Is the ISW Hamiltonian Diagonal in the Energy Basis?

[v_pino]

Homework Statement

Find the matrix elements of the Hamiltonian in the energy basis for the ISW. Is it
diagonal? Do you expect it to be diagonal?

Homework Equations

$H=\frac{p^2}{2m}+V$

$\frac{d}{dt}\langle Q \rangle = \frac{i}{\hbar} \langle[\hat H, \hat Q] \rangle + \langle \frac{\partial \hat Q}{\partial t} \rangle$

The Attempt at a Solution

How should I convert H into matrix?

 

[bjnartowt]

I assume by ISW you mean "infinite square well". Anyway, you're not asked to convert H into a matrix. You are asked to find its matrix elements. Some things: 1) Matrix elements are indexed scalars. 2) "Matrix elements" carries the connotation of "eigenvalues". 3) What happens when the Hamiltonian operator "hits" an energy-basis vector/ket? 4) What is the dual of this equation? Does that get the gears working?

 

[jlittle]

Thanks bjnartown, I searched and found this. I already had the correct math down, but your explanation is wonderful and I really learned a lot from it. I seriously created an account to tell you that, lol.