hilbert spaces old

Hilbert Spaces involved in the B&B paper PhL 1.19.05 There are a lot of Hilbert Spaces flying around all at once, and that causes immense confusion. So let's try to put this stuff now into some reasonable logic ordering, based on the discussion in document "Addendum... ". Space #1. We start with our abstract position operator eigenvalue equation X | xi > = xi | xi > but the xi are our N discrete points which are the nodes of a polynomial PN(xi). The states | xi > have these orthonormality and completeness properties, 1 = i=0N Ai |xi><xi| <xi|xj> = (1/Ai) i,j The analogy with the continuous world is this: 1 = dx w(x) | x > < x | < x | x' > = (1/w(x)) (x - x') The states | xi > form a basis in an N dimensional Hilbert space we might call "node space". The space is a Hilbert space because the | xi > are eigenvectors of a Hermitian operator X, more on this later. Space #2. This is what we have called " |n> space". We could define a general "function space" where we could denote by | f > an abstract function f which maps some space into some other space. When we write | n>, we are referring to the particular function | Pn> which is one of our basis functions. In the coordinate representation we write <xi | Pn> = <xi | n > = Pn(xi) Now our n-space does not allow ANY function, it only allows functions which are polynomials of degree N-1 or less. Thus, this space is an N dimensional space. Our completeness and orthogonality in n-space appears as follows: 1 = n | n> <n| <n|m> = n,m Consider now this expansion: | n > = 1 | n > = i=1N Ai |xi><xi|n > = i=1N Ai Pn(xi) |xi> This shows that | n > is just a linear combination of the |xi> states. So the vectors | n > are just an alternate basis in our "node space". They form an orthonormal basis. Here is a result we get by playing with our two different bases: <xi|xj> = (1/Ai) i,j = <xi n | n> <n| xj> = n <xi |n > <n| xj> = n Pn(xi) Pn(xj) From this we conclude that n Pn(xi) Pn(xj) = 0 i j n [ Pn(xi) ]2 = (1/Ai) = Si and we will see this summation used in various places. Space #3. This is the "continuous position space" with | x >. In this space we have as noted earlier 1 = dx w(x) | x > < x | < x | x' > = (1/w(x)) (x - x') X | x > = x | x > This is an infinite dimensional vector space. We could associate this space with | f >, the space of functions, which is of course also of infinite dimension. We often list these functions as x0, x1, x2 etc. We know that the space of polys of degree N-1 is an N dimensional subspace of the | f> space. The space and its subspace had the same identify operator "1". That is why we can do the following: As an application of this we can write < n | m > = < n | 1 | m > = < n | dx w(x) | x > < x | m > = dx w(x) <n|x><x|m> => n,m = dx w(x) Pn(x) Pm(x) So "continuous position space" is the space which has w(x) and (a,b) and which we use to define our Pn(x) ortho polys. Space #4: This is what we call the J-space, another N dimensional space. This is an EN space where we do things like our Jacobi matrix equation. The vectors in J-space are just N-tuples of numbers, j = [v1, v2, ..... vN ] = a vector in J space Here is a vector of particular interest in J-space, V(xi) = [P0(xi), P1(xi), ..... PN-1(xi) ] We think of (xi) as a parameter on this vector, so V(xi) is just an N-tuple of numbers. Here is our Jacobi matrix equation in J-space J V(xi) = xi V(xi) which we can write as m Jnm [V(xi)]m = xi [V(xi)]n which we can rewrite as m Jnm Pm(xi) = xi Pn(xi) or m Jnm < xi | m > = xi < xi | n > We can now replace xi < xi | n > = < xi | X | n > using an earlier fact, and then we can simply remove the projection from both sides, since must be true for any xi , and we get (changing name of dummy sum index) m' Jnm' | m' > = X | n > Now close both sides with <m | to get Jnm = <m | X | n > This shows that the "Jacobi matrix" matrix elements are the matrix elements of the position operator X taken in the | n> basis of our N dimensional Hilbert Space. Now one other fact about our vector V(xi). We can compute it's magnitude squared in J-space, || V(xi)||J-space = V(xi) V(xi) = n [ Pn(xi) ]2 = Si Sample Calculation Consider, where we now identify matrix J with matrix TN, m Jnm Pm(xi) = xi Pn(xi) or TN V(xi) = xi V(xi) The "vector sense" of this eigenvalue equation for a given fixed index "i" is that of our J-space. After all, TN is a matrix in J-space. The "classic result" for first order perturbation theory is this xi = where recall that the denominator is Si . This is how we obtain (2.13) of Jim's ".. 2 classes of log..." paper, or the corresponding equation on page 5 of the first B&B paper. Now that we have obtained the correct "official result" we can rewrite the above as follows numerator = ( V(xi),[TN]V(xi)) = nm [V(xi)]n[TN]nm[V(xi)]m = nm Pn(xi) [TN]nm Pm(xi) = nm <xi | n> <n |[TN] | m> <m|xi> = <xi | [TN] | | xi > denominator = <xi | xi> = Si and we would say as our perturbation theory result xi = Now, recall that the abstract operator TN is really X, so we are talking about X. This X operator really does change if we change our weight function. Think of it this way X() | xi, > = xi | xi, > 1 = i=0N Ai |xi, ><xi, | <xi, |xj, > = Si() i,j where is a parameter we can vary. We would then quote directly from 1st order perturbation theory to make this claim xi = I am certainly tempted to identify the J-space with the n-space, continue manana.