Setup

Consider a Particle in a Potential . We want to find the Energy of the Ground state of a Particle in this Potential.

Solution

We will find an upper bound for the Ground state energy by calculating the Energy for the given by . The associated Hamiltonian is given by . We perform the Wave function energy calculation as .

We will split the calculations into 2 parts. We start with :

Secondly we simplify :

Hence

= Since we want to find an upper bound for the Ground state Energy , we want to minimize , using the standard method:

We get the upper bound .