Recent content by AspiringResearcher

Hi all, I am an undergraduate junior majoring in materials science who would like some advice with respect to which courses to take for the fall semester of my senior year. Some background: I am a materials science student and I intend to study spintronics and topological insulators for my...

Homework Statement A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E. a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1. b) Calculate the probability current density...

YOU'RE MY HERO

Couldn't we let H = C^{\infty} and say there's an isomorphism (coordinatization w.r.t. the {|x\rangle} basis) between C^{\infty} and L^2?

This is a bizarre problem. I think my professor and Griffiths are not very instructive in this. In class, \psi(x) is always called a wavefunction, not just the evaluation of |\psi\rangle at x; if it were just the evaluation, then it should be called a scalar and not a vector

What I am trying to do here is more of a thought experiment involving the construction of an abstract set H whose elements are kets and is a Hilbert Space

I believe I have established an isomorphism Λ : L(H,H) \mapsto L(L^2,L^2) which satisfies what I was trying to do. I would appreciate a critique of my reasoning. Here, I am considering the set of quantum states/kets H, and the set of wavefunctions, L^2 as isomorphic Hilbert spaces. The...

Sorry if there is any confusion. Here, I am only discussing Hermitian operators whose eigenvectors span Hilbert space. Surely the spin operator's eigenfunctions don't span L^2, right?

I highly doubt that your answer is correct because your answer is always a rational number, whereas the sum of square roots is not necessarily rational.

I know how to execute a basis transformation using the completeness operator. I think you're missing the point - isn't it true, in general, that \psi(x) \neq |\psi\rangle Rather, \psi(x) = \langle e_x | \psi \rangle

The point is that an arbitrary operator \hat{Q} will have a different form in L^2 than in H. Perhaps I wasn't clear in the OP that H is the vector space of all kets. Kets and wavefunctions (members of L^2) are not the same thing; there is an isomorphism that relates them. I want to explicitly...

You have been helped quite a lot already. You should be able to use your knowledge and the help given to you to construct the solution from the ground up.

The method suggested by the hints in that PDF is equivalent to the solution that @haruspex and I propose.

By "most of the problem," I meant strategy formulation, which is the most important part of solving physics problems.

I will attempt to help you with a solution. First of all, I will comment, as many other users here have commented, that the problem statement and the diagram are not consistent. I will assume in this attempt of a solution that the diagram is the correct representation of the problem; the...