2.3 The Operators of Quantum Mechanics

Suppose the wave function of a particle in a one-dimensional nano-wire is a Gaussian:

$$\Psi = C e^{-x^2/\ell^2}$$

where $C$ and $\ell$ are constants that are not of interest now.

1

What function does the $x$-position operator produce from $\Psi$?

2

What function does the linear $x$-momentum operator produce from $\Psi$?

3

What function does the kinetic energy operator produce from $\Psi$?

4

If the potential energy of the particle $V=cx^2$ with $c$ a constant, then what does the Hamiltonian produce from $\Psi$?

5

What is the Laplacian in Cartesian coordinates?