# Quantum Mechanics and The Harmonic Oscillator

Form groups of three to discuss and find a model for the following challenge. You should address the challenge, its modelling, and implications through respectful peer-to-peer interaction. Once you have reached consensus, you should document in detail your solution process including problem data, modelling assumptions, theorems/equations used, etc.

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Your report must include:

Title, reflecting your work.Abstract, a summary of your approach and results.Introduction, providing background information on what is known of the topic.Results, documenting in detail your model and how it is used to reproduce known results.Discussion, what can be learnt from your model and results provide pictures.Conclusion, discuss the importance of your results and possible future work.Acknowledgements provide information about the contribution of each author.

Hand in a single pdf file created using latex using the IoP class and attach a well-documented single-run script in the language of your choice (python, Julia, Matlab, Mathematica, etc) producing the images in your document.

Challenge

Simple molecules are usually modelled as two-well potentials.A standard analytic form to describe such potentials in the literature is the modified Manning potential,𝑉(𝑥)=−𝑉1sech6𝑥−𝑉2sech4𝑥−𝑉3sech2𝑥,

with the requirements,𝑉1>0,𝑉2<0,𝑉3>0,and−𝑉32𝑉2<1,

plus the transformation 𝑧=tanh2𝑥and the eigenstate ansatz,𝜓(𝑧)=𝑒√𝑉12𝑧(1−𝑧)√−𝐸2𝜙(𝑧),

where E is the eigenvalue corresponding to the eigenstate.

This allows us to rewrite the time-independent Schrödinger equation in the form:(∑𝑎𝑗𝑧𝑗𝑑2𝑑𝑧23𝑗=1+∑𝑏𝑗𝑧𝑗𝑑𝑑𝑧2𝑗=0+∑𝑐𝑧𝑗1𝑗=0)𝜙(𝑧)=0.

Such that we can find eigenstates using numerical methods. I am fond of a particular method called grading that sometimes can provide analytic solutions. Try your hand at finding an eigenstate of the potential, use the following values for plots: 𝑉1=1,𝑉2=−6,𝑉3=6.Can you find solutions for different parameters?

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