## Monday, March 30, 2009

### The Quantum Harmonic Oscillator Ladder Operators

The Setup
As a first example, I'll discuss a particular pet-peeve of mine, which is something covered in many introductory quantum mechanics classes: The algebraic solution to quantum (1D) simple harmonic oscillator.1 The one-dimensional, time-independent SchrÃ¶dinger equation is:
$\mathcal{H}\Psi=E\Psi,$
(1)
where $\mathcal{H}$ is the Hamiltonian of the system. Explicitly, this Hamiltonian is
$\mathcal{H}=\frac{p^2}{2m}+V(x),$
(2)
where $p$ is the particle's momentum, $m$ is it's mass and $V(x)$ is the potential the particle is placed into. The potential associated with a classical harmonic oscillator is
$V(x)&=\frac{1}{2}kx^2$
$=\frac{1}{2}m\omega^2x^2,$
(3)
$\begin{split}
V(x)&=\frac{1}{2}kx^2\\
&=\frac{mx^2}{2\omega^2},
\end{split}$
where $\omega^2\equiv k/m$. For the sake of convenience, so we don't get bogged down with various factors,2 we'll consider $m=\omega=\hbar=1$. Then, if we substitute (3) back into (2) we write the Hamiltonian as:
$\mathcal{H}=\frac{p^2+x^2}{2}.$
(4)

Figure 1: How Einstein developed his famous $E=mc^2$ expression.

Now, at this point, many texts, (see [1] or [2] for example,) define, with no motivation other than future ``convenience'', two operators
$a \equiv (x+ip)/\sqrt{2}$
(5a)
$a^\dag \equiv (x-ip)/\sqrt{2}$
(5b)
and proceed to show how these can be used to simplify the Hamiltonian and easily solve the problem. While it is an elegant and quick solution, this presentation is completely useless. I find it highly unlikely that Dirac sat down to solve this problem and tried a whole series of random operators
$c \equiv (x^2+ip)/7$
$d \equiv (x^p-e^{ip})/\pi$
$f \equiv (\sqrt{x}e^{-x^2/8}+ip^3)/\sqrt\pi$
$g &\equiv (\sqrt{x p}-ip/x)/3$
and so on, along with their complex conjugates, until he lucked out with the solution in (5), see Fig. 1.

A better way
What is far more likely is the argument given by Griffiths in [3], which I'll loosely follow. He presents a rationale and a method for approaching this problem. Namely, he suggests factoring the Hamiltonian (4) into terms linear in $x$ and $p$. If we ignore the operator properties of $x$ and $p$ momentarily, and consider the classical quantities, we can factor the Hamiltonian
$\frac{(x+ip)}{\sqrt 2}\frac{(x- ip)}{\sqrt 2}=\mathcal{H}_{\textit{classical}}=\frac{(p+ ix)}{\sqrt 2}\frac{(p- ix)}{\sqrt 2}.$
(6)
Now we see a reason why (5) makes sense to try. Each term, either on the right or left side of (6),3 contains two terms which are complex conjugates of each other. If this were a classical problem, we could, in principle, make a change of variables converting the Hamiltonian to something of the form
$\mathcal{H}=\eta^2/2=\eta^*\eta/2,$
(7)
where $\eta$ is any of the combinations of $x$ and $p$ in (6). Although this is not a ``canonical transformation,'' the symmetric form4 of the Hamiltonian allows us to reduce the Hamiltonian from a function of two dynamical variables to a function of a single dynamical variable.

Switching back to quantum mechanics, we now see a rationale for choosing $a$ and $a^\dag$ as we did.5 Although deciding which variable to attach the $i$ to and its choice of sign is a guess,6 we now have a general method for approaching Hamiltonians that look like they might be easily factored classically -- try using the classical factorizations with quantum quantities and see what happens.

References
[1] J.J. Sakurai. Modern Quantum Mechanics. Addison-Wesley, San Francisco, CA, revised edition, 1993.
[2] F. Schwabl. Quantum Mechanics. Springer, 3rd edition, 2005.
[3] D.J. Griffiths. Introduction to Electrodynamics. Pearson Prentice Hall, 3rd edition, 1999.
[4] P.A.M. Dirac. The Principles of Quantum Mechanics. Oxford University Press, USA.

1 I'll be discussing this in the boring algebraic sense of symbols and whatnot, leaving off the geometric/visual interpretation of the algebra for another time.
2 Or maybe I'm just lazy
3 And, since this is a classical problem, the order does not matter, either.
4 Yes, it's only up to a constant which I've set to one, but you can still symmetrize things by changing to unitless variables, see [2].
5 Recall that the $\dag$ for a quantum-mechanical operator/matrix serves the role of the $*$ in (7).
6 It actually does not matter to the solution of the problem. The only change is which operator acts to `step up' the state. Dirac actually defined his operators with the $i$ attached to the $x$ variable, see [4].

## Monday, March 23, 2009

### Behind the Guesses

[Click here for a PDF of this post with nicer formatting]
Why I'm starting this blog
Figure 1: How physics does not work. Comic (cropped) from [1].

If you sit in enough physics, and to a lesser degree mathematics lectures, and/or read enough books, you're bound to encounter the phrase ``We guess the solution...''  Or, ``If we define ... it turns out that..."  Now, this is not inherently problematic.  After all, if the goal is only to solve some problem which does not have an obvious solution, or prove a random mathematical statement, these approaches will suffice.  Then, the student or reader, assuming they memorize the solution (or remember where to find it), will now be exquisitely equipped to solve exactly the same problem.

However, this approach does not teach students any skills, tools or methods for approaching new and exciting problems.  Additionally, I find, it is difficult to gain any meaningful understanding of the material through the ``Guess and Show'' method.  It's possible that for many people definition is the same as explanation, but from my personal experience, and from discussions with other students, it seems that such people are not the majority of physics students.

So I'm starting this blog to attempt to show some of the reasoning that goes into these guesses.  After all, the great physicists didn't just pull their solutions out of their butt, see Fig. 1.  Even if getting the correct solution ultimately was a result of guesswork, physicists don't sit around solving problems by randomly throwing darts at variables to see what sticks.

Now, I am not claiming that science should be taught according to the historical development of the theories.  In fact, usually doing that leads to needless confusion.  What I am saying, however, is that even if we know the answer already, it is still essential to explain some rationale as to why such an answer makes sense -- even if it's not the original idea behind the guess.

What to expect
So what can you expect to see on this blog?  Well, a lot of quantum mechanics, to be sure.  The conceptual difficulty and widespread lack of understanding,1 of this subject, together with the plethora of ``known solutions'' make it a prime target.  But I'll hopefully cover topics in electrodynamics, classical mechanics, thermodynamics, and so on.  Maybe even some mathematics, if I'm feeling up for it.

In general, I will not carry through the solution to the end; you can always follow my references to see the complete thing.  Instead I will try to provide some physical or mathematical reasoning behind some so-called ``guesses'' which ``turn out'' the correct solution. Additionally, with each post I will try to highlight a key methodological point that going behind the guesses gives us.  These points will be written in a different typeface to easily spot.

I do not expect to always get things right.  This is a learning experience for me, too.  If I make mistakes, please point them out in the comments or email me.  If I make corrections, I will leave the original, unedited version up as reference -- sometimes we learn more from mistakes!  I will try to post about once a month until I run out of ideas.  I suggest you use one of the subscription options on the side so you don't have to keep checking the site to see if I've updated.  So, if you have a particular solution (or problem) that's always bothered you, let me know, and I'll see if I can come up with something.  And if I can't I'll post it anyway, and we'll see if anyone out there in the vast Intertubes who stumbles across the blog can help out.

Each blog post will be have a link to a nicely formatted PDF with the same content.  Come back here this Monday (March 30, 2009) for my first real post.

References
[1] Z. Weiner. Saturday morning breakfast cereal [online]. March 2009. Available from: http://www.smbc-comics.com/index.php?db=comics&id=1452 [cited 9 March 2009].

1 I don't claim to understand quantum mechanics.  But at least I don't pretend to.