Consider possible terms in the (quantum) Hamiltonian for a one-dimensional system,

where \(\psi(x)\) is a continuous complex function of the spatial variable \(x\), and \(1 /\left(d^{2} / d x^{2}\right)\) is understood as the inverse of an operator, acting on the function. Can these terms be understood to be coming from local interactions in \(x\) (interactions defined at a single point \(x\) ), and if so, why? Use a calculation to explain your reasoning.

Transcribed Image Text:

H = =fdx 4' (x)e" # 4(x). I = [ dx 4" (x) 12/24 ( -v(x), d/dx (5.63)