Advice for High School Student's Quantum Mechanics Assignment

[Nakalay]

Hi

I have been given an interdisciplinary assignment in physics and mathematics. The product of this impending project should be a paper of approximately 15-20 pages. My chosen subject is quantum mechanics. I would very much like your ideas and advice concerning the organization and the choice of topics to investigate/expound on in this paper.

I am still in High School so I'm not too familiar with many of the more advanced mathematical tools associated with QM. However, I am quite conversant with topics such as vectors, calculus and differential equations. My teacher proposed that I include a chapter on complex numbers (their general properties in mathematics and the necessity (?) for them in the theory of quantum physics) but where would you suggest that I look for information on this subject?

Concerning the physics, which is my main focus, I would like to study the phenomenon of quantum tunneling, and I would also like to investigate such topics as the Schrödinger equation, wave functions and the Heisenberg uncertainty principle. I am not sure how to go about it, though, so I hope you can advise me on this topic.

I should have plenty of material; I have already been to the library where I borrowed the following books:

The principles of quantum mechanics / P.A.M. Dirac
Introductory quantum mechanics / Richard E. Liboff 
Quantum mechanics : a modern and concise introductory course / Daniel R. Bes
Quantum physics of atoms, molecules, solids, nuclei, and particles / Robert Eisberg
Principles of quantum mechanics / Ramamurti Shankar
QED : the strange theory of light and matter / Richard P. Feynman
Quantum mechanics demystified / David McMahon
In Search of Schrödinger's Cat / John Gribbin

And the following books are on their way:

Introduction to quantum mechanics / David J. Griffiths
Modern quantum mechanics / J. J. Sakurai
Quantum mechanics / Claude Cohen-Tannoudji
Understanding quantum physics : A user's manual / Michael A. Morrison
Quantum theory / David Bohm
Understanding quantum mechanics / Roland Omnès 
The strange world of quantum mechanics / Daniel F. Styer 
Schrödinger's kittens and the search for reality / John Gribbin
Quantum mechanics in simple matrix form / Thomas F. Jordan

How would you suggest I go about this assignment? Which topics are essential and how could I organize it? I will be very grateful for any advice. Thanks in advance.

 

[cesiumfrog]

Did you just request every book they had on the topic?

If you want to teach yourself QM, I suggest you complete all of the exercises in Griffiths (at least until you get to the 1D scattering/tunneling example, which is as much as you were aiming for). Then write 20 pages explaining what you've done so that others in your class can understand it. You won't need a second book on QM: a single textbook is sufficient (Griffiths especially so, since it is a standard, modern and favourite uni textbook), and you'll only confuse yourself if you try to learn multiple approaches before you properly understand one.

Alternatively, if you just want to write "about" quantum mechanics you'll want to flip through numerous popular accounts (including QED). In this case make sure your write-up consists largely of direct quotes (with proper citation as always) from numerous sources. This is to avoid making too many incorrect statements (never trick yourself into thinking you have actually "learnt" a topic just by reading stories "about" the topic).

 

[Nakalay]

Thanks for the reply.

Did you just request every book they had on the topic?

Well, I just wanted to be 100 % sure that I had all the books that I could possibly need.

Alternatively, if you just want to write "about" quantum mechanics you'll want to flip through numerous popular accounts (including QED). In this case make sure your write-up consists largely of direct quotes (with proper citation as always) from numerous sources. This is to avoid making too many incorrect statements (never trick yourself into thinking you have actually "learnt" a topic just by reading stories "about" the topic).

My aim is to condense some of the key aspects of QM into those 15-20 pages so as to give a somewhat clear image of this theory. For example if I wanted to focus explore tunneling, which quantum related topics would I need to account for to give a decent description of this phenomenon?

 

[Count Iblis]

The book "The principles of quantum mechanics / P.A.M. Dirac" is the best book ever written on Quantum Mechanics. This book probably won't be of much help for your assignment as it will take you some time to digest the contents.

The popular books by John Gribbin, Paul Davies, Heinz Pagels are probably more suitable for your project. E.g. Heinz Pagels in his book the Cosmic Code gives a very clear example of a Bell's inequality that involves almost no maths!

But the best way to actually learn quantum mechanics is by avoiding the books that use the historical approach starting with wave particle duality, Schrödinger equation etc. like most modern textbooks do and instead read a book that starts with the formalism first like Dirac does. Whenever you see that there is a gap in your knowledge that prevents you from understanding it, you must study the maths.

To understand quantum mechanics you need to understand the following topics:

Linear algebra: Vectors, vector spaces, inner products, matrices, eigenvectors, eigenvalues etc.

Calculus

Classical Mechanics: In particular Hamiltonian and Lagrangian formulation of classical mechanics. Sadly, this is not taught in high school.

When you read Diracs book then you'll encounter more advanced concepts like Hilbert spaces, the dual of a vector/Hilbert space etc. However, this is explained sufficiently in the book itself, so that only an elementary knowledge of linear algebra and calculus is enough to understand it.

 

[Nakalay]

Thanks for the reply, Count Iblis, I will definitely keep that in mind. But as you correctly assume I will not have time to study any book in detail during this impending project. That must, unfortunately, wait until after my project is done.

I would still very much like to know which topics I should touch on in my paper if I - for example - wanted to investigate the quantum tunneling phenomenon.

 

[Count Iblis]

I think the best thing is to study complex numbers a bit. In particular, discuss the formula exp(i x) = cos(x) + i sin(x) in your paper.

In quantum mechanics a particle with momentum P and energy E is described by the wavefunction: Exp[i (P/h-bar x - E/h-bar t)] and you need to be able to understand and explain this a bit.

You can then discuss tunneling. The kinetic energy in the "forbidden region" is negative. Classically a particle cannot be found there. But in quantum mechanics the wavefunction is not zero. The kinetic energy is P^2/(2m) and if this is negative that means that P is purely imaginary. The term Exp[i P/h-bar x] becomes of the form Exp(-a x). So the wavefunction which as a periodic function becomes an exponential function. On the other side of the potential barrier the wavefunction is still nonzero and it behaves like
Exp[i (P/h-bar x - E/h-bar t)] again but with a reduced amplitude.

On the side of the barrier where the particle is incident the wavefunction is actually of the form (omitting time dependece):

A Exp[i P/h-bar x] + B Exp[-i P/h-bar x]

The first term correpsonds to the particle with momentum P, so the second term corresponds to the particle with momentum - P which means that it is moving away from the barrier, i.e it describes the possibility of being reflected back by the barrier.

So, you see that you can already do a lot with only an elementary knowledge of complex numbers...

 

[cesiumfrog]

I think the complex numbers are a bit of a herring.

Despite the teacher already noting that they are used in physics and QM, I think that's only due to convenience rather than necessity (the e^ix stuff could be reformulated as a real 2D vector-valued sinusoidal function, for example). Properly explaining this topic would require much deeper understanding of the physics than what is required just to do some 1D tunnelling textbook-exercises.

On the other hand, if you're not yet familiar with complex numbers, you might be better off forgetting QM for now. Write a program to generate Mandelbrot fractals, use e^ix to describe propagation and interference of *classical* waves (eg. light through Young's double slit). Then come back to a modern QM textbook (I really advise against the "popular" stuff: so prone to giving miss-impressions).