
- 0
Hi physicsforums,
I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying [itex]L^2[/itex] space, state vectors, bra-ket notation, and operators, etc.
I have a few questions about the relationship between [itex]L^2[/itex], the space of square-integrable complex-valued functions [itex]\psi[/itex] and Hilbert Space, which is the space of state vectors [itex]|\psi\rangle[/itex]. To me, it is obvious that these two spaces are isomorphic. For the remainder of this problem, for clarity's sake, I will define the isomorphism between these two spaces: [tex] ζ : H \mapsto L^2 [/tex] defined by
[tex]∀ x ∈ R, ζ(|x \rangle) = \delta(x)[/tex]
where [itex]|x\rangle[/itex] is the eigenvector of [itex]\hat{x}[/itex] with eigenvalue x.
In linear algebra, for a vector space V, the set of linear operators from mapping V to itself is called [itex]L(V;V)[/itex]; this forms a vector space.
My question is this - every linear operator in [itex]L^2[/itex] has a corresponding linear operator in [itex]H[/itex], at least from what I've seen so far.
How do you construct an explicit isomorphism [itex] Λ : L(H;H) \mapsto L(L^2,L^2) [/itex] using [itex]ζ[/itex] as defined above? Please help me with this.
I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying [itex]L^2[/itex] space, state vectors, bra-ket notation, and operators, etc.
I have a few questions about the relationship between [itex]L^2[/itex], the space of square-integrable complex-valued functions [itex]\psi[/itex] and Hilbert Space, which is the space of state vectors [itex]|\psi\rangle[/itex]. To me, it is obvious that these two spaces are isomorphic. For the remainder of this problem, for clarity's sake, I will define the isomorphism between these two spaces: [tex] ζ : H \mapsto L^2 [/tex] defined by
[tex]∀ x ∈ R, ζ(|x \rangle) = \delta(x)[/tex]
where [itex]|x\rangle[/itex] is the eigenvector of [itex]\hat{x}[/itex] with eigenvalue x.
In linear algebra, for a vector space V, the set of linear operators from mapping V to itself is called [itex]L(V;V)[/itex]; this forms a vector space.
My question is this - every linear operator in [itex]L^2[/itex] has a corresponding linear operator in [itex]H[/itex], at least from what I've seen so far.
How do you construct an explicit isomorphism [itex] Λ : L(H;H) \mapsto L(L^2,L^2) [/itex] using [itex]ζ[/itex] as defined above? Please help me with this.