In the spirit of the Polymath Project, I’d like to present some thoughts on a mathematical physics problem that could be interesting. The problem is simple enough: compute the eigenvalues of the Hamiltonian for a perturbed quantum harmonic oscillator. But before we get started, I’ll just state that we should try to follow Timothy Gowers’s 12 ground rules at the end of his Polymath kick-off post. Also, I’ll credit Clark Alexander for walking me through this entire method, and encouraging this post.

### The Unperturbed Quantum Harmonic Oscillator — A Survey of Notation

So, we have the quantum harmonic oscillator, easily represented and solved by using raising and lowering operators:

where the notation and is employed, such that .

The spectrum of solutions to the Schrödinger Equation are familiar — the first three are shown in the following plot:

### The Quartic Perturbation to the Quantum Harmonic Oscillator — Rewriting the Hamiltonian

With the above notation, we can readily incorporate a perturbation of into our Hamiltonian:

can be normal ordered to the following result:

(1)

Intuitively, we know that the result of this perturbation will be a slight change in the resulting eigenstates making up the solution. A plot of the resulting change for an arbitrarily chosen lambda is shown below:

### Computing the Eigenvalues to First Order in in Easy Steps

**Step 1: Identify All Elements of the Lie Algebra**

Elements of the Lie Algebra at first order ( where ) are determined by performing commutations with and , as identified below. At first order, terms of order are ignored, so only one commutation is required.

The first commutator:

(2)

At this point, we can identify all the terms of the Lie Algebra to first order.

**Nonperturbative Terms** ( where )

(3)

**First Order Terms** ( where )

(4)

In this representation, we see the following:

(5)

This representation is complete for our purposes because it satisfies two conditions:

- can be completely represented by terms in the algebra.
- No two terms can be commuted to create a third non-trivial term not shown in the group. (Remember, ).

**Step 2: Construct a General Lie Group Element**

In principle, the Lie group element could be constructed from all terms in the Lie algebra, like so:

But, by nature of the Hammard lemma, we can choose to exclude all terms that commute with . So we construct as follows:

This gives us 6 constants we tune in order make this Lie group element a transformation of basis between perturbed and unperturbed eigenstates.

**Step 3: Use the Hammard Lemma to Compute our Lie Group Element**

It is our goal to choose a such that the following is true:

(6)

where

where .

To first order in this simplifies to:

(7)

Performing the commutator of Equation 7 and normal ordering, we get the following:

(8)

**Step : Tune so is a Number Operator**

From Equations 5 and 7,

(9)

Now, using our knowledge that must commute with , we know that cannot have terms involving or . Thus, the alphas must be tuned such that:

(10)

Which leaves:

(11)

Huzzah! We’ve done it. This is the perturbation to the eigenvalue introduced by the quartic term.

### What’s So Special About This Approach to Perturbation Theory?

While we were methodically plodding through, you might not have noticed, but there’s a few special aspects to this approach:

- The solution came down to performing a few commutators (Step 1), expanding the Hammard Lemma (Step 3), and solving a simple system of equations, (Step ).
- It stays just that simple for all orders in .
- It stays just that simple for all perturbations of the form

Perhaps there’s more here. Perhaps we could find a general form for the perturbation to order (in ) to the harmonic oscillator? Or more?

Feel free to comment with questions and thoughts. Where would you go from here?