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Rasalhague

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**Background.**I recently started reading

*A Quantum Mechanics Primer*by Daniel T. Gillespie. I'm up to the section on Banach spaces in Kreyszig's

*Introductory to Functional Analysis with Applications*, and have looked ahead to the chapter on applications to quantum mechanics, Ch. 11. I've also read the brief section in Griffel's

*Linear Algebra and its Applications*, Vol. 2, section 12K*, on quantum mechanics. I'm left with some questions.

**1.**Suppose a quantum mechanical system is being modeled with unit vectors of [itex]L^2(\mathbb{R}^3,\mathbb{C})[/itex] to represent states of the system. How is the state space usually defined. Is it

(a) the Hilbert space [itex]\cal{H} = L^2(\mathbb{R}^n,\mathbb{C})[/itex], including all the unreachable, impossible, undefined states whose magnitude is not 1, and allowing distinct mathematical states - i.e. points in the state space, state vectors - to (redundantly) represent the same physical state,

(b) a set of equivalence classes on the subset of [itex]L^2(\mathbb{R}^n,\mathbb{C})[/itex] consisting of all unit vectors, equivalent under the relation [itex](|z|=1) \Rightarrow (\Psi \sim z \Psi)[/itex], made into a Hilbert space by appending to the standard function-space definitions of scaling and vector addition a rule which says "rescale to unit length",

or something else?

**2.**While searching for the answer to this question, I came across what looks like an alternative formulation, using what's called a projective Hilbert space, [itex]\cal{P}\cal{H}[/itex], where the vectors comprise a set whose elements are equivalence classes of L

^{2}vectors in the aforementioned subset, equivalent under the relation

[tex](\forall z \in \mathbb{C}\setminus \left \{ 0 \right \})(\forall \Psi \in \cal{P}\cal{H})[\Psi \sim z \Psi].[/tex]

But it seems this is not "how quantum mechanics is ordinarily formulated" (Brody & Hughston, 1999:

*Geometric Quantum Mechanics*), and requires a more complicated-looking version of the Schrödinger equation. While (b) modified the usual definitions of scaling and vector addition to ensure the underlying set is closed under the vector space operations, I guess a projective Hilbert space must modify the definition of the inner product to ensure that [itex](\forall \Psi \in \cal{P}\cal{H})(\left \langle \Psi, \Psi \right \rangle = 1)[/itex]. Is this done by simply specifying that the function integrated to give the value of an inner product is chosen to be an element of the vector (the vector regarded as an equivalence class of functions) such that

[tex]\int_{\mathbb{R}^3} ff = 1.[/tex]

Is this projective Hilbert space what Kreyszig is describing on p. 574, "Hence we could equally well say that the

*state*of our system is a one-dimensional subspace, [itex]y \subset L^2[/itex]..."? Is "state" synonymous (here or generally) with his "state vector"? And

*is a projective Hilbert space actually a Hilbert space*?*

Oh, one more question: what is the

**y**in Brody and Hughston's projective Schrödinger equation (p. 4).

*EDIT: It seems not! Suppose there exists a projective Hilbert space which is a Hilbert space, and that

*a*is a vector of this space. Then

a = -a;

a + a = a + (-a) = a - a = 0;

a + a = 2 a = a = 0.

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