# Problem 2.

I was deceived by an incorrect result in Mathematica, which led me to believe that the second order energy perturbation was zero (whereas part (c) of the problem asked if it was greater or lesser than zero). I started starting writing this up to show my reasoning, but our Professor quickly provided an example after class showing how this zero must be wrong, and I didn’t have to show him any of this.

## Setup

Recall first the one dimensional particle in a box. Within the box we have to solve

and find

With

our general state, involving terms of each sign, takes the form

Inserting boundary conditions gives us

The determinant is zero

which provides our constraint on

We require for any integer , or

This quantizes the energy, and inverting 1.3 gives us

To complete the task of matching boundary value conditions we cheat and recall that the particular linear combinations that we need to match the boundary constraint of zero at were sums and differences yielding cosines and sines respectively. Since

So sines are the wave functions for since for integer . Similarly

Cosine becomes zero at , so our wave function is the cosine for .

Normalizing gives us

## Two non-interacting particles. Three lowest energy levels and degeneracies

Forming the Hamiltonian for two particles in the box without interaction, we have within the box

we can apply separation of variables, and it becomes clear that our wave functions have the form

Plugging in

supplies the energy levels for the two particle wavefunction, giving

Letting each range over for example we find

It’s clear that our lowest energy levels are

with degeneracies respectively.

## Ground state energy with interaction perturbation to first order.

With positive and an interaction potential of the form

The second order perturbation of the ground state energy is

where

and

to proceed, we need to expand the matrix element

So, for our first order calculation we need

For the second order perturbation of the energy, it is clear that this will reduce the first order approximation for each matrix element that is non-zero.

Attempting that calculation with \href{https://github.com/peeterjoot/physicsplay/blob/796c8e3739ae1a9ca26270a0e91384afba45661d/notes/phy456/problem\

This worksheet can be seen to be giving misleading results, by evaluating

Yet, the FullSimplify gives

I’m hoping that asking about this on stackoverflow will clarify how to use Mathematica correctly for this calculation.